Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.46.0-wmf.21 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 0 35598 2801476 1875611 2026-03-30T07:18:05Z Anikmolla786 3007678 Fixed typo: “great” → “greatest”. 2801476 wikitext text/x-wiki {{mathematics}} {{lesson}} {{unkel}} In this lesson we learn about the '''Euclidean algorithm''', an ancient algorithm for finding the [[greatest common divisor]] of two integers. The idea is as follows. Let p,q be two integers. Let p<q. The greatest common divisor of p and q, denoted (p,q), is the same as (p,q-p), or (p,q-2p),... and so on. Since there is such a q-kp which is smaller than p, the problem is reduced to the simpler one of calculating (q-kp, p). And so on. For more details and background, see [[w:Euclidean algorithm]], and [[w:Euclidean domain]]. == Try your hands on Euclidean algorithm== Substitute the following text into the sandbox: <pre> {{Subst:Euclidean algorithm|first integer|second integer}} </pre> ===Sandbox=== {{center top}}[[Image:Libya 5230 Wan Caza Dunes Luca Galuzzi 2007.jpg|150px]] <big><big><big><big><big>+</big></big></big></big></big> [[Image:Crystal Clear app kthememgr.png|120px]]{{center bottom}} The greatest common divisor (g.c.d.) of 4689 and 2346 is <br>( 4689,2346 ) <br>=(2346,2343) <br>=(2343,3) <br>=(3,0) <br>=(0,Division by zero) <br>The greatest common divisor is 3. {{Mathematics-stub}} [[Category:Algorithms]] [[Category:Arithmetic]] sb4rybjgzizkb5hymlt2bxdrt6k0r9w 0 65745 2801434 2722350 2026-03-30T05:35:50Z Anikmolla786 3007678 Removed duplicate “in the”. 2801434 wikitext text/x-wiki == Equations == === Truth table size-related equations === These equations determine the truth table size ''from'' the highest value of logic and the number of characteristics in the set and in the group for the '''bounded''' class<ref>See page 158, Table II of the '''primary reference'''</ref> of elements and denote the range of the truth table values for the set and the group.<ref>See page 172 of the '''primary reference'''.</ref> ==== Group size ==== : <math> Q = V^C </math>, where: ::* '''''Q''''' is the group size<ref>group - Columns C1 to C5 (0 to C, C=5)</ref> ::* '''''V''''' is the highest value of logic in the group, ::* '''''C''''' is the highest number of characteristics in the group. ==== Set size ==== : <math> R = V^K</math>, where: ::* '''''R''''' is the set size<ref>Target set - Columns C1 to C3 (0 to K, K=3)</ref> ::* '''''V''''' is the highest value of logic in the group, ::* '''''K''''' is the highest number of characteristics in the set. === Theoretical and Empirical Separatory Equations === ==== Element-related equations ==== ===== Maximum number of pairs of elements to separate ===== Maximum number of pairs of elements to separate refers to [[w:Triangular matrix|triangularization]] of the matrix to permit comparison of each element with every other element to determine the number of pairs that are separable or disjoint. Pairs are separable or disjoint whenever the logic values of the elements that make up a pair are different. In theory, therefore the maximum possible number of pairs that can be separated is determined by the following equation:<ref>See page 176 Table XI of the '''primary reference'''.</ref> : <math> p_{max} = \frac{\left[{G (G-1)}\right]}{2} </math>, where:<ref>[[w:Triangular number|Triangular Number]]</ref> ::*'''''p_max''''' is the maximum number of pairs to separate, and ::*'''''G''''' is the number of elements in the '''bounded''' class. ===== Order of elements ===== The elements are arranged in descending order according to their truth table value, i.e., the value calculated as the sum of each characteristic's logic state value times the highest value of logic raised to the power of the order of the characteristic.<ref>See [http://www.cstc.org/data/resources/60/convexp.html Number Systems] - essentially using the characteristic values to compute a network or memory address, followed by sorting.</ref> The element [[w:Positional notation|truth table value]] allows elements to be sorted and identified as unique or equivalent. : <math> e_i = \sum_{j=0}^C \left[v_{i,j} V^{(C-j)}\right]</math>, where: ::*'''''e_i''''' is the element truth table value in the group, ::*'''''V''''' is the highest value of logic in the group, ::*'''''v''''' is the value of logic of each characteristic in the group, ::*'''''j''''' is the '''''jth''''' characteristic index, where: :::'''''j = 0..C''''' and where: ::::*'''''C''''' is the number of characteristics in the group, ::*'''''i''''' is the '''''ith''''' element index, where: :::'''''i = 0..G''''' and where: ::::*'''''G''''' is the number of elements in the '''bounded''' class. ==== Characteristic-related equations ==== ===== Theoretical separation ===== ====== The general identification equation ====== :<math> S_j = \frac{1-{V^{-j}}}{1-V^{-C}}</math>, where:<ref>See page 153, page 167, Fig. 2. & page 175 of the '''primary reference'''</ref> ::*'''''S_j''''' is the theoretical separatory value per '''''jth''''' characteristic, ::*'''''C''''' is the highest number of characteristics in the group, ::*'''''V''''' is the highest value of logic in the group and ::*'''''j''''' is the '''''jth''''' characteristic index in the target set, where: :::'''''j = 0..K''''' and where: ::::*'''''K''''' is the number of characteristics in the target set. ====== Minimal number of characteristics to result in theoretical separation ====== : <math> t_{min} = \frac{\log G}{\log V}</math>, where:<ref>See page 157, ''Primary Schemes'' footnote of the '''primary reference'''</ref> ::*'''''t_min''''' is the minimal number of characteristics to result in theoretical separation, ::*'''''G''''' is the number of element in the '''bounded''' class and ::*'''''V''''' is the highest value of logic in the group. ===== Empirical separation ===== ====== Target set truth table values ====== <math> t_i = \sum_{j=0}^K v_{i,j} V^{(K-j)}</math>, where:<ref>As the characteristic with the greatest separatory value is moved to the next most significant position, K is incremented to expand the target set size from two characteristics to the number of characteristics in the group or to the number of characteristics which result in 100% separation. For the initial target set with one characteristic the separatory value is computed individually for each characteristic in the group to find the initial characteristic with the highest separatory value.</ref> :*'''''<big>t_i</big>''''' is the [[w:Positional notation|truth table value]] of the target set, ::*'''''<big>v_i,j</big>''''' is the element's attribute value, :::*'''''i''''' is the '''''i'''''th element's index value, where, ::::'''''i = 0...G'''''' where '''''G''''' is the number of elements in the '''bounded''' class, and, :::*'''''j''''' is the '''''j'''''th characteristic's index value, where, ::::'''''j = 0...K''''' and where, :::*'''''K''''' is the number of characteristics in the target set, ::::*'''''V''''' is highest value of logic in the group, ::::*'''''V^(K-j)''''' is the positional value of the '''''j'''''th characteristic. <math> n_{t_i} = n_{t_i} + 1 </math>, where, :<big>'''''n'''''_{'''''t'''''_'''''i'''''}</big> contains the [[w:multiset|multiset]] count for each [[w:Positional notation|truth table value]]<ref>coefficient of association, <math> coa = \frac{n_{t_i}}{n_R} </math>, see page 172 of the '''primary reference'''</ref>. ====== Separatory equation ====== ====== Initial separation ====== <math> S_j = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}</math>, where:<ref>Must be applied initially to each characteristic. (The equations have been implemented in [[w:Mathcad]] and using the Visual Basic programming language. See [[Optimal_classification#Application_Example|Application Example - Flag Recognition]]</ref> :*'''''S_j''''' is the initial empirical separatory value for each characteristic, where, ::'''''j = 0...C''''' and is the index of the '''''j'''''th characteristic in the group and '''''C''''' is the number of characteristics in the group, and, ::'''''l = 0...R''''' and is the truth table value of the '''''j'''''th characteristic, where '''''R''''' is the truth table size, where: ::::'''''R''''' = '''V⁰''', and, <!-- :::'''<math> R = V^{0} </math>''', where: --> ::::*'''''V''''' is the highest value of logic in the group and, :::;*'''''0''''' is the target set exponent for a single characteristic, and, :*'''''G''''' is the number of elements in the '''bounded''' class. ====== Subsequent separation ====== <math> S = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}</math>, where: :*'''''S_j''''' is the initial empirical separatory value for each characteristic, where, ::'''''l = 0...R''''' and is the target set truth table index value, where '''''R''''' is the target set truth table size value, where: ::::'''''R''''' = '''V^K''', and, <!-- :::'''<math> R = V^{K} </math>''', where: --> ::::*'''''V''''' is the highest value of logic in the group and, :::;*'''''K''''' is the number of characteristics in the target set, and, :*'''''G''''' is the number of elements in the '''bounded''' class. ==Notes== {{reflist}} [[Category:Optimal classification]] 4z0xw02n213z0twgwj3u5pjx4fh3wvs 0 72403 2801531 2477300 2026-03-30T09:38:02Z Anikmolla786 3007678 Fixed verb form: “begun” → “began”. 2801531 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had sean many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseantic league towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his fist voyage to the New World in August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned back to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] py9adm3zew30qpexcl250jtt7s02mdu 2801532 2801531 2026-03-30T09:39:13Z Anikmolla786 3007678 Fixed typo: “sean” → “seen”. 2801532 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseantic league towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his fist voyage to the New World in August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned back to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] kkk4tww169diu2in8hwvsxg4apjt97a 2801533 2801532 2026-03-30T09:40:24Z Anikmolla786 3007678 Corrected “Hanseantic” → “Hanseatic”. 2801533 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his fist voyage to the New World in August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned back to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] ahyebhbn3iy4nl9vakhfs4yfgp68xn6 2801534 2801533 2026-03-30T09:41:35Z Anikmolla786 3007678 Fixed typo: “fist” → “first”. 2801534 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World in August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned back to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] 2vem3g6fo5ivuetfcwzeu2wuwmhqgv2 2801535 2801534 2026-03-30T09:42:36Z Anikmolla786 3007678 Changed preposition “in” to “on” for specific date. 2801535 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned back to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] muc3el2jltwlc78pub8e922tifpbysy 2801536 2801535 2026-03-30T09:44:58Z Anikmolla786 3007678 Removed redundant “back”. 2801536 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships have wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] 107j8cdhz3ldgfdprt14ge4r78g3h6p 2801537 2801536 2026-03-30T09:46:32Z Anikmolla786 3007678 Corrected verb tense: “have wrecked” → “had wrecked”. 2801537 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New Word. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] s3lws0vplkq67xuhkdeujr56j2sdlfy 2801538 2801537 2026-03-30T09:50:45Z Anikmolla786 3007678 Fixed typo: “Word” → “World”. 2801538 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Deiogo de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] onegk1d7tqyvby9x7dn9i9818wvb13f 2801541 2801538 2026-03-30T09:51:50Z Anikmolla786 3007678 Corrected name: “Deiogo” → “Diego”. 2801541 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpse of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] 57u5am4e2nmbqafdkc595g8tllsx4sc 2801542 2801541 2026-03-30T09:52:45Z Anikmolla786 3007678 Fixed subject‑verb agreement: “corpse” → “corpses”. 2801542 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpses of his men. Nearby Tainos told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] t4id3wfuvwjobl9yftxsn4rybntafrw 2801543 2801542 2026-03-30T09:53:32Z Anikmolla786 3007678 Standardized to “Taíno” (consistent with earlier usage). 2801543 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpses of his men. Nearby Taíno told him that the Natives retaliated against them because they mistreated the Tainos. Comlumbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] bs9j1repzpshy1iwcp7s5y46c36vmha 2801544 2801543 2026-03-30T09:54:28Z Anikmolla786 3007678 Fixed typo: “Comlumbus” → “Columbus”. 2801544 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpses of his men. Nearby Taíno told him that the Natives retaliated against them because they mistreated the Tainos. Columbus did not give up, thus went and built another settlement further east called <b>La Islabella</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] ig6dwu8tuyikhgpfdgrv05kww9aln08 2801545 2801544 2026-03-30T09:55:52Z Anikmolla786 3007678 Corrected settlement name: “La Islabella” → “La Isabela”. 2801545 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpses of his men. Nearby Taíno told him that the Natives retaliated against them because they mistreated the Tainos. Columbus did not give up, thus went and built another settlement further east called <b>La Isabela</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Islabella ==Unit 2 Vocabulary== [[Category:History of Haiti]] m536vpx8mzmiorebcr03tdyxo75iamq 2801546 2801545 2026-03-30T09:57:01Z Anikmolla786 3007678 Corrected spelling in vocabulary list. 2801546 wikitext text/x-wiki [[Colonial Haiti|Course]] / [[Colonial Haiti/Unit 2|Next Unit]] ==Age of Discovery== ====Hispaniola==== {{main|History of Haiti}} The recorded history of Haiti began on December 5, 1492 when the European navigator Christopher Columbus happened upon a large island in the region of the western Atlantic Ocean that later came to be known as the Caribbean Sea. It was inhabited by the <b>Taíno</b>, an Arawakan people, who variously called their island Ayiti, <b>Bohio</b>, or <b>Kiskeya</b>. Columbus promptly claimed the island for the Spanish Crown, and renamed it La Isla Española ("the Spanish Island"), or Hispañola (later Anglicized as <b>Hispaniola</b>). ====Portugal Leads Exploration==== {{main|History of Portugal (1415–1542)}} [[Image:Faroe stamp 226 Discovery of America - Kristoffur Kolumbus.jpg|thub|left|200px]] The exploration age began as a result of trade, need for fresh conquests, and technological advances witnessed in Portugal. 15th century Portugal had seen many breakthroughs in the fields of mathematics and naval technology, thus making it the scientific center of that time. In light of their location on the European map, the Portuguese were a seafaring people. It's no wonder that most trade routes were by sea, delivering goods to England, Flanders, Italy, and the Hanseatic League towns. By land, trade would be more of a challenge for them, as they'd face the task of crossing Castile and Aragon territory (now modern-day Spain). ====Columbus and New World==== {{main|Christopher Columbus}} Christopher Columbus, Italian Cristoforo Colombo, was a <b>Genoese</b> navigator. He made his first voyage to the New World on August 3, 1492 with three ships: Santa Maria, Pinta, and Nina. In December 5, 1492, he arrived in northern Hispaniola, and later had to abandon the Santa Maria since she crashed ashore. He returned to Spain, since his voyage was charted by the monarch thereof, with twenty-five natives (only seven survived the voyage) and left 39 men who founded the settlement of La Navidad. ====La Navidad==== Christopher, after one of his ships had wrecked, ordered his men to have the ship dismantled providing building materials for the construction of a small fortress. This would be the very first European settlement in the New World. Christopher, however, returned to Spain, leaving Diego de Arana as governor. When Columbus came back from Spain during his second voyage, on November 27, 1493 he arrived hoping to see a bustling village. To his surprise, only thing that was left of the village were corpses of his men. Nearby Taíno told him that the Natives retaliated against them because they mistreated the Tainos. Columbus did not give up, thus went and built another settlement further east called <b>La Isabela</b>, after his queen. ==Unit 1 Vocabulary== *Taíno *Bohio, Kiskeya *Hispaniola *Genoese *La Navidad & La Isabela ==Unit 2 Vocabulary== [[Category:History of Haiti]] exuyb17jjttjk3d3m8qui9xnb29wygl 4 75745 2801319 2800941 2026-03-29T14:11:50Z Juandev 2651 /* Please publish my post */ Reply 2801319 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) a01zc83yon6pr6mdclzxz8uu27tflxg 2801324 2801319 2026-03-29T16:24:51Z Codename Noreste 2969951 /* Please publish my post */ Reply 2801324 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) c0d59ksm4e69lfgusxyrd9lqxllqnum 2801478 2801324 2026-03-30T07:19:03Z Senad Dizdarević 3062367 /* New user limit */ new section 2801478 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) ey8po2ckzl22021os2r9n3mibmo8ema 2801491 2801478 2026-03-30T07:30:03Z Juandev 2651 /* New user limit */ Reply 2801491 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) 94hel7wb2sy91fez4fbpkroloyqtt6j 2801496 2801491 2026-03-30T07:53:52Z Senad Dizdarević 3062367 /* New user limit */ Reply 2801496 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) 2k0htldl1sa7dcmrlai853uwgk36dzp 0 81560 2801313 2390746 2026-03-29T12:45:29Z Anikmolla786 3007678 Fixed grammar: “And important part” → “An important part”. 2801313 wikitext text/x-wiki {{center top}}<big>'''Welcome to the Lesson of Analytical Trigonometry'''</big>{{center bottom}} {{center top}}''Part of the [[School:Olympiads|School of Olympiads]]''{{center bottom}} This topic deals with the '''analytical''' aspects of '''Trigonometry'''. Widely this topic covers Trigonometric Identities and Equations. An important part of this topic is trigonometry through Complex Numbers by the use of De Moivre's Law and its application.<br> {|class="wikitable" style="background-color:#FFFFFF;" !colspan="2"| Function !colspan="2"| Inverse function !colspan="2"| Reciprocal !colspan="2"| Inverse reciprocal |- | sine | sin | arcsine | arcsin | cosecant | csc | arccosecant | arccsc |- | cosine | cos | arccosine | arccos | secant | sec | arcsecant | arcsec |- | tangent | tan | arctangent | arctan | cotangent | cot | arccotangent | arccot |} =Theorems= ==Identities== ===Basic Relationships=== {|class="wikitable" style="background-color:#FFFFFF" ! Pythagorean trigonometric identity |<math>\sin^2 \theta + \cos^2 \theta = 1\,</math> |- ! Proofs of trigonometric identities|Ratio identity |<math>\tan \theta = \frac{\sin \theta}{\cos \theta}</math> |} {| class="wikitable" style="background-color:#FFFFFF;text-align:center" |+ '''Each trigonometric function in terms of the other five.''' ! Function ! <math>(\sin \theta)</math> ! <math>(\cos \theta)</math> ! <math>(\tan \theta)</math> ! <math>(\csc \theta)</math> ! <math>(\sec \theta)</math> ! <math>(\cot \theta)</math> |- ! <math> \sin \theta =</math> | <math> \sin \theta\ </math> | <math>\pm\sqrt{1 - \cos^2 \theta}\ </math> | <math>\pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}\ </math> | <math> \frac{1}{\csc \theta}\ </math> | <math>\pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}\ </math> | <math>\pm\frac{1}{\sqrt{1 + \cot^2 \theta}}\ </math> |- ! <math> \cos \theta =</math> | <math>\pm\sqrt{1 - \sin^2\theta}\ </math> | <math> \cos \theta\ </math> | <math>\pm\frac{1}{\sqrt{1 + \tan^2 \theta}}\ </math> | <math>\pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\ </math> | <math> \frac{1}{\sec \theta}\ </math> | <math>\pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}\ </math> |- ! <math> \tan \theta =</math> | <math>\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\ </math> | <math>\pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}\ </math> | <math> \tan \theta\ </math> | <math>\pm\frac{1}{\sqrt{\csc^2 \theta - 1}}\ </math> | <math>\pm\sqrt{\sec^2 \theta - 1}\ </math> | <math> \frac{1}{\cot \theta}\ </math> |- ! <math> \csc \theta =</math> | <math> \frac{1}{\sin \theta}\ </math> | <math>\pm\frac{1}{\sqrt{1 - \cos^2 \theta}}\ </math> | <math>\pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}\ </math> | <math> \csc \theta\ </math> | <math>\pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}\ </math> | <math>\pm\sqrt{1 + \cot^2 \theta}\ </math> |- ! <math> \sec \theta =</math> | <math>\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}\ </math> | <math> \frac{1}{\cos \theta}\ </math> | <math>\pm\sqrt{1 + \tan^2 \theta}\ </math> | <math>\pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}\ </math> | <math> \sec \theta\ </math> | <math>\pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}\ </math> |- ! <math> \cot \theta =</math> | <math>\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\ </math> | <math>\pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}\ </math> | <math> \frac{1}{\tan \theta}\ </math> | <math>\pm\sqrt{\csc^2 \theta - 1}\ </math> | <math>\pm\frac{1}{\sqrt{\sec^2 \theta - 1}}\ </math> | <math> \cot \theta\ </math> |} {|class="wikitable" style="background-color:#FFFFFF" |+ '''Historic Shorthands''' |- ! Name(s) ! Abbreviation(s) ! Value |- | versed sine, versine | <math>\textrm{versin}\,\theta\ </math><br><math>\textrm{vers}\,\theta\ </math> | <math> 1 - \cos \theta\ </math> |- | versed cosine, vercosine,<br>coversed sine, coversine | <math>\textrm{vercos}\,\theta\ </math><br><math>\textrm{coversin}\,\theta\ </math><br><math>\textrm{cvs}\,\theta\ </math> | <math>1 - \sin \theta\ </math> |- | haversed sine, haversine | <math>\textrm{haversin}\,\theta\ </math><br><math>\textrm{hav}\,\theta\ </math> | <math>\tfrac{1}{2} \textrm{versin}\ \theta\ </math> |- | haversed cosine, havercosine,<br>hacoversed sine, hacoversine,<br>cohaversed sine, cohaversine | <math>\textrm{havercos}\,\theta\ </math><br><math>\textrm{hacoversin}\,\theta\ </math><br><math>\textrm{cohav}\,\theta\ </math> | <math>\tfrac{1}{2} \textrm{vercos}\,\theta\ </math> |- | exterior secant, exsecant | <math>\textrm{exsec}\,\theta\ </math> | <math>\sec \theta - 1\ </math> |- | exterior cosecant, excosecant | <math>\textrm{excsc}\,\theta\ </math> | <math>\csc \theta - 1\ </math> |} {|class="wikitable" style="background-color: #FFFFFF" |+'''Symmetries''' ! Reflected in <math>\theta=0 </math> ! Reflected in <math>\theta= \pi/2</math><br>(co-function identities) ! Reflected in <math>\theta= \pi</math> |- |<math> \begin{align} \sin(-\theta) &= -\sin \theta \\ \cos(-\theta) &= +\cos \theta \\ \tan(-\theta) &= -\tan \theta \\ \csc(-\theta) &= -\csc \theta \\ \sec(-\theta) &= +\sec \theta \\ \cot(-\theta) &= -\cot \theta \end{align} </math> |<math> \begin{align} \sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\ \cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\ \tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\ \csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\ \sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\ \cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta \end{align} </math> |<math> \begin{align} \sin(\pi - \theta) &= +\sin \theta \\ \cos(\pi - \theta) &= -\cos \theta \\ \tan(\pi - \theta) &= -\tan \theta \\ \csc(\pi - \theta) &= +\csc \theta \\ \sec(\pi - \theta) &= -\sec \theta \\ \cot(\pi - \theta) &= -\cot \theta \\ \end{align} </math> |} {|class="wikitable" style="background-color: #FFFFFF" |+'''Periodicity and Shifts''' !Shift by π/2 !Shift by π <br> Period for tan and cot !Shift by 2π <br> Period for sin, cos, csc and sec |- |<math> \begin{align} \sin(\tfrac{\pi}{2} + \theta) &= +\cos \theta \\ \cos(\tfrac{\pi}{2} + \theta) &= -\sin \theta \\ \tan(\tfrac{\pi}{2} + \theta ) &= -\cot \theta \\ \csc(\tfrac{\pi}{2} + \theta) &= +\sec \theta \\ \sec(\tfrac{\pi}{2} + \theta) &= -\csc \theta \\ \cot(\tfrac{\pi}{2} + \theta) &= -\tan \theta \end{align} </math> |<math> \begin{align} \sin(\pi + \theta) &= -\sin \theta \\ \cos(\pi + \theta) &= -\cos \theta \\ \tan(\pi + \theta) &= +\tan \theta \\ \csc(\pi + \theta) &= -\csc \theta \\ \sec(\pi + \theta) &= -\sec \theta \\ \cot(\pi + \theta) &= +\cot \theta \\ \end{align} </math> |<math> \begin{align} \sin(2\pi + \theta) &= +\sin \theta \\ \cos(2\pi + \theta) &= +\cos \theta \\ \tan(2\pi + \theta) &= +\tan \theta \\ \csc(2\pi + \theta) &= +\csc \theta \\ \sec(2\pi + \theta) &= +\sec \theta \\ \cot(2\pi + \theta) &= +\cot \theta \end{align} </math> |} ===Angle Sum Identities=== ==Complex Numbers, De Moivre's Law and Argand Plane== =Examples= =Resources= ==Textbooks== ==Practice Questions== <quiz display=simple points="1/1!"> {&nbsp; |type="{}"} <math>sin\frac\pi6=</math>{ 1/2|0.5_3 } {&nbsp; |type="{}"} <math>cos\frac\pi3=</math>{ 1/2|0.5_3 } {&nbsp; |type="{}"} <math>tan\frac\pi4=</math>{ 1_3 } </quiz> ==Related WebApps== ==Active participants== The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed). ---- [[Category:Secondary_Math_Lessons]] hcjd5rqb8x6typf50as0vq19o98djgw 0 98508 2801452 2800270 2026-03-30T06:17:58Z IronGargoyle 23939 Remove AI nonsense 2801452 wikitext text/x-wiki <noinclude> * A collection of images related to [[motivation and emotion]] from [[commons:|Wiki Commons]] * Images may be useful for [[Motivation and emotion/Book|book chapter]]s</noinclude> {{center top}} <gallery> File:1944 JonWhitcomb USNavy (3214638694).jpg File:2010 - A year plenty of Hopes.jpg File:Adam and Eve (UK CIA P-1947-LF-77).jpg File:AG LEADER.JPG File:Alienation.jpg File:Alma-Tadema Unconscious Rivals 1893.jpg File:Amarguraubeda.JPG File:Angelo Bronzino 003.jpg File:Arianna e la sua lente.JPG File:B&W Happiness.jpg File:BB-Bea.jpg File:Bipolar Dyptych 1 365.jpg File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg File:Child's Angry Face.jpg File:Concert exercise.jpg File:Contempt.jpg File:Crying child with blonde hair.jpg File:Cycling Time Trial effort.jpg File:De mulieribus claris - Marcia.png File:Depression.jpg File:Disgust expression cropped.jpg File:Disgust1.jpg File:Drill sergeant screams.jpg File:Doctor Who (13).jpg File:Edward Lear A Book of Nonsense 57.jpg File:Eeg registration.jpg File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg File:Emo boy 03 in rage.jpg File:Emotions wheel.png File:Empathy and the brain.png File:Eros bobbin Louvre CA1798.jpg File:Expression of the Emotions Figure 1.png File:Expression of the Emotions Figure 20.png File:Faradarmani.gif File:Forestay-Eye-Round-seizings-Bulls-eye.jpg File:Folsom Hula Hoop.png File:Fragile Emotion cropped.jpg File:Georg Friedrich Kersting Kinder am Fenster.jpg File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG File:Grass of Happiness.jpg File:Gustave Courbet - Le Désespéré (1843).jpg File:Inside my head.jpg File:Interest.jpg File:It's a Braun definitely.jpg File:Just love cropped.jpg File:Little girl buried up to her neck in sand.jpg File:Lotus Reflects the Sun.jpg File:Love Rush!.jpg File:Love's Passing.jpg File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif File:Mary Magdalene Crying Statue.jpg File:Mascia Ferri in Speed Skydiving.jpg File:Mood dice.svg File:Motivation & Emotion.png File:Motivation and Emotion Scrabble.jpg File:Motivation Laptop.svg File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg File:Noun emotion 1325508.svg File:One hand handstand.jpg File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg File:Person holding clock in front of head.jpg File:Picswiss_UR-25-03.jpg File:Plato Pio-Clemetino Inv305.jpg File:Plutchik-wheel.svg File:Portrait gemma and mehmet.jpg File:Rock climbing (B&W).jpg File:Sadness 2.jpg File:Sadness.jpg File:Schadenfreude.png File:Sépulcre Arc-en-Barrois 111008 12.jpg File:Sigmund Freud LIFE.jpg File:Smiling Brazilian girl (black and white).jpg File:Smiling girl.jpg File:Sophia (robot).jpg File:Souvenir Seller - Moscow - Russia.JPG File:SURPRISE.jpg File:Thorunn3.jpg File:Tug-of-war.jpg File:Two people laughing.jpg File:Two people with brain cogs turning.png File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg File:Vault figure.jpg File:Vincent Van Gogh - Sorrow.JPG File:WebKit logo.svg File:Week-end pleasure.jpg File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg File:Worried little girl.jpg File:Yakunchikova Fear.jpg File:You may now kiss the bride.jpg </gallery> {{center bottom}} {{center top}} For more images, search <span class="plainlinks">[http://commons.wikimedia.org/w/index.php?title=Special:Search&fulltext=Search&ns0=1&ns1=1&ns2=1&ns3=1&ns4=1&ns5=1&ns6=1&ns7=1&ns8=1&ns9=1&ns10=1&ns11=1&ns12=1&ns13=1&ns14=1&ns15=1&ns100=1&ns101=1&ns102=1&ns103=1&redirs=0&search= wiki commons].</span> {{center bottom}} <noinclude> [[Category:Motivation and emotion]] [[Category:Galleries]] </noinclude> e4o6effn8tgumpjn8khqelf5gremkax 0 99734 2801314 1646377 2026-03-29T12:57:59Z ארשת 3039075 Stub vs sub 2801314 wikitext text/x-wiki This is a stub page for references to education material on multiplication and division of positive and negative whole numbers. == Multiplication and Division of Integers == Multiply divide the numbers as usual (disregarding it being positive or negative). Then define if the result is negative or posite using the following rules: {| |- | +n || x || +n || = || +n |- | -n || x || -n || = || +n |- | +n || x || -n || = || -n |- | -n || x || +n || = || -n |- | +n || ÷ || +n || = || +n |- | -n || ÷ || -n || = || +n |- | +n || ÷ || -n || = || -n |- | -n || ÷ || +n || = || -n |} +n stands for a positive integer and -n for a negative integer. There is also cheeky way to help you remember... *If a good thing happens to a good person, you're happy (+ x + = +) *If a bad thing happens to a good person, you're sad (- x + = -) *If a bad thing happens to a bad person, you're happy (- x - = +) *If a good thing happens to a bad person, your sad (+ x - = -) [[Category:Mathematics]] smh9su8o33xzmnzi0uu20g36rpdw17c 2801315 2801314 2026-03-29T12:58:34Z Quinlan83 2913823 Reverted edits by [[Special:Contributions/ארשת|ארשת]] ([[User_talk:ארשת|talk]]) to last version by [[User:MaintenanceBot|MaintenanceBot]] using [[Wikiversity:Rollback|rollback]] 1646377 wikitext text/x-wiki This is a sub page for references to education material on multiplication and division of positive and negative whole numbers. == Multiplication and Division of Integers == Multiply divide the numbers as usual (disregarding it being positive or negative). Then define if the result is negative or posite using the following rules: {| |- | +n || x || +n || = || +n |- | -n || x || -n || = || +n |- | +n || x || -n || = || -n |- | -n || x || +n || = || -n |- | +n || ÷ || +n || = || +n |- | -n || ÷ || -n || = || +n |- | +n || ÷ || -n || = || -n |- | -n || ÷ || +n || = || -n |} +n stands for a positive integer and -n for a negative integer. There is also cheeky way to help you remember... *If a good thing happens to a good person, you're happy (+ x + = +) *If a bad thing happens to a good person, you're sad (- x + = -) *If a bad thing happens to a bad person, you're happy (- x - = +) *If a good thing happens to a bad person, your sad (+ x - = -) [[Category:Mathematics]] nkgmi7b8v1bd1crf0mxokoqthxyx3ry 0 122726 2801563 2190001 2026-03-30T10:48:02Z Anikmolla786 3007678 Fixed author name: “Lukeoff” → “Lukoff”. 2801563 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researches assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawaet al, 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et at. 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et at. 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] ob6e2kmjohbjdrariqw1jajmp3lffh8 2801564 2801563 2026-03-30T10:49:29Z Anikmolla786 3007678 Added missing space after author name. 2801564 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researches assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et at. 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et at. 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] i63z4clybguzyzoq0mj0d1m6kxsaoew 2801567 2801564 2026-03-30T10:50:21Z Anikmolla786 3007678 Added missing “by”. 2801567 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researches assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et at. 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et at. 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] qorbf8d1xlxjuejrpf197oj0azxu2l5 2801568 2801567 2026-03-30T10:51:15Z Anikmolla786 3007678 Added missing article “a”. 2801568 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researches assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et at. 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et at. 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] s7llby9b73443l44eymlpw34479f412 2801571 2801568 2026-03-30T10:52:17Z Anikmolla786 3007678 Changed “researches” to “researchers”. 2801571 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et at. 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et at. 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] qzvcw6jimazhuloe7v48zlw0mzsbhlr 2801572 2801571 2026-03-30T10:53:46Z Anikmolla786 3007678 Fixed “et at.” → “et al.”. 2801572 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Spychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] cluptuj6nt6lg6qrsuj8rv298fn89nr 2801574 2801572 2026-03-30T10:55:08Z Anikmolla786 3007678 Fixed typo in journal title. 2801574 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to god amd implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] izov0tlzx5g7ahdrrxdwciovt8df1ba 2801575 2801574 2026-03-30T10:56:32Z Anikmolla786 3007678 Fixed typo “amd” → “and”; capitalized “God”. 2801575 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology for Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] fb194b2t2zo7wqp45ma1gsofqc94my7 2801577 2801575 2026-03-30T10:59:02Z Anikmolla786 3007678 Corrected journal name: “for” → “of”. 2801577 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, AH.. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] 4qfn0iy35z9i7fab5czea408wbbcamz 2801579 2801577 2026-03-30T11:02:53Z Anikmolla786 3007678 Fixed double period and initial format. 2801579 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, A. H. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] 4cu8jtma73b5eursq5koq5z6w5c2btn 2801580 2801579 2026-03-30T11:03:57Z Anikmolla786 3007678 Added hyphen to “non-kin”. 2801580 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non-kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human’s spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, A. H. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] ngyh4pviv5cnjspt3d4nsx4bi18yyti 2801581 2801580 2026-03-30T11:04:47Z Anikmolla786 3007678 Changed “human’s” to “human” (adjective). 2801581 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non-kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experienced can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, A. H. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] h73kcxtknqoc8hhzo1omklwqztcnwe8 2801582 2801581 2026-03-30T11:05:50Z Anikmolla786 3007678 Fixed noun form: “experienced” → “experience”. 2801582 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non-kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experience can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that these results suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, A. H. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] its06ug09aqzi910wfh0u525fcywjwj 2801584 2801582 2026-03-30T11:07:27Z Anikmolla786 3007678 Removed redundant “these results suggest”. 2801584 wikitext text/x-wiki {{title|Spiritual and religious motivation:<br>What is the nature of human’s spiritual motivation and religiosity?}} {{MECR|http://www.screenr.com/HmFs}} {{TOCright}} ==Overview== [[File:The Creation of Adam.jpg|250px|thumb|The creation of Adam]] Religiosity refers to an individual’s dedication to organised systems of beliefs about the sacred and religious activities and rituals. However, spirituality and religiosity historically have tended to overlap as spirituality has often been experienced through religion (Hill & Pargament, 2008). ==Transcendent actualisation: Humanistic and transpersonal psychology== [[File: Maslow hierarchy of needs.jpg|2565px|thumb||left]] Behaviourism and Freudian Psychoanalysis in the middle of the twentieth century were the two dominant forces in psychology (Stanislav, 2008). However, Psychologists such as Carl Rogers and Abraham Maslow critiqued these approaches to psychology as they believed that human behaviour can not only be accounted for by unconscious instincts and interactions with the environment but also by human growth and potential (Maslow, 1969). Carl Rogers suggested that human beings have a tendency to self-actualise that is, become everything that one is capable of becoming (Rogers, 1951). An individual who strives to reach his or her optimal sense of satisfaction or self-actualise is as described by Rogers (1951), as fully functioning. Abraham Maslow (1969) suggested that to reach self actualisation one must satisfy a hierarchy of needs including: physiological needs, safety needs, belongingness and love needs, esteem needs and need for self actualisation. Satisfying these needs as suggested by Maslow will lead an individual to personal growth and self actualisation (Maslow, 1954). However, Maslow suggested that the self actualisation needs to incorporate a spiritual component and in 1967 Maslow and colleagues founded the Association of Transpersonal Psychology (Stanislav, 2008). Maslow referred to transpersonal psychology as the fourth force of psychology, the first and second being Psychoanalysis and Behaviourism and the third humanistic psychology (Stanislav, 2008). This field of psychology would incorporate the spiritual component of self actualisation (Stanislav, 2008). Maslow in 1971 introduced the concept of transcendent actualisation which refers to an individual awareness and harmony of a spiritual centre or inner being which can affect human predisposition to love, will, altruistic awareness and discovery of profound meaning in life (Hamel, Leclerc & Lefrancois, 2003). Hamel et al (2003) suggests that transcendent actualisation can be understood through different levels of consciousness and motivational characteristics. These researchers suggest that transpersonal actualisation occurs in three stages: Prepersonal, Personal and Transpersonal Growth levels. The Prepersonal Growth Level is characterised by ordinary or subjective consciousness which is when one’s consciousness being shaped from ones drives and environmental influences (Hamel et al. 2003). This growth level is also characterised by extrinsic motivations that is, conforming to and imitating others actions and desires (Hamel et al. 2003). Hamel et al (2003) suggest that the Personal Growth Level is characterised by authentic consciousness which refers to the capability to understand and control elements of one’s personality such as expectations and beliefs. Also, the researchers suggest that it is characterised by intrinsic motivations which are motivations based on one’s internal desires and interests to develop ones personality. Lastly, the Transpersonal Growth Level refers to a stage characterised by objective consciousness or metacognition which involves a heightened awareness of stimuli based on spiritual perceptions, higher order feelings, ethical reflection and inspiration (Hamel et al. 2003). The researchers also suggest that this growth level is characterised by metamotivations which are based on an awareness of profound or spiritual values in everyday life and the harmonizing of the different components of one’s personality. ==Perceived closeness to God: Attachment theory== To know and connect with a God is the central function of many religious traditions and is the ultimate goal of some spiritual individuals (Hill & Pargament, 2008). Hill and Pargament (2008) suggest that God to many individuals can be seen as someone who provides comfort and security or an attachment figure. Human beings have a genetic predisposition to form an attachment bonds with one’s primary caregiver at a young age (Bowlby 1969 as cited in Bottoms et al. 2010). The primary caregiver’s sensitivity to the child’s needs affects the child’s internal working models or cognitive representations of itself and others (Bowlby 1969 as cited in Bottoms et al. 2010). Fujikawa, Halcrow, Hall and Hill (2009) in their study examined 483 undergraduate students’ human and spiritual attachment patterns. The researchers assessed two hypotheses: firstly, that God substitutes as an attachment figure in individuals with negative human attachment patterns and secondly, that attachment patterns to human attachment figures corresponds to God attachment patterns. The results supported the correspondence and the compensation models of attachment patterns with God (Fujikawa et al. 2009). The Correspondence model holds that adult attachment to God corresponds to or reflects internal working models developed from one’s human attachment figures (Fujikawa et al. 2009). The Compensation model suggests that God attachment or religious beliefs compensate for insecure attachments with one’s primary human caregiver (Fujikawa et al., 2009). In a similar study, Reinert (2005) examined 75 Roman Catholic seminarians’ early primary caregiver attachments, attachment to God and later sense of self in two time periods eight months apart. As expected the researchers found that low self-esteem and higher levels of shame were found in participants with reported previous avoidant attachments with their primary caregiver. Interestingly, the results showed that over the eight month period the seminarians had an increase in self-esteem and the researchers suggest that this can be attributed to attachment to God compensating for previous attachment deficits. Also, the results suggested that previous secure human attachment with a primary caregiver was associated with a greater connection with God and previous anxious and avoidant attachment with a primary caregiver was associated with an instable relationship with God (Reinert, 2005). Research such as Anderson, Hall, Lewis Hall and Reiner (2010) has shown that an attachment to God can have health benefits such as perceived stress. Anderson et al. (2010) in their study of 276 undergraduate students found that adult attachment and God attachment anxiety was significantly related to perceived stress levels. Also, research such as Bottoms, Hernandez and Salerno (2010) assessed if different attachment classifications to God can affect an individual’s alcohol use and coping strategies. The researchers suggest that participants with a secure attachment to God as compared to insecure attachments to God were more comfortable when dealing with problems to turn to God for support. Secure God attachment provides an additional support which may lead to regulating negative emotions in a healthier way as compared to people without this support (Bottoms et al. 2010). The results also show that the participants with positive spiritual coping styles and secure attachments with God drink alcohol significantly less as compared to individuals with avoidant God attachments (Bottoms et al. 2010). ==Religious support: Extrinsic motivations== The nature of human’s spiritual motivation and religiosity can not only be explained by intrinsic motivations but also extrinsic motivations (Hill & Pargament, 2008). Intrinsic motivations of religiosity and spirituality can include living according to one’s beliefs and learning about one’s religion. Extrinsic motivations for religion can include securing social status and being part of a group or community (Hill & Pargament, 2008). Research suggests that religions can promote connectedness and caring among its members (Hill & Pargament, 2008; Cohen & Hill, 2007). Cohen and Hill (2007) suggest that religions differ in their individualistic and collectivists characteristics. Moreover, in their study Cohen and Hill (2007) found that American Protestants are more individualistic as compared to American Jews and Catholics who are more socially and community oriented. [[File: Wydrome2000.jpg|320px|thumb|World youth day 2000 (Rome)]] Ellison and George (1994) assessed 2956 individuals in North Carolina to examine whether individuals who attend church frequently have more social resources then people who do not attend church frequently. The results of the study suggest the following: that frequent church attendees have more non-kin social networks, more telephone and in person contacts, receive more supportive or helpful transactions such as money, goods and services and report more nurturing social interactions or sense of being cared for than do people who are not frequent church attendees (Ellison & George, 1994). In a review of literature regarding the health benefits of religion Ellison and Levin (1998) suggest that the increased social resources individuals engaging in a religion receive can lead to health benefits. The researchers suggest that religion is a means by which individuals with a common faith, values and interests are brought together and can develop friendships. Also, religions offer informal support which can include things like pastoral advice and counselling (Ellison and Levin, 1998). This support can lead to positive influences on self esteem and self efficacy, coping mechanisms and positive emotions such as forgiveness. ==The mystical experience== [[File:Rossakiewicz Prayer.jpg|100px|thumb]] The mystical experience has been referred to in literature as an altered state of consciousness, meaning an experience that is different to the ordinary day to day experiences (Heriot-Maitland, 2008). A mystical experience involves a state of consciousness in which an individual comes in contact with an abstract power far greater than the individual, a realisation of an ultimate reality and a heightened state of awareness (Heriot-Maitland, 2008; Lukoff, 1985). Lukeoff (1985) suggests that the mystical experience is a desirable state which can be a life altering experience. The mystical experience has historically been experienced through practices such as meditation or contemplation, prayer and chanting (Heriot-Maitland, 2008; Forman, 1998). The Mystical experience in literature has often been compared to psychosis or schizophrenia due to its altered state of consciousness characteristics and unusualness (Buckley, 1981; Heriot-Maitland, 2008; Lukoff, 1985). Moreover, the mystical experience can include hearing voices, Hallucinations and time distortions similar to psychotic episodes. However, the mystical experience differs from psychosis in terms of individuals being able to control the onset of a mystical experience entry out of such states (Heriot-Maitland, 2008). Also, mystical experiences can often lead to positive life consequences as compared to negative outcomes of psychosis (Heriot-Maitland, 2008). ==Brain imaging research== [[File:MRI_brain.jpg|200px|thumb||left]] Recent research has examined the neurological underpinnings of the mystical, spiritual and religious experience (Beauregard & Paquette, 2006). Beauregard and Paquette (2006) in their study used functional magnetic resonance imaging (fMRI) to determine the neural correlates of the mystical experience. The researchers asked the Carmelite nuns to remember and try to relive their most intense mystical experience whilst data was being collected by fMRI scans. The results of this study suggest that several brain regions and systems are activated in the mystical experience suggesting that the mystical experience can be attributed to several brain regions. (Beauregard & Paquette, 2006). In a study Aglioti, Fabbro, Skrap and Urgesi, (2010) examined patients self transcendence personality scores before and after brain surgery. The researchers found that damage to the left and right inferior parietal regions of the brain increased self transcendence personality scores. Aglioti et al. (2010) suggest that the left and right inferior parietal regions are important mediators of the personality trait self transcendence. Spiritual and religious experience effects on hippocampal volume change have also been studied as Hayward, Koenig, Owen, Payne and Steffens (2011) examined hippocampal volume change in individuals who experienced a life altering religious experience. The results showed that a life changing religious experience can influence hippocampal atrophy later in life (Hayward et al., 2011). The researchers suggest that the hippocampal atrophy observed in individuals with life changing religious experiences may be attributed to by the stress such life experiences produce. Furthermore, this study shows that engaging in one’s religiosity can affect brain physiology (Hayward et al., 2011). ==Summary and conclusions== An individual’s spiritual motivation and religiosity can be vastly different from person to person and there are many motivational forces that drive human spirituality and religiosity. An individual may be on the journey of self actualisation, seeking an attachment with a God, seeking to be a part of a community of people with similar beliefs and values, seeking the mystical experience and have a neurological predisposition to self transcendence traits. ==References== {{Hanging indent|1= Anderson, T.L., Hall, T.W., Lewis Hall, M.E., & Reiner, S.R. (2010). Adult attachment, god attachment and gender in relation to perceived stress. Journal of Psychology and Theology, 3, 175-185. Aglioti, S.M., Fabbro, F., Skrap, M., & Urgesi, C. (2010). The spiritual brain: Selective cortical lesions modulate human self-transcendence. Neuron, 65, 309-319. Beauregard, M., & Paquette, V. (2006). Neural correlates of a mystical experience in Carmelite nuns. Neuroscience Letters, 405, 186-190. Bottoms, B.L., Hernandez, G., & Salerno, J.M. (2010). Attachment to god, spiritual coping, and alcohol use. The International Journal for the Psychology of Religion, 20, 97-108. Buckley, P. (1981). Mystical experience and schizophrenia. Schizophrenia Bulletin, 3, 516-521. Cohen, A.B., & Hill, P.C. (2007). Religion as culture: Religious individualism and collectivism among American catholics, jews, and protestants. Journal of Personality, 75. Ellison, C.G., & George, L.J. (1994). Religious involvement, social ties, and social support in a southeastern community. Journal for the Scientific Study of Religion, 33, 46-61. Ellison, C.G., & Levin, J.S. (1998). The religion-health connection: Evidence, theory and future direction. Journal of Health, Education and Behaviour, 25, 700-720. Forman, R.K. (1998). What does mysticism have to teach us about consciousness? Journal of Consciousness Studies, 2, 185-201. Fujikawa, A., Halcrow, S.R., Hall, T.W., & Hill, P.C. (2009). Attachment to God and implicit spirituality: clarifying correspondence and compensation models. Journal of Psychology and Theology, 4, 227-242. Hamel, S., Leclerc, G., & Lefrancois, R. (2003).A psychological outlook on the concept of transcendent actualisation. The International Journal for the Psychology of Religion, 13, 3-15. Hayward, R.D., Koenig, H.G., Owen, A.D., Payne, M.E., & Steffens, D.C. (2011). Religious factors and hippocampal atrophy in late life. PLoS ONE, 6. Heriot-Maitland, C.P. (2008). Mysticism and madness: different aspects of the same human experience. Mental Health, Religion and Culture, 11, 301-325. Hill, P.C., & Pargament, K.I. (2008). Advances in the conceptualization an measurement of religion and spirituality: Implications for physical and mental health research. Psychology of Religion and Spirituality, 1, 3-17. Lukoff, D. (1985). The diagnosis of mystical experience with psychotic features. The Journal of Transpersonal Psychology, 2, 155-181. Maslow, A. H. (1969). The farther reaches of human nature. Journal of Transpersonal Psychology, 1-9. Reinert, D.F. (2005). Spirituality, self-representations, and attachment to parents: a longitudinal study of roman catholic college seminarians. Journal of Counselling and Values, 226-238. Rogers, C. R. (1951). Client-centered therapy: its current practice, implications and theory. London: Constable. Stanislav, (2008). A brief history of transpersonal psychology. The Journal of Transpersonal Studies, 1-21.}} [[Category:Motivation and emotion/Book/2011]] [[Category:Motivation and emotion/Book/Religiosity]] [[Category:Motivation and emotion/Book/Spirituality]] sze54vo7dgy4gdz46zeplswirk5dgw6 0 139384 2801390 2800962 2026-03-30T01:37:03Z Young1lim 21186 /* Adder */ 2801390 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260330.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] o4bfi4kaoqbzf5hp3m29j2az8nrug8m 0 164590 2801503 1597892 2026-03-30T08:31:26Z Anikmolla786 3007678 Fixed typo: “distingushed” → “distinguished”. 2801503 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctiual performance if it benefits pwrsons with disabilities. Under current US law, the term "work" seems to be somewhat ambigupus, but may have been inxluded to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have diffrent parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 42ljrge3curiz3e6iyb3tb5bhtese3i 2801504 2801503 2026-03-30T08:32:13Z Anikmolla786 3007678 Fixed typo: “instinctiual” → “instinctual”. 2801504 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits pwrsons with disabilities. Under current US law, the term "work" seems to be somewhat ambigupus, but may have been inxluded to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have diffrent parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} pts08lsex6zz5uw6p7ngoi3s44ztfkl 2801505 2801504 2026-03-30T08:35:45Z Anikmolla786 3007678 Fixed typo: “pwrsons” → “persons”. 2801505 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambigupus, but may have been inxluded to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have diffrent parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 862t32vjpnx4iy8n724pxtbsi60dl4r 2801506 2801505 2026-03-30T08:36:39Z Anikmolla786 3007678 Fixed typo: “ambigupus” → “ambiguous”. 2801506 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been inxluded to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have diffrent parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} sv4pc6py6jvdgehnbbso5wfk9qumkli 2801507 2801506 2026-03-30T08:37:25Z Anikmolla786 3007678 Fixed typo: “inxluded” → “included”. 2801507 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have diffrent parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 4dfxwyky3w3vwknwdil08hqlro00y1z 2801508 2801507 2026-03-30T08:38:28Z Anikmolla786 3007678 Fixed typo: “diffrent” → “different”. 2801508 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interprwetation ary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} hq5iy10kz321u00nn2q7siwtq93kgfq 2801509 2801508 2026-03-30T08:39:28Z Anikmolla786 3007678 Fixed typo: “interprwetation” → “interpretation”. 2801509 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and diffwerent Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 7cq0336ux5dg3krxzl2hm7mettol8i1 2801510 2801509 2026-03-30T08:40:46Z Anikmolla786 3007678 Fixed typo: “diffwerent” → “different”. 2801510 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to peeform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} aw5sf3mmqjkcr372wlyaty1o4a91k2u 2801511 2801510 2026-03-30T08:41:42Z Anikmolla786 3007678 Fixed typo: “peeform” → “perform”. 2801511 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks ordo work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} fom3r4v50pswdtuqbvkvhcnjngdwltl 2801512 2801511 2026-03-30T08:42:32Z Anikmolla786 3007678 Added missing space: “ordo” → “or do”. 2801512 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different deom peoemance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 55ztdx8q6p4l5ig7lcwajjo3zcdbvoh 2801513 2801512 2026-03-30T08:43:21Z Anikmolla786 3007678 Fixed typo: “deom” → “from”. 2801513 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who my have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 02s7a9cou2wwd33r781ouc5apd33isn 2801514 2801513 2026-03-30T08:44:55Z Anikmolla786 3007678 Fixed typo: “my” → “may”. 2801514 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seems that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} q9m2yn6rmbbv5dqffyrvjsuyeuo78u6 2801515 2801514 2026-03-30T08:45:45Z Anikmolla786 3007678 Fixed subject‑verb agreement: “seems” → “seem”. 2801515 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by srvice dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} eps130hb9ju9ovjbaqwbz75af3455lh 2801516 2801515 2026-03-30T08:46:31Z Anikmolla786 3007678 Fixed typo: “srvice” → “service”. 2801516 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to " non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 8dpudm10cpqnvxvypo323vwyyb3kngf 2801517 2801516 2026-03-30T08:49:54Z Anikmolla786 3007678 2801517 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular wold be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} dz9ultyqyl0rhdoftkv0zqplobyfjnl 2801518 2801517 2026-03-30T08:51:21Z Anikmolla786 3007678 Fixed typo: “wold” → “would”. 2801518 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, American vernacular would be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} ggflfey3zwrud0n66k2qruok6m12ajp 2801519 2801518 2026-03-30T08:55:09Z Anikmolla786 3007678 Fixed typo: “America” → “the USA”. For better understanding. Or people can think that's America continent. 2801519 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to tjankthem for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} oq4h5lt4micxx95kcpfiyjjp2qx6mzn 2801520 2801519 2026-03-30T08:56:28Z Anikmolla786 3007678 Fixed typo: “tjankthem” → “thank them”. 2801520 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses cann be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 2py0ufq1whxb8ll4jh88tpvz0vdeg19 2801521 2801520 2026-03-30T08:57:24Z Anikmolla786 3007678 Fixed typo: “cann” → “can”. 2801521 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue drom the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 24t0zuztiq6dm3wkg70d9dxjegku7db 2801522 2801521 2026-03-30T08:58:07Z Anikmolla786 3007678 Fixed typo: “drom” → “from”. 2801522 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which sttangers will not.Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} tk8si4felr65zrbtnshdkbtyc02pesz 2801523 2801522 2026-03-30T08:59:24Z Anikmolla786 3007678 Fixed typo: “sttangers” → “strangers”. 2801523 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitationa due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 7io59birsyz4mmtdtn6tl271y3of314 2801524 2801523 2026-03-30T09:00:27Z Anikmolla786 3007678 Fixed typo: “limitationa” → “limitations”. 2801524 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the partof third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 96tqj1hrbpuigkngv1sre7uouavmtee 2801525 2801524 2026-03-30T09:01:22Z Anikmolla786 3007678 Added missing space: “partof” → “part of”. 2801525 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the part of third parties. aA situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} 4cj882ivoae8hopz7kgjgi5v6qmjk4j 2801526 2801525 2026-03-30T09:02:40Z Anikmolla786 3007678 Removed extra “a”. 2801526 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the part of third parties. A situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at Least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} l1tw2a9hn1xap49wmotf1eu2tslria6 2801527 2801526 2026-03-30T09:03:31Z Anikmolla786 3007678 Corrected capitalization: “at Least” → “at least”. 2801527 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the part of third parties. A situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[posiyivr training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} s9a52fbiacljnhmjmqwj0zfxs64n2fm 2801528 2801527 2026-03-30T09:06:01Z Anikmolla786 3007678 Fixed typo: “posiyivr” → “positive”. 2801528 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the part of third parties. A situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[positive training]] is more likely toevidence non-violence. See also [[K9]] {{CourseCat}} ha8vtxnoaeac1u9vdk6k7v946d21v1s 2801529 2801528 2026-03-30T09:07:11Z Anikmolla786 3007678 Added missing space: “toevidence” → “to evidence”. 2801529 wikitext text/x-wiki Service dogs are distinguished under US law by performance of "tasks or work". This is as opposed to either "mere presence" of a comforting animal, or the performances based upon instinct. The regulations are reviewed every ten years, and some voices seek to have the definition to include instinctual performance if it benefits persons with disabilities. Under current US law, the term "work" seems to be somewhat ambiguous, but may have been included to allow some wiggle room in cases where it is clear the animal is working to benefit a qualified handler, but it is difficult to pinpoint a specific task per se. Research question: how have different parties differentiated between SD tasks and "work". Does interpretation vary widely amongst DOJ experts, advocates and owner-handlers? Are there clear misconceptions in play? Have judicial opinions in state courts and different Federal circuit courts varied in their usage of the terminology? ==Advanced== The [[ADA regulations]] define a service dog as one trained to perform tasks or do work. Homework Research question: How is 'work" different from performance of "tasks"? Without doing any research, what do you believe the distinctions may be? How are the terms defined in various general dictionaries? ...legal dictionaries? How are the terms used in DOJ discussions? ...in the world of service dog discussion? ...disability specialists? ...lawmakers? ...public commenters who may have influenced legislative or regulatory history? Are there distinct differences in these usages? Does it seem that usages by service dog trainers and handlers diverge from usage by DOJ sources, judges, or lawmakers? Might such differences reflect misconceptions? ........... In the [[DOJ addendum]], reference is made early in the discussions to "non-violent protection or rescue work". Setting aside the analysis of whether this language may inform an analysis of "protection work", it is clear the department's duly diligent analysis conceptualizes "rescue work". Rescue implies an extreme event such as a fall, a medical crisis, a disaster or being somehow stuck. We can be stuck socially, and if someone comes along and provides an opportunity to exit the situation, the USA vernacular would be to thank them for "rescuing" us. Some writers on psychiatric disabilities discuss specifically training the assistance animal to provide an exit cue or response. These trained responses can be based upon trained or innate recognition of a dangerous situation. A different task altogether, but which is also discussed, is for the animal to respond to a cue from the owner which the animal recognizes as a command but which strangers will not. Such situations can arise as a direct result of functional limitations due to a disability or also from dissonance, discrimination or even malice on the part of third parties. A situation of this nature can activate a psychiatric state such as agitation, anger, or paranoia. A task can be defined of leading the handler to exit or take other measures based on either the handler's assessment or the canine's assessment. This can be characterized as either rescue work, protection work, or simply as "protection". Using the term "protection work" carries a different connotation than use of the bare term "protection". The latter could be interpreted to suggest a crime deterrent based upon (a) the mere presence of the animal or (b) based on violent "attack dog" training. These are both big [[no-no's]]. By using protection [[work]] we emphasize at least that this is not a [[mere presence crime deterrent]] situation. This usage still leaves the open question of whether the [[protection work]] in question is "attack dog" training or "[[non-violent protection]]" work. But that half of the battle can be won with training records showing bite inhibition, socialization and other features consistent with regulations and industry best practices. Much of the so-called "attack" training relies on old school [[alpha dog]] dominance hierarchy, and training records clearly consistent with new school [[positive training]] is more likely to evidence non-violence. See also [[K9]] {{CourseCat}} ttuq6c9n91dlf5w8qcglb4qugt041b6 0 172073 2801485 1807979 2026-03-30T07:26:24Z Anikmolla786 3007678 Fixed typo in father’s last name: “Man” → “Mann”. 2801485 wikitext text/x-wiki ==Overview== '''Charles Riborg Mann''' was a teacher, researcher, and physicist during World War I. He contributed to War Department decisions on education and U.S. universities during the war. ==Biography== Charles Mann was born in Orange, New Jersey on July 12th, 1869. His parents were Charles Holbrook Mann and Clausine Borchsenius Mann. He married Adrienne Amalie Graf on June 25, 1896. They had two children, Rihorg Graf and Adrienne<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. He had earned his Bachelors and Masters degrees from [[Wikipedia:Columbia University|Columbia University]] by 1891, and he earned his Ph.D. from the [[Wikipedia:Humboldt University of Berlin|University of Berlin]] in 1895. He worked at the [[Wikipedia:University of Chicago|University of Chicago]] from 1896 to 1914 and was promoted from research assistant to associate in physics to instructor before moving up to assistant professor and finally to associate professor<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. In 1914, he became involved in the Joint Committee on Engineering Education of National Engineering Societies and the [[Wikipedia:Carnegie Foundation for the Advancement of Teaching|Carnegie Foundation for the Advancement of Teaching]] in New York City collecting information about the engineering and physics education throughout the country. In 1918, he was appointed a civilian advisor to the Committee on Education and Special Training in the [[Wikipedia:United States Department of War|United States' War Department]]. There, Mann was involved in creating the Student's Army Training Corps. He corresponded with government representatives from the War Department and with university administrators around the country<ref>Mann</ref>. The committee’s work contributed to the development of relations between the U.S. military and the country’s university systems. Mann sent many reports and telegrams detailing his opinions and actions on the topic of training students for the military while allowing them to continue to attend college<ref>Breen</ref>. After the Students’ Army Training Corps was dissolved, he was appointed the permanent chairman of the Civilian Advisory Board to the War department and also served as the director of the American Council on Education from 1922-1934. He was president emeritus from 1934 until his death on September 10th, 1942 at age seventy-three. ==Social Issues== ===The Students' Army Training Corps=== As the United States entered World War I in 1917, it began to prepare itself for the conflict. One method of recruiting and training new soldiers was to turn to the readily available population of young men in the country’s university system. The Students’ Army Training Corps (S.A.T.C.) was created in the summer of 1918 by the War Department after advisement from the Committee on Education and Special Training. This served the dual purpose of providing military training to students in any U.S. university with more than 100 male students in attendance while identifying successful students as officer candidates and of continuing to educate these students in areas that would be useful to the war effort<ref>Mann</ref>. According to [[Wikipedia:The Daily Tar Heel|''The Daily Tar Heel'']] (the official newspaper of [[Wikipedia:University of North Carolina at Chapel Hill|The University of North Carolina at Chapel Hill]], one of the schools with chapters of the S.A.T.C.), “The instruction for S.A.T.C. members will be partly military and partly in allied subjects that have a value as a means of training officers and experts to meet the needs of the service." Members of the corps were drawn from most all of the disciplines available for study at the time with emphasis on European languages, manual design, and hard sciences<ref>"The New Order Brings Many Campus Changes"</ref>. During the existence of the S.A.T.C., which lasted until the end of the war, buildings on campuses were converted into mess halls, barracks, and armories. According to the Washington Post, student candidates were “required to live virtually under army conditions.”<ref>"Training Corps at Colleges"</ref> However, the war was won soon after the initiation of the program and by December of 1918, Mann sent out the announcement to the hundreds of involved U.S. universities that the program was no longer in effect. In the following years many requested the final report of the Corps' effectiveness, which Mann provided<ref>Mann</ref>. ===Higher Education in the Early 20th Century=== Charles Mann was a life-long academic. Growing up, he had access to greater education than most people at the time. Being born in the northern part of the U.S. made this possible, though he did leave the country at times while pursuing his education. He died an impressively educated man, holding two post-doctoral degrees: an [[Wikipedia:Doctor of Science|Sc.D.]] and an [[Wikipedia:Legum Doctor|LL.D.]] in addition to his Ph.D.<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. Higher education had just begun to blossom when the war began. Just after the turn of the century, “junior colleges” (now called community colleges) started to spring up all over the country, growing from only about twenty in 1909 to almost 300 by the end of the war. Four-year colleges were also growing, but not as quickly and predominantly in smaller areas more rural than where junior colleges were established. These smaller programs continued to be more popular for most of the early 1900s because of their open enrollment, low cost of attendance, and emphasis on technical education. ==References== {{Reflist}} {{Refbegin}} * Breen, W. J. ''Uncle Sam at Home: Civilian Mobilization, Wartime Federalism, and the Council of National Defense, 1917-1919''. Westport, Conn.: Greenwood Press, 1984. Web. * Faughnan, Michael J. "You're in the Army Now: The Students' Army Training Corps at Selected Virginia Universities in 1918." Ph.D. The College of William and Mary, 2008. United States -- Virginia: ''ProQuest Dissertations & Theses Full Text, ProQuest Dissertations & Theses Global.'' Web. * "The New Order Brings Many Campus Changes." ''The Daily Tarheel,'' XXVII ed.: 1. 02 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492>. * Mann, Charles Riborg. Box 1, Folders 1-9. Charles Riborg Mann papers. Southern Historical Collection, University of North Carolina Chapel Hill Libraries, Chapel Hill, NC. 2 Feb 2015. * "Over 1400 Students and Alumni are in Service." ''The Daily Tarheel,'' XXVII ed.: 1. 09 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492;>. * "Training Corps at Colleges." ''The Washington Post (1877-1922):'' 6. Jul 26, 1918 1918. ''ProQuest Historical Newspapers: The Washington Post.'' Web. <http://search.proquest.com/docview/145623893?accountid=14244>. * ''Who Was Who in America : A Companion Volume to Who's Who in America.'' 2 Vol. Chicago: Marquis, 1950. Web. {{Refend}} [[Category:World War I -- Life Histories|Mann, Charles Riborg]] tvpdqogokzto3obgfa7f3re7z7xlyl2 2801487 2801485 2026-03-30T07:27:15Z Anikmolla786 3007678 Corrected son’s name: “Rihorg” → “Riborg” (matches father’s middle name). 2801487 wikitext text/x-wiki ==Overview== '''Charles Riborg Mann''' was a teacher, researcher, and physicist during World War I. He contributed to War Department decisions on education and U.S. universities during the war. ==Biography== Charles Mann was born in Orange, New Jersey on July 12th, 1869. His parents were Charles Holbrook Mann and Clausine Borchsenius Mann. He married Adrienne Amalie Graf on June 25, 1896. They had two children, Riborg Graf and Adrienne<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. He had earned his Bachelors and Masters degrees from [[Wikipedia:Columbia University|Columbia University]] by 1891, and he earned his Ph.D. from the [[Wikipedia:Humboldt University of Berlin|University of Berlin]] in 1895. He worked at the [[Wikipedia:University of Chicago|University of Chicago]] from 1896 to 1914 and was promoted from research assistant to associate in physics to instructor before moving up to assistant professor and finally to associate professor<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. In 1914, he became involved in the Joint Committee on Engineering Education of National Engineering Societies and the [[Wikipedia:Carnegie Foundation for the Advancement of Teaching|Carnegie Foundation for the Advancement of Teaching]] in New York City collecting information about the engineering and physics education throughout the country. In 1918, he was appointed a civilian advisor to the Committee on Education and Special Training in the [[Wikipedia:United States Department of War|United States' War Department]]. There, Mann was involved in creating the Student's Army Training Corps. He corresponded with government representatives from the War Department and with university administrators around the country<ref>Mann</ref>. The committee’s work contributed to the development of relations between the U.S. military and the country’s university systems. Mann sent many reports and telegrams detailing his opinions and actions on the topic of training students for the military while allowing them to continue to attend college<ref>Breen</ref>. After the Students’ Army Training Corps was dissolved, he was appointed the permanent chairman of the Civilian Advisory Board to the War department and also served as the director of the American Council on Education from 1922-1934. He was president emeritus from 1934 until his death on September 10th, 1942 at age seventy-three. ==Social Issues== ===The Students' Army Training Corps=== As the United States entered World War I in 1917, it began to prepare itself for the conflict. One method of recruiting and training new soldiers was to turn to the readily available population of young men in the country’s university system. The Students’ Army Training Corps (S.A.T.C.) was created in the summer of 1918 by the War Department after advisement from the Committee on Education and Special Training. This served the dual purpose of providing military training to students in any U.S. university with more than 100 male students in attendance while identifying successful students as officer candidates and of continuing to educate these students in areas that would be useful to the war effort<ref>Mann</ref>. According to [[Wikipedia:The Daily Tar Heel|''The Daily Tar Heel'']] (the official newspaper of [[Wikipedia:University of North Carolina at Chapel Hill|The University of North Carolina at Chapel Hill]], one of the schools with chapters of the S.A.T.C.), “The instruction for S.A.T.C. members will be partly military and partly in allied subjects that have a value as a means of training officers and experts to meet the needs of the service." Members of the corps were drawn from most all of the disciplines available for study at the time with emphasis on European languages, manual design, and hard sciences<ref>"The New Order Brings Many Campus Changes"</ref>. During the existence of the S.A.T.C., which lasted until the end of the war, buildings on campuses were converted into mess halls, barracks, and armories. According to the Washington Post, student candidates were “required to live virtually under army conditions.”<ref>"Training Corps at Colleges"</ref> However, the war was won soon after the initiation of the program and by December of 1918, Mann sent out the announcement to the hundreds of involved U.S. universities that the program was no longer in effect. In the following years many requested the final report of the Corps' effectiveness, which Mann provided<ref>Mann</ref>. ===Higher Education in the Early 20th Century=== Charles Mann was a life-long academic. Growing up, he had access to greater education than most people at the time. Being born in the northern part of the U.S. made this possible, though he did leave the country at times while pursuing his education. He died an impressively educated man, holding two post-doctoral degrees: an [[Wikipedia:Doctor of Science|Sc.D.]] and an [[Wikipedia:Legum Doctor|LL.D.]] in addition to his Ph.D.<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. Higher education had just begun to blossom when the war began. Just after the turn of the century, “junior colleges” (now called community colleges) started to spring up all over the country, growing from only about twenty in 1909 to almost 300 by the end of the war. Four-year colleges were also growing, but not as quickly and predominantly in smaller areas more rural than where junior colleges were established. These smaller programs continued to be more popular for most of the early 1900s because of their open enrollment, low cost of attendance, and emphasis on technical education. ==References== {{Reflist}} {{Refbegin}} * Breen, W. J. ''Uncle Sam at Home: Civilian Mobilization, Wartime Federalism, and the Council of National Defense, 1917-1919''. Westport, Conn.: Greenwood Press, 1984. Web. * Faughnan, Michael J. "You're in the Army Now: The Students' Army Training Corps at Selected Virginia Universities in 1918." Ph.D. The College of William and Mary, 2008. United States -- Virginia: ''ProQuest Dissertations & Theses Full Text, ProQuest Dissertations & Theses Global.'' Web. * "The New Order Brings Many Campus Changes." ''The Daily Tarheel,'' XXVII ed.: 1. 02 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492>. * Mann, Charles Riborg. Box 1, Folders 1-9. Charles Riborg Mann papers. Southern Historical Collection, University of North Carolina Chapel Hill Libraries, Chapel Hill, NC. 2 Feb 2015. * "Over 1400 Students and Alumni are in Service." ''The Daily Tarheel,'' XXVII ed.: 1. 09 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492;>. * "Training Corps at Colleges." ''The Washington Post (1877-1922):'' 6. Jul 26, 1918 1918. ''ProQuest Historical Newspapers: The Washington Post.'' Web. <http://search.proquest.com/docview/145623893?accountid=14244>. * ''Who Was Who in America : A Companion Volume to Who's Who in America.'' 2 Vol. Chicago: Marquis, 1950. Web. {{Refend}} [[Category:World War I -- Life Histories|Mann, Charles Riborg]] gosdi8b5scizxnyxamtgbrbaeel6wsf 2801489 2801487 2026-03-30T07:28:20Z Anikmolla786 3007678 Added missing apostrophes for possessive forms: “Bachelor’s” and “Master’s”. 2801489 wikitext text/x-wiki ==Overview== '''Charles Riborg Mann''' was a teacher, researcher, and physicist during World War I. He contributed to War Department decisions on education and U.S. universities during the war. ==Biography== Charles Mann was born in Orange, New Jersey on July 12th, 1869. His parents were Charles Holbrook Mann and Clausine Borchsenius Mann. He married Adrienne Amalie Graf on June 25, 1896. They had two children, Riborg Graf and Adrienne<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. He had earned his Bachelor’s and Master’s degrees from [[Wikipedia:Columbia University|Columbia University]] by 1891, and he earned his Ph.D. from the [[Wikipedia:Humboldt University of Berlin|University of Berlin]] in 1895. He worked at the [[Wikipedia:University of Chicago|University of Chicago]] from 1896 to 1914 and was promoted from research assistant to associate in physics to instructor before moving up to assistant professor and finally to associate professor<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. In 1914, he became involved in the Joint Committee on Engineering Education of National Engineering Societies and the [[Wikipedia:Carnegie Foundation for the Advancement of Teaching|Carnegie Foundation for the Advancement of Teaching]] in New York City collecting information about the engineering and physics education throughout the country. In 1918, he was appointed a civilian advisor to the Committee on Education and Special Training in the [[Wikipedia:United States Department of War|United States' War Department]]. There, Mann was involved in creating the Student's Army Training Corps. He corresponded with government representatives from the War Department and with university administrators around the country<ref>Mann</ref>. The committee’s work contributed to the development of relations between the U.S. military and the country’s university systems. Mann sent many reports and telegrams detailing his opinions and actions on the topic of training students for the military while allowing them to continue to attend college<ref>Breen</ref>. After the Students’ Army Training Corps was dissolved, he was appointed the permanent chairman of the Civilian Advisory Board to the War department and also served as the director of the American Council on Education from 1922-1934. He was president emeritus from 1934 until his death on September 10th, 1942 at age seventy-three. ==Social Issues== ===The Students' Army Training Corps=== As the United States entered World War I in 1917, it began to prepare itself for the conflict. One method of recruiting and training new soldiers was to turn to the readily available population of young men in the country’s university system. The Students’ Army Training Corps (S.A.T.C.) was created in the summer of 1918 by the War Department after advisement from the Committee on Education and Special Training. This served the dual purpose of providing military training to students in any U.S. university with more than 100 male students in attendance while identifying successful students as officer candidates and of continuing to educate these students in areas that would be useful to the war effort<ref>Mann</ref>. According to [[Wikipedia:The Daily Tar Heel|''The Daily Tar Heel'']] (the official newspaper of [[Wikipedia:University of North Carolina at Chapel Hill|The University of North Carolina at Chapel Hill]], one of the schools with chapters of the S.A.T.C.), “The instruction for S.A.T.C. members will be partly military and partly in allied subjects that have a value as a means of training officers and experts to meet the needs of the service." Members of the corps were drawn from most all of the disciplines available for study at the time with emphasis on European languages, manual design, and hard sciences<ref>"The New Order Brings Many Campus Changes"</ref>. During the existence of the S.A.T.C., which lasted until the end of the war, buildings on campuses were converted into mess halls, barracks, and armories. According to the Washington Post, student candidates were “required to live virtually under army conditions.”<ref>"Training Corps at Colleges"</ref> However, the war was won soon after the initiation of the program and by December of 1918, Mann sent out the announcement to the hundreds of involved U.S. universities that the program was no longer in effect. In the following years many requested the final report of the Corps' effectiveness, which Mann provided<ref>Mann</ref>. ===Higher Education in the Early 20th Century=== Charles Mann was a life-long academic. Growing up, he had access to greater education than most people at the time. Being born in the northern part of the U.S. made this possible, though he did leave the country at times while pursuing his education. He died an impressively educated man, holding two post-doctoral degrees: an [[Wikipedia:Doctor of Science|Sc.D.]] and an [[Wikipedia:Legum Doctor|LL.D.]] in addition to his Ph.D.<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. Higher education had just begun to blossom when the war began. Just after the turn of the century, “junior colleges” (now called community colleges) started to spring up all over the country, growing from only about twenty in 1909 to almost 300 by the end of the war. Four-year colleges were also growing, but not as quickly and predominantly in smaller areas more rural than where junior colleges were established. These smaller programs continued to be more popular for most of the early 1900s because of their open enrollment, low cost of attendance, and emphasis on technical education. ==References== {{Reflist}} {{Refbegin}} * Breen, W. J. ''Uncle Sam at Home: Civilian Mobilization, Wartime Federalism, and the Council of National Defense, 1917-1919''. Westport, Conn.: Greenwood Press, 1984. Web. * Faughnan, Michael J. "You're in the Army Now: The Students' Army Training Corps at Selected Virginia Universities in 1918." Ph.D. The College of William and Mary, 2008. United States -- Virginia: ''ProQuest Dissertations & Theses Full Text, ProQuest Dissertations & Theses Global.'' Web. * "The New Order Brings Many Campus Changes." ''The Daily Tarheel,'' XXVII ed.: 1. 02 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492>. * Mann, Charles Riborg. Box 1, Folders 1-9. Charles Riborg Mann papers. Southern Historical Collection, University of North Carolina Chapel Hill Libraries, Chapel Hill, NC. 2 Feb 2015. * "Over 1400 Students and Alumni are in Service." ''The Daily Tarheel,'' XXVII ed.: 1. 09 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492;>. * "Training Corps at Colleges." ''The Washington Post (1877-1922):'' 6. Jul 26, 1918 1918. ''ProQuest Historical Newspapers: The Washington Post.'' Web. <http://search.proquest.com/docview/145623893?accountid=14244>. * ''Who Was Who in America : A Companion Volume to Who's Who in America.'' 2 Vol. Chicago: Marquis, 1950. Web. {{Refend}} [[Category:World War I -- Life Histories|Mann, Charles Riborg]] objj6c4zalq889kpjvnqjab0622l280 2801490 2801489 2026-03-30T07:30:01Z Anikmolla786 3007678 Removed stray semicolon from URL in reference. 2801490 wikitext text/x-wiki ==Overview== '''Charles Riborg Mann''' was a teacher, researcher, and physicist during World War I. He contributed to War Department decisions on education and U.S. universities during the war. ==Biography== Charles Mann was born in Orange, New Jersey on July 12th, 1869. His parents were Charles Holbrook Mann and Clausine Borchsenius Mann. He married Adrienne Amalie Graf on June 25, 1896. They had two children, Riborg Graf and Adrienne<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. He had earned his Bachelor’s and Master’s degrees from [[Wikipedia:Columbia University|Columbia University]] by 1891, and he earned his Ph.D. from the [[Wikipedia:Humboldt University of Berlin|University of Berlin]] in 1895. He worked at the [[Wikipedia:University of Chicago|University of Chicago]] from 1896 to 1914 and was promoted from research assistant to associate in physics to instructor before moving up to assistant professor and finally to associate professor<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. In 1914, he became involved in the Joint Committee on Engineering Education of National Engineering Societies and the [[Wikipedia:Carnegie Foundation for the Advancement of Teaching|Carnegie Foundation for the Advancement of Teaching]] in New York City collecting information about the engineering and physics education throughout the country. In 1918, he was appointed a civilian advisor to the Committee on Education and Special Training in the [[Wikipedia:United States Department of War|United States' War Department]]. There, Mann was involved in creating the Student's Army Training Corps. He corresponded with government representatives from the War Department and with university administrators around the country<ref>Mann</ref>. The committee’s work contributed to the development of relations between the U.S. military and the country’s university systems. Mann sent many reports and telegrams detailing his opinions and actions on the topic of training students for the military while allowing them to continue to attend college<ref>Breen</ref>. After the Students’ Army Training Corps was dissolved, he was appointed the permanent chairman of the Civilian Advisory Board to the War department and also served as the director of the American Council on Education from 1922-1934. He was president emeritus from 1934 until his death on September 10th, 1942 at age seventy-three. ==Social Issues== ===The Students' Army Training Corps=== As the United States entered World War I in 1917, it began to prepare itself for the conflict. One method of recruiting and training new soldiers was to turn to the readily available population of young men in the country’s university system. The Students’ Army Training Corps (S.A.T.C.) was created in the summer of 1918 by the War Department after advisement from the Committee on Education and Special Training. This served the dual purpose of providing military training to students in any U.S. university with more than 100 male students in attendance while identifying successful students as officer candidates and of continuing to educate these students in areas that would be useful to the war effort<ref>Mann</ref>. According to [[Wikipedia:The Daily Tar Heel|''The Daily Tar Heel'']] (the official newspaper of [[Wikipedia:University of North Carolina at Chapel Hill|The University of North Carolina at Chapel Hill]], one of the schools with chapters of the S.A.T.C.), “The instruction for S.A.T.C. members will be partly military and partly in allied subjects that have a value as a means of training officers and experts to meet the needs of the service." Members of the corps were drawn from most all of the disciplines available for study at the time with emphasis on European languages, manual design, and hard sciences<ref>"The New Order Brings Many Campus Changes"</ref>. During the existence of the S.A.T.C., which lasted until the end of the war, buildings on campuses were converted into mess halls, barracks, and armories. According to the Washington Post, student candidates were “required to live virtually under army conditions.”<ref>"Training Corps at Colleges"</ref> However, the war was won soon after the initiation of the program and by December of 1918, Mann sent out the announcement to the hundreds of involved U.S. universities that the program was no longer in effect. In the following years many requested the final report of the Corps' effectiveness, which Mann provided<ref>Mann</ref>. ===Higher Education in the Early 20th Century=== Charles Mann was a life-long academic. Growing up, he had access to greater education than most people at the time. Being born in the northern part of the U.S. made this possible, though he did leave the country at times while pursuing his education. He died an impressively educated man, holding two post-doctoral degrees: an [[Wikipedia:Doctor of Science|Sc.D.]] and an [[Wikipedia:Legum Doctor|LL.D.]] in addition to his Ph.D.<ref>''Who Was Who in America : A Companion Volume to Who's Who in America.''</ref>. Higher education had just begun to blossom when the war began. Just after the turn of the century, “junior colleges” (now called community colleges) started to spring up all over the country, growing from only about twenty in 1909 to almost 300 by the end of the war. Four-year colleges were also growing, but not as quickly and predominantly in smaller areas more rural than where junior colleges were established. These smaller programs continued to be more popular for most of the early 1900s because of their open enrollment, low cost of attendance, and emphasis on technical education. ==References== {{Reflist}} {{Refbegin}} * Breen, W. J. ''Uncle Sam at Home: Civilian Mobilization, Wartime Federalism, and the Council of National Defense, 1917-1919''. Westport, Conn.: Greenwood Press, 1984. Web. * Faughnan, Michael J. "You're in the Army Now: The Students' Army Training Corps at Selected Virginia Universities in 1918." Ph.D. The College of William and Mary, 2008. United States -- Virginia: ''ProQuest Dissertations & Theses Full Text, ProQuest Dissertations & Theses Global.'' Web. * "The New Order Brings Many Campus Changes." ''The Daily Tarheel,'' XXVII ed.: 1. 02 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492>. * Mann, Charles Riborg. Box 1, Folders 1-9. Charles Riborg Mann papers. Southern Historical Collection, University of North Carolina Chapel Hill Libraries, Chapel Hill, NC. 2 Feb 2015. * "Over 1400 Students and Alumni are in Service." ''The Daily Tarheel,'' XXVII ed.: 1. 09 Oct 1918 1918. ''NewsPapers.com.'' Web. 19 Jan 2015 <http://www.newspapers.com/image/67881492>. * "Training Corps at Colleges." ''The Washington Post (1877-1922):'' 6. Jul 26, 1918 1918. ''ProQuest Historical Newspapers: The Washington Post.'' Web. <http://search.proquest.com/docview/145623893?accountid=14244>. * ''Who Was Who in America : A Companion Volume to Who's Who in America.'' 2 Vol. Chicago: Marquis, 1950. Web. {{Refend}} [[Category:World War I -- Life Histories|Mann, Charles Riborg]] 0hlt5pbjy0cy8rbb83ys2c24e3hdppl 0 199237 2801468 2171610 2026-03-30T07:11:23Z Anikmolla786 3007678 Fixed plural “speeds” and “than those” for consistency. 2801468 wikitext text/x-wiki [[Image:Dew collection icon.png|right|80px|]] __NOTOC__ Dew can bring substantial amount of water when the other resources (groundwater, rain, fog) are lacking. Rainwater harvesting structures can also be used to collect dew at night, which condenses on a surface from where water droplets drip into a gutter that goes to a reservoir. Quantity from dew varies according to location but at times has been shown to constitute a significant proportion of normal rainfall. Dew collection is a possible supplementary source of water in arid and semi- arid areas, but has so far not been widely applied and is an area of further research. ===Suitable conditions=== * Large areas of unused land. * Areas with large diurnal temperature range (at least 12 degrees Celsius from day to night). * Areas with low winds (winds cause evaporation). It is advised to have an upper limit of 4 m/s at 10m. * The average wind speeds where dew forms correspond to rather large speeds (1-2 m/s), higher than those found in continental locations of 0.1–0.2 m/s. * Dew yields are seen to rise when cloud cover diminishes because the radiative cooling increases. * The largest dew yields correspond to the highest humidity and the lowest cooling temperature. * The condensing material must be adequate (thermally isolated, water resistant, on a slope). ====Do not collect dew in these circumstances:==== * On or near poisonous plants * On plants or objects that are chemically treated or sprayed * In areas where obvious animal defecation has taken place * Near roadsides {| border="1" cellpadding="5" cellspacing="0" align="center" |- ! width="50%" style="background:#efefef;" | Advantages ! style="background:#f0f8ff;" | Disadvantages |- | valign="top" | - Water quality can be good <br> - Unaffected by drought <br> - Can possibly be a supplementary water source to rainwater harvesting <br> - Low costs especially for dew irrigation ($1 per plant) <br> - Possible to do on household level <br> | valign="top" | - Relatively small water quantities can be harvested <br> - Variability in water collection, which also varies according to the season. Need to supplement water from other sources. <br> |} ===Resilience to changes in the environment=== Dew collection is not affected by drought or changing precipitation, only if there are changes in humidity. Less humidity means less dew. ===Construction, operation and maintenance=== [[Image:Dew water from metal roofs.jpg|thumb|right|200px|Properties that have metal roofing as this one , can be used to harvest dew water simply by adding gutters and for increased output a layer of insulation underside. Without the insulation the output is nearly half of that from plastic condensers.]] The best time to gather dew is in the early morning before the sun has touched your collection area. The catchment area needs to be large enough to collect the required amount of water, while also being practical to where it is collected. Typical collection rates vary according to the site – in Morocco rates of collection were 18.9 litres/m2/yr (in an area of 215mm annual rainfall) whereas in Spain rates increased to a range of 41.5 to 71.1 litres/ m2/yr (in an area of between 246 – 324mm annual rainfall). The mesh area constituting the catchment area should be made of a fine material that loses heat rapidly. A black fabric can be used, and a U.V. stabilized polyethylene is sometimes used as well. Efficiency of collection: it seems that elevated surfaces collect 14% more water compared to one on the ground. It may also be helpful to use galvanized roof sheets that can be painted with special [http://www.opur.fr/ OPUR] paint – this paint enhances infrared cooling and remain hydrophilic due to photocatalytic reaction with ultraviolet light. Such a system was created in Morocco where painted sheets were underlain with a 2cm thick polystyrene insulation, and the roof pitch was 30 degrees. The system uses radiative cooling at night. Foil can also be used but is prone to damage. ====Height==== The higher the collector the faster the air moved at that elevation and thus more dew collected. More research needs to be done on the relationship between dew collector height and water collection efficiency. Dew collection can also be used to create a form of micro irrigation where plastic trays are used to funnel dew to plant roots. ===Costs=== Costs vary widely, but the materials needed in each locality will determine total costs to build a system. <br> A simple system consists of a large flat angled surface, pipes or gutters to transport the dripping, and a collection tank. ===Field experiences=== * [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf FOG AND DEW COLLECTION PROJECTS IN CROATIA.] ===Manuals, videos, and links=== [http://www.opur.fr/ International Organization for Dew Utilization (OPUR)] ===Acknowledgements=== * CARE Nederland, Desk Study: [http://akvopedia.org/wiki/Resilient_WASH_systems_in_drought-prone_areas Resilient WASH systems in drought-prone areas]. October 2010. * M. Mileta, D., Beysens and V. Nikolayev, et al. [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf Fog and Dew Collection Projects in Croatia] Croatia and France, 2004. * [http://www.tacticalintelligence.net/blog/dew-collection.htm Dew Collection for Survival Water] Tactical Intelligence. July 2011. * Original [http://akvopedia.org Akvopedia.org] article and more images: [http://akvopedia.org/wiki/Dew_collection_and_storage akvopedia.org/wiki/Dew_collection_and_storage] ===References=== <references/> {{CourseCat}} 83rmopsqd3vuhetyktb292bccr9veg0 2801469 2801468 2026-03-30T07:12:24Z Anikmolla786 3007678 Changed “U.V.” to “UV” and added hyphen for compound adjective. 2801469 wikitext text/x-wiki [[Image:Dew collection icon.png|right|80px|]] __NOTOC__ Dew can bring substantial amount of water when the other resources (groundwater, rain, fog) are lacking. Rainwater harvesting structures can also be used to collect dew at night, which condenses on a surface from where water droplets drip into a gutter that goes to a reservoir. Quantity from dew varies according to location but at times has been shown to constitute a significant proportion of normal rainfall. Dew collection is a possible supplementary source of water in arid and semi- arid areas, but has so far not been widely applied and is an area of further research. ===Suitable conditions=== * Large areas of unused land. * Areas with large diurnal temperature range (at least 12 degrees Celsius from day to night). * Areas with low winds (winds cause evaporation). It is advised to have an upper limit of 4 m/s at 10m. * The average wind speeds where dew forms correspond to rather large speeds (1-2 m/s), higher than those found in continental locations of 0.1–0.2 m/s. * Dew yields are seen to rise when cloud cover diminishes because the radiative cooling increases. * The largest dew yields correspond to the highest humidity and the lowest cooling temperature. * The condensing material must be adequate (thermally isolated, water resistant, on a slope). ====Do not collect dew in these circumstances:==== * On or near poisonous plants * On plants or objects that are chemically treated or sprayed * In areas where obvious animal defecation has taken place * Near roadsides {| border="1" cellpadding="5" cellspacing="0" align="center" |- ! width="50%" style="background:#efefef;" | Advantages ! style="background:#f0f8ff;" | Disadvantages |- | valign="top" | - Water quality can be good <br> - Unaffected by drought <br> - Can possibly be a supplementary water source to rainwater harvesting <br> - Low costs especially for dew irrigation ($1 per plant) <br> - Possible to do on household level <br> | valign="top" | - Relatively small water quantities can be harvested <br> - Variability in water collection, which also varies according to the season. Need to supplement water from other sources. <br> |} ===Resilience to changes in the environment=== Dew collection is not affected by drought or changing precipitation, only if there are changes in humidity. Less humidity means less dew. ===Construction, operation and maintenance=== [[Image:Dew water from metal roofs.jpg|thumb|right|200px|Properties that have metal roofing as this one , can be used to harvest dew water simply by adding gutters and for increased output a layer of insulation underside. Without the insulation the output is nearly half of that from plastic condensers.]] The best time to gather dew is in the early morning before the sun has touched your collection area. The catchment area needs to be large enough to collect the required amount of water, while also being practical to where it is collected. Typical collection rates vary according to the site – in Morocco rates of collection were 18.9 litres/m2/yr (in an area of 215mm annual rainfall) whereas in Spain rates increased to a range of 41.5 to 71.1 litres/ m2/yr (in an area of between 246 – 324mm annual rainfall). The mesh area constituting the catchment area should be made of a fine material that loses heat rapidly. A black fabric can be used, and a UV-stabilized polyethylene is sometimes used as well. Efficiency of collection: it seems that elevated surfaces collect 14% more water compared to one on the ground. It may also be helpful to use galvanized roof sheets that can be painted with special [http://www.opur.fr/ OPUR] paint – this paint enhances infrared cooling and remain hydrophilic due to photocatalytic reaction with ultraviolet light. Such a system was created in Morocco where painted sheets were underlain with a 2cm thick polystyrene insulation, and the roof pitch was 30 degrees. The system uses radiative cooling at night. Foil can also be used but is prone to damage. ====Height==== The higher the collector the faster the air moved at that elevation and thus more dew collected. More research needs to be done on the relationship between dew collector height and water collection efficiency. Dew collection can also be used to create a form of micro irrigation where plastic trays are used to funnel dew to plant roots. ===Costs=== Costs vary widely, but the materials needed in each locality will determine total costs to build a system. <br> A simple system consists of a large flat angled surface, pipes or gutters to transport the dripping, and a collection tank. ===Field experiences=== * [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf FOG AND DEW COLLECTION PROJECTS IN CROATIA.] ===Manuals, videos, and links=== [http://www.opur.fr/ International Organization for Dew Utilization (OPUR)] ===Acknowledgements=== * CARE Nederland, Desk Study: [http://akvopedia.org/wiki/Resilient_WASH_systems_in_drought-prone_areas Resilient WASH systems in drought-prone areas]. October 2010. * M. Mileta, D., Beysens and V. Nikolayev, et al. [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf Fog and Dew Collection Projects in Croatia] Croatia and France, 2004. * [http://www.tacticalintelligence.net/blog/dew-collection.htm Dew Collection for Survival Water] Tactical Intelligence. July 2011. * Original [http://akvopedia.org Akvopedia.org] article and more images: [http://akvopedia.org/wiki/Dew_collection_and_storage akvopedia.org/wiki/Dew_collection_and_storage] ===References=== <references/> {{CourseCat}} eo22c2bp5r3zfj3tealmpxyakmymytr 2801472 2801469 2026-03-30T07:15:11Z Anikmolla786 3007678 Standardized unit formatting and removed stray space. 2801472 wikitext text/x-wiki [[Image:Dew collection icon.png|right|80px|]] __NOTOC__ Dew can bring substantial amount of water when the other resources (groundwater, rain, fog) are lacking. Rainwater harvesting structures can also be used to collect dew at night, which condenses on a surface from where water droplets drip into a gutter that goes to a reservoir. Quantity from dew varies according to location but at times has been shown to constitute a significant proportion of normal rainfall. Dew collection is a possible supplementary source of water in arid and semi- arid areas, but has so far not been widely applied and is an area of further research. ===Suitable conditions=== * Large areas of unused land. * Areas with large diurnal temperature range (at least 12 degrees Celsius from day to night). * Areas with low winds (winds cause evaporation). It is advised to have an upper limit of 4 m/s at 10m. * The average wind speeds where dew forms correspond to rather large speeds (1-2 m/s), higher than those found in continental locations of 0.1–0.2 m/s. * Dew yields are seen to rise when cloud cover diminishes because the radiative cooling increases. * The largest dew yields correspond to the highest humidity and the lowest cooling temperature. * The condensing material must be adequate (thermally isolated, water resistant, on a slope). ====Do not collect dew in these circumstances:==== * On or near poisonous plants * On plants or objects that are chemically treated or sprayed * In areas where obvious animal defecation has taken place * Near roadsides {| border="1" cellpadding="5" cellspacing="0" align="center" |- ! width="50%" style="background:#efefef;" | Advantages ! style="background:#f0f8ff;" | Disadvantages |- | valign="top" | - Water quality can be good <br> - Unaffected by drought <br> - Can possibly be a supplementary water source to rainwater harvesting <br> - Low costs especially for dew irrigation ($1 per plant) <br> - Possible to do on household level <br> | valign="top" | - Relatively small water quantities can be harvested <br> - Variability in water collection, which also varies according to the season. Need to supplement water from other sources. <br> |} ===Resilience to changes in the environment=== Dew collection is not affected by drought or changing precipitation, only if there are changes in humidity. Less humidity means less dew. ===Construction, operation and maintenance=== [[Image:Dew water from metal roofs.jpg|thumb|right|200px|Properties that have metal roofing as this one , can be used to harvest dew water simply by adding gutters and for increased output a layer of insulation underside. Without the insulation the output is nearly half of that from plastic condensers.]] The best time to gather dew is in the early morning before the sun has touched your collection area. The catchment area needs to be large enough to collect the required amount of water, while also being practical to where it is collected. Typical collection rates vary according to the site – in Morocco rates of collection were 18.9 litres/m²/yr (in an area of 215mm annual rainfall) whereas in Spain rates increased to a range of 41.5 to 71.1 litres/m²/yr (in an area of between 246 – 324mm annual rainfall). The mesh area constituting the catchment area should be made of a fine material that loses heat rapidly. A black fabric can be used, and a UV-stabilized polyethylene is sometimes used as well. Efficiency of collection: it seems that elevated surfaces collect 14% more water compared to one on the ground. It may also be helpful to use galvanized roof sheets that can be painted with special [http://www.opur.fr/ OPUR] paint – this paint enhances infrared cooling and remain hydrophilic due to photocatalytic reaction with ultraviolet light. Such a system was created in Morocco where painted sheets were underlain with a 2cm thick polystyrene insulation, and the roof pitch was 30 degrees. The system uses radiative cooling at night. Foil can also be used but is prone to damage. ====Height==== The higher the collector the faster the air moved at that elevation and thus more dew collected. More research needs to be done on the relationship between dew collector height and water collection efficiency. Dew collection can also be used to create a form of micro irrigation where plastic trays are used to funnel dew to plant roots. ===Costs=== Costs vary widely, but the materials needed in each locality will determine total costs to build a system. <br> A simple system consists of a large flat angled surface, pipes or gutters to transport the dripping, and a collection tank. ===Field experiences=== * [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf FOG AND DEW COLLECTION PROJECTS IN CROATIA.] ===Manuals, videos, and links=== [http://www.opur.fr/ International Organization for Dew Utilization (OPUR)] ===Acknowledgements=== * CARE Nederland, Desk Study: [http://akvopedia.org/wiki/Resilient_WASH_systems_in_drought-prone_areas Resilient WASH systems in drought-prone areas]. October 2010. * M. Mileta, D., Beysens and V. Nikolayev, et al. [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf Fog and Dew Collection Projects in Croatia] Croatia and France, 2004. * [http://www.tacticalintelligence.net/blog/dew-collection.htm Dew Collection for Survival Water] Tactical Intelligence. July 2011. * Original [http://akvopedia.org Akvopedia.org] article and more images: [http://akvopedia.org/wiki/Dew_collection_and_storage akvopedia.org/wiki/Dew_collection_and_storage] ===References=== <references/> {{CourseCat}} i4jn7t5jjzh7e5nn2kum89bs64ofgos 2801474 2801472 2026-03-30T07:16:04Z Anikmolla786 3007678 Added hyphen to “micro‑irrigation”. 2801474 wikitext text/x-wiki [[Image:Dew collection icon.png|right|80px|]] __NOTOC__ Dew can bring substantial amount of water when the other resources (groundwater, rain, fog) are lacking. Rainwater harvesting structures can also be used to collect dew at night, which condenses on a surface from where water droplets drip into a gutter that goes to a reservoir. Quantity from dew varies according to location but at times has been shown to constitute a significant proportion of normal rainfall. Dew collection is a possible supplementary source of water in arid and semi- arid areas, but has so far not been widely applied and is an area of further research. ===Suitable conditions=== * Large areas of unused land. * Areas with large diurnal temperature range (at least 12 degrees Celsius from day to night). * Areas with low winds (winds cause evaporation). It is advised to have an upper limit of 4 m/s at 10m. * The average wind speeds where dew forms correspond to rather large speeds (1-2 m/s), higher than those found in continental locations of 0.1–0.2 m/s. * Dew yields are seen to rise when cloud cover diminishes because the radiative cooling increases. * The largest dew yields correspond to the highest humidity and the lowest cooling temperature. * The condensing material must be adequate (thermally isolated, water resistant, on a slope). ====Do not collect dew in these circumstances:==== * On or near poisonous plants * On plants or objects that are chemically treated or sprayed * In areas where obvious animal defecation has taken place * Near roadsides {| border="1" cellpadding="5" cellspacing="0" align="center" |- ! width="50%" style="background:#efefef;" | Advantages ! style="background:#f0f8ff;" | Disadvantages |- | valign="top" | - Water quality can be good <br> - Unaffected by drought <br> - Can possibly be a supplementary water source to rainwater harvesting <br> - Low costs especially for dew irrigation ($1 per plant) <br> - Possible to do on household level <br> | valign="top" | - Relatively small water quantities can be harvested <br> - Variability in water collection, which also varies according to the season. Need to supplement water from other sources. <br> |} ===Resilience to changes in the environment=== Dew collection is not affected by drought or changing precipitation, only if there are changes in humidity. Less humidity means less dew. ===Construction, operation and maintenance=== [[Image:Dew water from metal roofs.jpg|thumb|right|200px|Properties that have metal roofing as this one , can be used to harvest dew water simply by adding gutters and for increased output a layer of insulation underside. Without the insulation the output is nearly half of that from plastic condensers.]] The best time to gather dew is in the early morning before the sun has touched your collection area. The catchment area needs to be large enough to collect the required amount of water, while also being practical to where it is collected. Typical collection rates vary according to the site – in Morocco rates of collection were 18.9 litres/m²/yr (in an area of 215mm annual rainfall) whereas in Spain rates increased to a range of 41.5 to 71.1 litres/m²/yr (in an area of between 246 – 324mm annual rainfall). The mesh area constituting the catchment area should be made of a fine material that loses heat rapidly. A black fabric can be used, and a UV-stabilized polyethylene is sometimes used as well. Efficiency of collection: it seems that elevated surfaces collect 14% more water compared to one on the ground. It may also be helpful to use galvanized roof sheets that can be painted with special [http://www.opur.fr/ OPUR] paint – this paint enhances infrared cooling and remain hydrophilic due to photocatalytic reaction with ultraviolet light. Such a system was created in Morocco where painted sheets were underlain with a 2cm thick polystyrene insulation, and the roof pitch was 30 degrees. The system uses radiative cooling at night. Foil can also be used but is prone to damage. ====Height==== The higher the collector the faster the air moved at that elevation and thus more dew collected. More research needs to be done on the relationship between dew collector height and water collection efficiency. Dew collection can also be used to create a form of micro‑irrigation where plastic trays are used to funnel dew to plant roots. ===Costs=== Costs vary widely, but the materials needed in each locality will determine total costs to build a system. <br> A simple system consists of a large flat angled surface, pipes or gutters to transport the dripping, and a collection tank. ===Field experiences=== * [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf FOG AND DEW COLLECTION PROJECTS IN CROATIA.] ===Manuals, videos, and links=== [http://www.opur.fr/ International Organization for Dew Utilization (OPUR)] ===Acknowledgements=== * CARE Nederland, Desk Study: [http://akvopedia.org/wiki/Resilient_WASH_systems_in_drought-prone_areas Resilient WASH systems in drought-prone areas]. October 2010. * M. Mileta, D., Beysens and V. Nikolayev, et al. [http://balwois.com/balwois/administration/full_paper/ffp-587.pdf Fog and Dew Collection Projects in Croatia] Croatia and France, 2004. * [http://www.tacticalintelligence.net/blog/dew-collection.htm Dew Collection for Survival Water] Tactical Intelligence. July 2011. * Original [http://akvopedia.org Akvopedia.org] article and more images: [http://akvopedia.org/wiki/Dew_collection_and_storage akvopedia.org/wiki/Dew_collection_and_storage] ===References=== <references/> {{CourseCat}} 7498txnj7gcc6hud5cmipvq8o0un9qw 0 203702 2801548 1597104 2026-03-30T10:11:18Z Anikmolla786 3007678 Removed personal training note. 2801548 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical sights] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the Downtown affects the beauty of this town because it is located next to lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main road, and they are the artery of the town. Many stores and coffee shops were in here and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted and made the economy of this town was the best in this region, and it started with a type of wood and transformed to banks, shopping stores, and other necessities that people need. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} j8hiihj8ac0vagkj4kavp0mvhi3qao9 2801551 2801548 2026-03-30T10:21:49Z Anikmolla786 3007678 Fixed subject‑verb agreement and article. 2801551 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the Downtown affects the beauty of this town because it is located next to lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main road, and they are the artery of the town. Many stores and coffee shops were in here and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted and made the economy of this town was the best in this region, and it started with a type of wood and transformed to banks, shopping stores, and other necessities that people need. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} 2mb6qrhfrxeg3hdfsuf52sx6i8wmxod 2801552 2801551 2026-03-30T10:22:52Z Anikmolla786 3007678 Lowercased “downtown” and capitalized “Lake”. 2801552 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main road, and they are the artery of the town. Many stores and coffee shops were in here and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted and made the economy of this town was the best in this region, and it started with a type of wood and transformed to banks, shopping stores, and other necessities that people need. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} 94hscnnjpy330k63jls9l0ezgd1xj28 2801553 2801552 2026-03-30T10:25:11Z Anikmolla786 3007678 Fixed grammar: “road” → “roads”, “artery” → “arteries”. 2801553 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were in here and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted and made the economy of this town was the best in this region, and it started with a type of wood and transformed to banks, shopping stores, and other necessities that people need. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} ra49jjdgeix5ijm098f1a4f7aeqle2r 2801554 2801553 2026-03-30T10:27:31Z Anikmolla786 3007678 Fixed run‑on sentence and missing verb. 2801554 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted and made the economy of this town was the best in this region, and it started with a type of wood and transformed to banks, shopping stores, and other necessities that people need. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} iobhrg94dqshg38mc5v5hy3yfp511yt 2801555 2801554 2026-03-30T10:29:38Z Anikmolla786 3007678 Reworded unclear and ungrammatical sentence. 2801555 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted business and made the economy of this town the best in the region. It started with a type of wood and transformed into banks, shopping stores, and other necessities. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historial Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} 80fx86tjbm4bst50jlhntdx7oqufqyb 2801556 2801555 2026-03-30T10:30:31Z Anikmolla786 3007678 Fixed typo: “Historial” → “Historical”. 2801556 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted business and made the economy of this town the best in the region. It started with a type of wood and transformed into banks, shopping stores, and other necessities. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historical Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial which because one of the best theaters in the whole world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} n5v3tuqy7e0pg8tdoaf1mj9e1x64h41 2801557 2801556 2026-03-30T10:31:24Z Anikmolla786 3007678 Corrected garbled sentence. 2801557 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted business and made the economy of this town the best in the region. It started with a type of wood and transformed into banks, shopping stores, and other necessities. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historical Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial that has become one of the best theaters in the world. == External Links == [http://www.Google.com Google] == References == == References == <ref>http://yahoo.com </ref> {{CourseCat}} e1ji6lmpam5mhfr074sh2v7x5jn89zl 2801559 2801557 2026-03-30T10:34:04Z Anikmolla786 3007678 Removed duplicate References heading and properly numbered the citations. 2801559 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted business and made the economy of this town the best in the region. It started with a type of wood and transformed into banks, shopping stores, and other necessities. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historical Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial that has become one of the best theaters in the world. == External Links == [http://www.Google.com Google] == References == <ref>http://yahoo.com </ref> {{CourseCat}} nt5ar13i473t5xzdey4qs0mjl6p0pv7 2801562 2801559 2026-03-30T10:41:14Z Anikmolla786 3007678 Removed duplicate References heading and properly numbered the citations. 2801562 wikitext text/x-wiki [[File:WIMap-doton-Menomonie.png|thumb]] == '''Introduction:''' == Downtown Menomonie is a [http://www.dunnhistory.org/ historical site] that attract many people to view due to the history of this town. There are many historical buildings that are so captivating such as Mabel Tainter Theater. Also, the location of the downtown affects the beauty of this town because it is located next to Lake Menomin. Furthermore, Downtown Menomonie contains many memorials which lead to sub-histories to its history. == '''Mine Street & Broadway Street of''' '''Menomonie:''' == “Downtown Menomonie and its Main Street were part of the original plat of the city that was laid out in 1859.” “The stores, shops, banks and offices along Broadway, Main Street and Crescent Street have provided a wide variety of goods and services for more than 100 years.” “Main Street’s distinctive buildings and lakefront setting are among the cornerstones of planning for the future.” “Many of the first buildings along Main Street, Broadway and Crescent were simple wood-frame structures. By 1900, many had been replaced by two- or three-story brick buildings, often with ornate facades typical of the then-popular Italinate and Queen Anne styles.” <ref><nowiki>http://www.mainstreetmenomonie.org/history/</nowiki></ref> Walking down Main Street and Broadway Street is enough to picture the history and the impact of the two main roads, and they are the arteries of the town. Many stores and coffee shops were here, and some of them still. Some buildings are the same since 80 years ago.  Wealthy people conducted business and made the economy of this town the best in the region. It started with a type of wood and transformed into banks, shopping stores, and other necessities. ==  '''The Mabel Tainter:''' == “The Mabel Tainter Memorial [[Menomonie, Wisconsin History/kbmarie|Theater]], built in 1889 as a memorial of the daughter of Captain [[Menomonie, Wisconsin History/Andrew Tainter/SarahBeth19|Andrew and Bertha Tainter]], is Downtown Menomonie’s crown jewel.” “The theater is a fully functional Victorian-era theater with a year-round performing arts season featuring nationally recognized artists.” “The theater is listed on the National Register of Historic Places, a charter member of the League of Historic American Theatres, and a designated Wisconsin Historical Marker Site.” “The Memorial provided the community a reading room and library, club rooms, and an auditorium. It also served as the home of the original Menomonie Unitarian Society, whose minister, Henry Doty Maxson, inspired the Tainters to have it built as a center dedicated to the city's moral and social welfare. The Mabel Tainter Memorial continues to provide a cultural and educational focus for the entire community.” === <ref><nowiki>http://www.hmdb.org/marker.asp?marker=9854&Result=1</nowiki></ref> === The theater has provided many facilities to the community and has an impacted on downtown Menomonie. For example, there are many rooms in the theater that conduct meetings and gatherings. Also, the first library in Menomonie was in the theater which was built in the theater, and the books were given by Mr. Tainter from his home, and they were about 3000 books. Furthermore, the library was free for Dunn county people. [[File:Mabel Tainter Theatre.jpg|alt=Red Apple|left|thumb|Red Apple]] picture another picture Apple.<ref name=":0">Google. (2015) Retrieved from: http://www.google.com </ref> == memorials == Mabel Tainter is a memorial that has become one of the best theaters in the world. == External Links == <ref>Google </ref> {{CourseCat}} <references /> av6pxhhroffwb92to5pox8czel8bdrq 0 206558 2801480 1503532 2026-03-30T07:23:10Z Anikmolla786 3007678 Added missing comma and period; capitalized “This”. 2801480 wikitext text/x-wiki In order to help students in their documenting of lab reports, we are not required to write a certain length and also are not held to grammatical issues very closely. This will help prevent cheating in the class because the students aren't held tightly to their work on the lab reports. the slope is 2.8 ft/s/s and the uncertainty is .2 ft/s/s In this lab the students used the spit wads that were thrown in front of the basketball to find the average acceleration of the basketball by measuring the distance between the spit wads. 4ec0fan35hwpeewkootnmvb2oiloyp6 2801483 2801480 2026-03-30T07:24:15Z Anikmolla786 3007678 Capitalized “The”; added leading zero to “.2”; added missing period. 2801483 wikitext text/x-wiki In order to help students in their documenting of lab reports, we are not required to write a certain length and also are not held to grammatical issues very closely. This will help prevent cheating in the class because the students aren't held tightly to their work on the lab reports. The slope is 2.8 ft/s/s and the uncertainty is 0.2 ft/s/s. In this lab the students used the spit wads that were thrown in front of the basketball to find the average acceleration of the basketball by measuring the distance between the spit wads. qjjuh9zdcfdm1squv2narpaneq63o5t 0 213879 2801332 2801225 2026-03-29T18:03:07Z Ireicher2 2797952 /* Instructions */ small edit 2801332 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox |name = Digital Media Concepts |bodystyle = |titlestyle = |abovestyle = background:#cfc; |subheaderstyle = |title = Test Infobox |above = Above text |subheader = Subheader above image |subheader2 = Second subheader |imagestyle = |captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] |caption = Caption displayed below Example-serious.jpg |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 = Header defined alone | label1 = | data1 = |header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = |header3 = | label3 = | data3 = Data defined alone |header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) |header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) |belowstyle = background:#ddf; |below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] im90n6r3bsydtvngqou08dwgz5t6eip 2801334 2801332 2026-03-29T18:25:50Z Ireicher2 2797952 /* Spring 2026 */ added page 2801334 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:25, 29 March 2026 (UTC)[[Digital Media Concepts/StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox |name = Digital Media Concepts |bodystyle = |titlestyle = |abovestyle = background:#cfc; |subheaderstyle = |title = Test Infobox |above = Above text |subheader = Subheader above image |subheader2 = Second subheader |imagestyle = |captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] |caption = Caption displayed below Example-serious.jpg |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 = Header defined alone | label1 = | data1 = |header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = |header3 = | label3 = | data3 = Data defined alone |header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) |header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) |belowstyle = background:#ddf; |below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] sc94n78lpgdymorq7vbldd0e3nitdce 2801337 2801334 2026-03-29T18:27:03Z Ireicher2 2797952 /* Spring 2026 */ added page 2801337 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox |name = Digital Media Concepts |bodystyle = |titlestyle = |abovestyle = background:#cfc; |subheaderstyle = |title = Test Infobox |above = Above text |subheader = Subheader above image |subheader2 = Second subheader |imagestyle = |captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] |caption = Caption displayed below Example-serious.jpg |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 = Header defined alone | label1 = | data1 = |header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = |header3 = | label3 = | data3 = Data defined alone |header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) |header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) |belowstyle = background:#ddf; |below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] h956fh1pz7w37et3stn8jam9d408w9a 2801424 2801337 2026-03-30T05:02:06Z SwarG07 3062365 /* Spring 2026 */ 2801424 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}}* [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 05:01, 30 March 2026 (UTC) [[Digital Media Concepts/The Evolution of the Digital Album: Kanye West and the Stem Player]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] 0tz26om024wefd8ortetxqwdba4abgc 2801429 2801424 2026-03-30T05:17:41Z SwarG07 3062365 /* Spring 2026 */ 2801429 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|}}[[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 05:17, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]]{{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] fu807hd2te0d9hg9qc3l3gvmlmtdihn 2801430 2801429 2026-03-30T05:22:54Z Ireicher2 2797952 rollback changes 2801430 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox |name = Digital Media Concepts |bodystyle = |titlestyle = |abovestyle = background:#cfc; |subheaderstyle = |title = Test Infobox |above = Above text |subheader = Subheader above image |subheader2 = Second subheader |imagestyle = |captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] |caption = Caption displayed below Example-serious.jpg |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 = Header defined alone | label1 = | data1 = |header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = |header3 = | label3 = | data3 = Data defined alone |header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) |header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) |belowstyle = background:#ddf; |below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] h956fh1pz7w37et3stn8jam9d408w9a 2801447 2801430 2026-03-30T06:01:57Z SwarG07 3062365 /* Spring 2026 */ 2801447 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] * [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 06:01, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]]{{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] dl0y5owa8uo7dso0eq2e0hhdp1rn4cx 2801560 2801447 2026-03-30T10:37:49Z Aarna.Makam 3062364 /* Digital Artists Wiki Project Assignment/ Virtual Concerts & Holographic-Style Events */ 2801560 wikitext text/x-wiki {{DISPLAYTITLE:Digital Media Concepts/Virtual concerts and hologram-style live events}} == Virtual Concerts & Hologram-Style Live Events == Virtual concerts and hologram-style live events use projections and digital technology to replace artists or perform with them. They can be entirely virtual, where audiences access them through VR headsets, other devices, or gaming platforms, or they can be in person. In the last case, the artist or production team uses special effects like holograms to create the impression that someone who isn't actually there is performing through 3D projections. There are also performances that use motion capture and digital avatars, where real performers' movements are recorded and turned into lifelike digital versions on stage. This can include a solo hologram act or a real person interacting with a digital version during the performance. Several technologies make these performances immersive and create stunning visuals, including AR, VR, motion capture, and projection effects. == Virtual Concerts VS. Hologram Performances == '''1. Virtual Concerts''' Virtual concerts are highly accessible music events where a performer takes on a digital persona and performs in fully online or digital spaces. These events can use VR or AR [https://nime.org/proceedings/2024/nime2024_53.pdf] and are streamed live, allowing audiences from all over the world to join in real time. Some concerts have even taken place within video games, such as Roblox or ''Sky: Children of the Light [https://www.thatskygame.com/aurora/]''. Audience members share the same virtual space while streaming simultaneously, creating an immersive experience that feels more interactive than watching a prerecorded video. Organizers are attentive to visuals and graphics, enhancing the way the music is experienced. Individual experiences can be more or less enhanced depending on the device used, such as a VR headset, phone, or TV screen. '''2. Hologram Performances'''<br>Hologram-style concerts take place in person and use a lot of stage technology to create vivid performers and surreal special effect that don't physically exist or aren't actually there. They rely on projections and other effects to make everything look as accurate and realistic as possible, so the audience can become immersed in the show and experience it as if a real person is performing or as if something is happening right in front of them. There have been cases where artists who aren't present at the concert still appear to perform, for example ABBA Voyage [https://www.londontheatre.co.uk/theatre-news/news/what-is-it-like-inside-the-updated-abba-voyage-experience], as well as artists who have passed away appearing as virtual performers [https://interestingengineering.com/innovation/how-it-works-13-famous-people-brought-back-to-life-as-holograms], one such example being Michael Jackson [https://www.forbes.com/sites/zackomalleygreenburg/2014/06/24/man-in-the-mirror-how-virtual-michael-jackson-came-to-be/]. IN some cases, entire concerts can even be centered around a fully virtual or fictional character, along with other variations.<br> == Examples == '''''Sky: Children of the Light''''' collaborates with AURORA [https://www.thatskygame.com/aurora/] for live, free virtual concerts with beautiful visuals, starting in 2022 and returning from time to time. AURORA shows up in her Sky avatar and takes her audience through a magical, immersive concert. Different places within the game get a makeover full of decorations before the event to hype up players and encourage them to reserve a seat. Players can chat with one another and use reactions to show their excitement, making the area itself interactive and not just the concert. Players are also offered skins and outfits related to the artist and her avatar. '''Hatsune Miku''' is an example of a virtual pop star that has holographic concerts and collaborates with other artists [https://www.virtualhumans.org/human/hatsune-miku]. "She’s a 16 year old android singer, coming from the near distant future where music is long lost. Her voice was sampled from Japanese voice actress and singer Saki Fujita…It’s a tool for experimentation, lyrics and melodies can be completely changed instantly and there’s no need for rerecording. Her voice (Saki Fujita’s technically) has a very high range, meaning that it can perfectly be used for a wide range of genres; pop, EDM, classical and rock." [https://www.per-spex.com/articles/2024/10/23/the-world-is-yours-hatsune-miku]. Yamaha makes voice synthesizers for her. She performs at lives shows in different countries across the world, sometimes alongside her fellow vocaloids. == Technology Involved == Virtual concerts and hologram-style live events use a variety of digital production tools to make the show as seamless, appealing, and interactive as possible. Hologram-style concerts specifically, use animation3D modeling, projection systems, and illusion techniques like Pepper's Ghost, which has evolved over time. "‘Current performances typically use tightly stretched, transparent films and high-resolution, laser-powered projectors that provide very crisp, true-to-life images while still allowing audiences to see the background stage details.’" [https://education.cosmosmagazine.com/cosmos-magazine-how-do-you-stage-a-holographic-concert-article/]. They also use CGI and motion tracking. "Footage of a hologram’s subject is created by filming a body double in front of a greenscreen, as well as motion-capturing their performance. The video of the actor is then augmented with a digital avatar of the subject, puppeted by their motion-tracking data. The combination of these elements creates a realistic video which can then be projected onto a transparent surface to turn it into a ‘hologram’." [https://education.cosmosmagazine.com/cosmos-magazine-how-do-you-stage-a-holographic-concert-article/] For virtual concerts, the experience can be especially memorable because of how accessible they are across many online services and platforms. They can use avatars, game engines, livestreaming, VR, graphic design, 3D modeling, and more to create a space where viewers can have an unforgettable experience regardless of their physical location. VR and AR headsets can make the experience even more immersive and magical. Companies like Wave [https://wavexr.com/] and AMAZE VR [https://www.amazevr.com/] focus on providing, shared, live, and interactive concert experiences. ==External Links== https://avalliance.com/one-step-further-hologram-technology-for-live-events/ https://thebabelflute.com/holograms-the-future-of-performance/ https://www.npr.org/2024/03/17/1239078698/whats-behind-the-so-called-hologram-celebrity-concerts ==References== # AmazeVR. “AmazeVR.” ''AmazeVR'', www.amazevr.com/. # “Aurora - That Sky Game.” ''Thatskygame.com'', 2024, www.thatskygame.com/aurora/. # Bosanquet, Theo. “What Is It like inside the Updated “ABBA Voyage” Experience?” ''London Theatre'', 10 June 2025, www.londontheatre.co.uk/theatre-news/news/what-is-it-like-inside-the-updated-abba-voyage-experience. # Frame, Ciaran. ''Concerts of the Future: Designing an Interactive Musical Experience in VR''. # “Man in the Translucent Material: How Virtual Michael Jackson Came to Be.” ''Forbes.com'', 2026, www.forbes.com/sites/zackomalleygreenburg/2014/06/24/man-in-the-mirror-how-virtual-michael-jackson-came-to-be/. Accessed 30 Mar. 2026. # Schmidt-Rees, Hannah. “The World Is Yours - Hatsune Miku.” ''PERSPEX'', 31 Oct. 2024, www.per-spex.com/articles/2024/10/23/the-world-is-yours-hatsune-miku. # “Wave.” ''Wave'', wavexr.com/. # “Who Is Hatsune Miku? @Miku_expo, Explained.” ''Www.virtualhumans.org'', www.virtualhumans.org/human/hatsune-miku. # Yeo, Amanda. “COSMOS Magazine: How Do You Stage a Holographic Concert? (Article) - COSMOS Education.” ''COSMOS Education'', 16 Mar. 2022, education.cosmosmagazine.com/cosmos-magazine-how-do-you-stage-a-holographic-concert-article/. # Young, Chris. “How It Works: 13 Famous People Brought back to Life as Holograms.” ''Interestingengineering.com'', 12 Mar. 2020, interestingengineering.com/innovation/how-it-works-13-famous-people-brought-back-to-life-as-holograms. [[Category:Art]] [[Category:Digital art]] __FORCETOC__ [[Category:Technology]] [[Category:3D modeling]] [[Category:Virtual reality]] [[Category:Holograms]] fy32z78yyh5ixvtw5ckt8saezza2f8n 2801570 2801560 2026-03-30T10:52:15Z Aarna.Makam 3062364 Undid revision [[Special:Diff/2801560|2801560]] by [[Special:Contributions/Aarna.Makam|Aarna.Makam]] ([[User talk:Aarna.Makam|talk]]) 2801570 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] * [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 06:01, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]]{{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] dl0y5owa8uo7dso0eq2e0hhdp1rn4cx 2801576 2801570 2026-03-30T10:57:54Z Aarna.Makam 3062364 /* Spring 2026 */ 2801576 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Aarna.Makam|Aarna.Makam]] ([[User talk:Aarna.Makam|discuss]] • [[Special:Contributions/Aarna.Makam|contribs]]) 10:57, 30 March 2026 (UTC)[[Digital Media Concepts/Virtual Concerts & Hologram-Style Live Events ]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] * [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 06:01, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]]{{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] h2rx70yyk5oe0w15emuadsajuupq9xc 2801578 2801576 2026-03-30T11:02:04Z Aarna.Makam 3062364 /* Spring 2026 */ 2801578 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Aarna.Makam|Aarna.Makam]] ([[User talk:Aarna.Makam|discuss]] • [[Special:Contributions/Aarna.Makam|contribs]]) 11:01, 30 March 2026 (UTC)[[Digital Media Concepts/Virtual Concerts & Hologram-Style Live Events]] * [[User:Aarna.Makam|Aarna.Makam]] ([[User talk:Aarna.Makam|discuss]] • [[Special:Contributions/Aarna.Makam|contribs]]) 10:57, 30 March 2026 (UTC)[[Digital Media Concepts/Virtual Concerts & Hologram-Style Live Events ]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] * [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 06:01, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]]{{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] mzm8p5njkos2snkibz59lye3cpgzfll 0 214209 2801501 1791902 2026-03-30T08:16:50Z Anikmolla786 3007678 Fixed typo: “suffers” → “sufferers”. 2801501 wikitext text/x-wiki {{Title|Sexual assault non-reporting motivation:<br> What motivates non-reporting of sexual assault by victims?}} {{MECR3|1=https://www.youtube.com/watch?v=SbPEjN0lqEE}} __TOC__ == Overview == {{Warning|'''Warning:''' The contents of this page may be a trigger for victims of sexual assault. If the contents of this page are distressing please seek support.}} Sexual assault remains one of the most under-reported serious crime types despite numerous studies that indicate a high prevalence of sexual assault in today's society. It is often referred to as a hidden crime because of the high rate of nondisclosure. The motivations behind non-reporting of sexual assault are complex, and include the influence of societal constructs and the victim's mental and physical well-being. Victims will often report negative experiences of initial disclosure which dissuades them from further disclosure and seeking prosecution of their perpetrator through the justice system. This chapter will outline contributing motivations behind the non-reporting of sexual assault by victims. It will explain relevant theories such as learned helplessness and social stigma theory. The symptoms of post-traumatic stress disorder (PTSD) will be outlined. PTSD is a psychological disorder that affects a large number of sexual assault victims. The disclosure process model will be explained from the perspective of a sexual assault victim and the chapter will conclude with the societal factors that motivate non-reporting of sexual assault. Victims are likely to experience a complex combination of all of the motivations as set out in this chapter that will contribute to their unwillingness to disclose their assault. == Definitions == Sexual assault is as ''"an act of a sexual nature carried out against a person's will through the use of physical force, intimidation or coercion, and includes any attempt to do this. This includes rape, attempted rape, aggravated sexual assault (assault with a weapon), indecent assault, penetration by objects, forced sexual activity that did not end in penetration and attempts to force a person into sexual activity"'' (Australian Bureau of Statistics, 2012). == Sexual assault statistics == * One in five women will experience sexual assault in their lifetime. * One in thirteen men will experience sexual assault in their lifetime. * Females aged between 15 and 19 years were seven times more likely to have been a victim of sexual assault compared to the overall Australian population. * 70 per cent of sexual assaults are committed by a perpetrator known to the victim and are often a family member, friend or acquaintance. * One in ten adult women who are sexually assaulted are victims to their current or past intimate partner. * Most sexual assaults occurred at a residential location and did not involve the use of a weapon. * Victims will wait an average of 11 months before disclosing their assault (Australian Bureau of Statistics, 2012). == Disclosure process model == The Disclosure Process Model (DPM) explains the motivations and timing behind interpersonal verbal disclosure and how it can be beneficial to those who have a concealable identity including victims of sexual assault (Chaudoir & Fisher, 2010). The DPM aims to encapsulate the disclosure process as a single process including the initial decision as to whether disclosure will occur. It also includes the processes involving the outcome of the disclosure and whether it will occur again. The DPM theorises that disclosure can have serious effects on the individual, interpersonal relationships and wider societal relationships, through three mediating processes: * Alleviation of inhibition * Social support * Changes in social information (Chaudoir, Fisher, & Simoni, 2011). The DPM represents one disclosure process in what could be an ongoing number of decisions to disclose by a sexual assault victim over their lifetime (Chaudoir et al., 2011). It is theorised that a victim of sexual assault will go through the DPM whether the disclosure is informal to a friend or family member, or part of the formal process involved in reporting their assault as part of the legal process to prosecution. === Antecedent goals === The first step of the disclosure as purported by the DPM involves the setting by the individual of antecedent goals that are represented by avoidance and approach goals (Chaudoir & Fisher, 2010). Approach goals are made with the outlook to pursue what the victim perceives as positive outcomes, in the case of a victim of sexual assault this may include the following: * Increase understanding by another about the victim's personal situation * Strengthening interpersonal relationships with others through disclosure * Education of others on the nature and prevalence of sexual assault (Chaudoir et al., 2011) Avoidance goals are designed to avoid negative outcomes that the victim might recognise as having the potential to be punishing or have long term effects on their well-being and functioning (Chaudoir et al., 2011). In the case of a sexual assault victim, this may include the following: * Social rejection and/or social stigma * The breakdown of relationships both intimate and social * Reveals the victim as vulnerable to exploitation or revictimisation (Chaudoir et al., 2011) Victims of sexual assault who decide to disclose with approach goals as their motivator are more likely to benefit from the resulting disclosure as opposed to those who decide to disclose despite having avoidance goals in the initial stage (Chaudoir et al., 2011). Avoidance goals can stop the victim from progressing past the first stage of the DPM and the avoidance of negative outcomes may be strong enough to motivate the victim to never disclose their incident of sexual assault. === Disclosure event === The disclosure event involves the verbal communication by the victim to the receiver about their identity of a sexual assault victim. According to the DPM theory, the extent and nature of the disclosure event is characterised by the breadth, depth, duration and emotional content of the disclosure (Chaudoir & Fisher, 2010). The victim will receive and gauge the response of the recipient of the disclosure as being a positive, negative or neutral response. The manner in which a victim communicates during the disclosure can be influenced by the antecedent goal, with victims who had an initial approach goal being more likely to communicate more effectively and receive positive responses from the recipient of the disclosure (Chaudoir et al., 2011). === Mediating processes and disclosure outcomes === The long-term effects of disclosure can be both negative and positive. The negative effects can include psychological distress for the victim, increasingly risky sexual behaviour or increased drug or alcohol use. The positive effects can include increased support from social or official support networks, social and personal awareness of the nature of sexual assault. In the longer term, the disclosure may serve to change stereotypes of sexual assault victims in the victim’s social circle (Chaudoir et al., 2011). === Feedback loop === A single disclosure event can shape future instances of a victim disclosing their sexual assault experience. The victim may take the results of the previous disclosures, particularly if they were negative and decide to conceal the incident and never disclose again. Those who had an overly positive experience are likely to continue to cycle through the DPM for future potential disclosure opportunities (Chaudoir et al., 2011). == Social stigma theory == [[File:A poster produced by the U.S. Navy supports the Sexual Assault Prevention and Response (SAPR) program at Joint Base Pearl Harbor-Hickam, Hawaii, Oct. 30, 2013 131030-N-XD424-001.jpg|thumb|left|''Figure 1.'' A poster produced by the U.S. Navy supports the Sexual Assault Prevention and Response (SAPR) program]] The social stigma theory comes from the work of Erving Goffman who theorised that "stigma is an attribute that extensively discredits an individual, reducing him or her from a whole and usual person to a tainted discounted one" (Major & O'Brien, 2005). In response to his early work, further research into social stigma theory expanded the explanation to include the notion that "stigmatisation occurs when a person possesses (or is believed to possess) some attribute or characteristic that conveys a social identity that is devalued in a particular social context" (Major & O'Brien, 2005). In the context of the non-disclosure of sexual assault by victims, social stigma theory has an influence on the motivations of a victim to avoid disclosure of the incident so as to avoid social stigma. The concept of social stigma is not a phenomenon that exists only within an individual’s concept of self, but is a socially created one with the idea that in order to live successfully in a society, those who possess undesirable attributes are to be avoided (Herek, 2007). Victims of sexual assault are likely to avoid disclosure to escape the social identity stigma that goes with being a victim of sexual assault; these can include being perceived as sexually promiscuous, being a less than ideal partner for social exchange and being vulnerable to exploitation (Miller, Canales, Amacker, Backstrom, & Gidycz, 2011). == Learned helplessness theory == [[File:-The Blackmail.png|thumb|''Figure 2.'' Victims can feel they are trapped]] The learned helplessness theory is used to explain the phenomenon of individuals being exposed to uncontrollable and inescapable situations such as a sexual assault and as a result developing an inability to act or extract themselves from similar future environments or occurrences (Maier & Seligman, 1976). The learned helplessness theory originated from the findings of Maier and Seligman who exposed dogs to inescapable electric shocks and then observed that during the next exposure to the shocks the dogs didn't attempt to escape and were observed to be frozen in their inability to act and extract themselves from the situation (Wortman & Brehm, 1975). The observations and similar research conducted into the behaviour of individuals repeatedly exposed to traumatic situations has demonstrated that learned helplessness has de-motivational consequences on human behaviour (Maier & Seligman, 1976). The theory has serious implications in the context of nondisclosure of sexual assault by victims. Victims of sexual assault, particularly those who experience repeated instances of abuse are likely to develop learned helplessness. The victim may not disclose the assault as a result of experiencing feelings that they have no alternatives or choices to better their situation. Victims reported that during the period of sexual abuse they felt emotionally unsupported, without financial or socially supportive options and a general feeling of being unable to better their situation (Sable, Danis, Mauzy, & Gallagher, 2006). In many instances of repeated sexual assault by a family member, victims did not disclose the instances of assault until a lengthy period of time had passed and the perpetrator had died. The victims expressed that they had felt helpless at the time of the ongoing assault and that it wasn't until significant time had passed that they felt able to disclose the details of the sexual assault to someone (Sable et al., 2006). == Post-traumatic stress disorder == An experience of a traumatic event such as a sexual assault can lead to development of post-traumatic stress disorder (PTSD). The symptoms can become so serious that the victim is unable to disclose the assault. This is due to a number of factors, including fear of flashbacks, emotional detachment from friends and family leading to reduction in social support and a general inability to complete everyday tasks as a result of the PTSD (Klump, 2006). The Diagnostic and Statistical Manual of Mental Disorders V criteria for PTSD (American Psychiatric Association, 2013) are outlined as follows: ;Criteria A: stressor Individual is exposed to: death, threatened death, actual or threatened serious injury, or actual or threatened sexual assault: (one symptom required for diagnosis) * Direct exposure * Witnessing an event * Indirect exposure through learning that a close family member or friend has experienced trauma * Frequent or indirect exposure to traumatic elements of the event, often as part of professional duties. Examples include, emergency first responders, individuals working with details of child abuse (American Psychiatric Association, 2013). ;Criteria B: intrusion symptoms The traumatic event is re-experienced in the following ways: (one symptom required for diagnosis) * Re-occurring, involuntary, and intrusive memories * Disturbing nightmares. Note: Children may have frightening dreams that do not contain elements of the trauma * Flashbacks, which may range from brief episodes to complete loss of consciousness * Intense short or longer term distress after exposure to traumatic reminders * Marked physiological reactivity after exposure to trauma-related stimuli (American Psychiatric Association, 2013). ;Criteria C: avoidance Persistent avoidance of distressing trauma-related stimuli after the event: (one symptom required for diagnosis) * Avoiding thoughts or feelings connected to the traumatic event * Avoiding people or situations connected to the traumatic event (American Psychiatric Association, 2013). [[File:PTSD.png|thumb|''Figure 3.'' PTSD represented visually]] ;Criteria D: negative alterations in mood or cognitions === A recently added criteria that encapsulates symptoms reported by sufferers of PTSD. * Memory problems associated with the event * Negative thoughts about themselves or the world * Distorted sense of blame relating to the event * Ruminate on severe emotions such as shame or horror related to the event * Isolated or detached from others * Loss of interest in pre-event activities (American Psychiatric Association, 2013). ;Criteria E: increased arousal The brain remains alert and hypervigilant: * Difficulty concentrating * Increase in irritability and temper * Difficulty sleeping * Hypervigilance * Easily startled (American Psychiatric Association, 2013). Victims of sexual assault may experience all or a large number of the common symptoms of sexual assault. Disclosure of the sexual assault can be hindered not only by PTSD but also by additional mental health issues. These may have developed as a result of the traumatic event and/or after experiencing the long term effects of PTSD. Common additional mental health issues can include depression and anxiety, and can be emphasised by heavy use of drugs and alcohol (Klump, 2006). == Societal factors == {{expand}} === Informal support networks === Two thirds of sexual assault victims disclose the incident informally to at least one person they trust (Ahrens, 2006; Campbell, Ahrens, Sefl, Wasco, & Barnes, 2001). However, negative reactions and subsequent actions of the recipient of the information can often influence further disclosure of the incident (Ahrens, 2006). The most common reactions that prevent disclosure to additional sources of assistance or in order to begin legal proceedings are blaming victims for the assault occurring or doubting the victim’s disclosure (Ullman, 1996). In addition to this, victims reported that they were encouraged by informal support networks to maintain secrecy and were discouraged from seeking legal assistance (Ahrens, 2006). {{RoundBoxTop}}Case Study: "Karen was abandoned by her boyfriend at a coffee shop during a cross-country road trip. The manager offered her a room at the adjacent motel, but when the night watch-man brought her food, he raped her. When Karen tried to tell her sister about the assault, her sister didn’t seem to identify the experience as rape. This unsupportive interchange caused Karen to question the efficacy of disclosure. Her sister’s response was so unsympathetic that she never spoke to her about the assault again. Believing that there was no-one in her life who would support her, Karen did not disclose again for 19 years." (Filipas & Ullman, 2001).{{RoundBoxBottom}} === Relationship to the perpetrator === Victims were less likely to disclose an incident of sexual assault if the perpetrator was someone that was known to them or someone with whom they were in a close familial or intimate relationship. The following statistics reveal the true extent of the effect of the perpetrator’s relationship to the victim on nondisclosure of the sexual assault (Willis, 2011): * 25 per cent of victims disclosed sexual assault if the perpetrator was a stranger * 17 per cent of victims disclose sexual assault if the perpetrator was a former partner * 1 per cent of victims disclose sexual assault if the perpetrator was a current partner (Willis, 2011). The motivation for non-disclosure of sexual assault committed by a known offender can be severely hindered by the victim’s fear of repercussions from the perpetrator and the family and friends of both parties. The perpetrator may have the ability to exert power over the victim over a long period of time, by threatening financial hardship and/or breakdown of the familial unit. These strong motivators for nondisclosure are likely to encourage the victim to deal with the incidence of sexual assault privately and to avoid pursuing prosecution through the legal system (Willis, 2011). === Revictimisation === A number of studies into victims of sexual assault indicate that following disclosure by victims of assault to a confidant or as part of a public awareness campaign, in some instances the victim was targeted by an additional perpetrator. This perpetrator reportedly used the methods that the victim had described as part of their disclosure (Sable, Danis, Mauzy, & Gallagher, 2006). Victims often reported feeling extremely vulnerable to further attacks after disclosure and that they would be less motivated to report subsequent incidences due to the threat and occurrence of revictimisation (Mason, Ullman, Long, Long, & Starzynski, 2009). Those who had disclosed sexual assault, reported that often they were coping with additional issues such as increased alcohol and drug use and had become increasingly promiscuous as part of coping with the event (Mason et al., 2009). Often because of the additional issues that they were experiencing, victims were vulnerable to revictimisation and experienced further incidences of sexual assault. Victims felt that because of their history of being a sexual assault victim, the drug and alcohol use, and the promiscuity, they were unable to disclose any of their sexual assault incidents (Mason et al., 2009). Revictimisation may additionally come through the disclosure process itself. The victim may have to endure invasive medical procedures, give lengthy and detailed statements of the sexual assault and may come into contact with the perpetrator as part of the prosecution process. The victim may feel that they have just recovered from the ordeal only to be forced to go through it again through revictimisation during the legal process (Willis, 2011). === Knowledge of services === Victims of sexual assault are often unaware of the services available to them after their experience. The threat of a negative response as a result of previous experiences of disclosure and social stigma often prevents the victim from conducting research into what options are available to them. Without an informative support network such as a rape crisis centre or a victim support network, victims are likely to feel that disclosing the assault will only lead to undesirable outcomes. The victim may feel that the prosecution process is too daunting to face on their own (Sable, Danis, Mauzy, & Gallagher, 2006). === Perceptions of police === Perceptions of police, and the justice system as a whole, may severely hinder disclosure of a sexual assault. The victim may not feel assured that they will be believed and that the outcome of any legal action may not reach a desired outcome. Police are often under resourced and face the challenge of gathering enough evidence for the matter to appear before court. The victim may feel that the police do not have enough empathy towards them and are simply stepping through the required process. Sexual assault is often a difficult offence to prosecute. Recent studies have found that in the Australian Capital Territory one third of cases that proceeded to prosecution resulted in a conviction and one in ten apprehended offenders was acquitted. Sexual assault victims may not feel that a disclosure as part of the prosecution process will be beneficial to them (Willis, 2011). Disclosure can additionally be affected by cultural aversions towards police. Disclosure is unlikely to occur if the victim has had negative interactions with the police in their community, or if the victim comes from a cultural background that has an inherent distrust of law enforcement bodies. Victims may be encouraged to deal with the issue within the community and not disclose the incident further (Willis, 2011). == The role of media and popular culture == Media and popular culture have often portrayed sexual assaults as violent crimes involving weapons, and as being committed in public spaces by strangers. Statistics reported by sources such as the Australian Bureau of Statistics show that this is indeed largely a false representation (Australian Bureau of Statistics, 2012). The majority of sexual assault victims will be assaulted by someone that is known to them and the methods used by the perpetrator often do not involve a weapon. The false representation of how sexual assault occurs perpetuates the idea to victims that what they experienced was not a 'true' representation of rape or an attack as portrayed by popular culture. This means that the victim can often decide to not report their assault as they feel that they will not be believed, or they experience self-doubt that their attack is not worth reporting (Dowler, Fleming, & Muzzatti, 2006). In addition to the portrayal of how a sexual assault occurs, media and popular culture can shape societal views on how acceptable such an act is. Through media and popular culture, sexual assault can often be sensationalised to appear alluring and in the extreme desirable or exciting. University of California Los Angeles media researcher Dr Neil Malamuth stated that ''"media portrayals can promote the view that women desire violence; they can transform sensitive individuals' view of rape to make it seem more acceptable and not such an abhorrent act."'' With this in mind, a victim is less likely to report their attack as it may be viewed as just 'something that happens' and a minor incident in everyday life that should be accepted as such (Media Values, 2015). == Conclusion == The motivations behind a victim not reporting their sexual assault can be complex. The circumstances of the assault and the victim's own perceptions and emotional responses mean they can be overwhelmed and indecisive. They may not be aware of support services available to them.  Individuals need access to a positive and supportive audience. They need to be assisted to understand and accept they are not to blame for an incident. This is as an important first step in encouraging disclosure. A victim should be made aware of the support services available and be assisted to understand their options and the processes that may take place if they decide to disclose (Sable, Danis, Mauzy, & Gallagher, 2006). Victims may need encouragement to seek help for peripheral or consequential issues such as serious drug or alcohol use or symptoms of post-traumatic stress disorder. This will help them to recover from their ordeal fully or as part of pursuing a prosecution. Some victims may decide to disclose their sexual assault, but express a desire to keep their details confidential for example, to avoid social stigma. The victim’s wishes should be respected and they should be reassured that every effort will be taken to ensure that this is adhered while they decide how they proceed. == See also == [[Motivation and emotion/Book/2013/Emotional impacts of sexual assault]] [[Motivation and emotion/Book/2014/Sexual harassment and emotion]] == References == {{Hanging indent|1= Ahrens, C. E. (2006). Being Silenced: The Impact of Negative Social Reactions on the Disclosure of Rape. ''American Journal of Community Psychology, 38''(3), 263-274. doi:10.1007/s10464-006-9069-9 American Psychiatric Association. (2013). ''Diagnostic and statistical manual of mental disorders'' (5th ed.). Arlington, VA: American Psychiatric Publishing. Australian Bureau of Statistics. (2012). 4906.0 - Personal Safety, Australia, 2012. Retrieved from http://www.abs.gov.au/AUSSTATS/abs@.nsf/Lookup/4906.0Main+Features12012?OpenDocument. Campbell, R., Ahrens, C. E., Sefl, T., Wasco, S. M., & Barnes, H. E. (2001). Social reactions to rape victims: Healing and hurtful effects on psychological and physical health outcomes. ''Violence and victims, 16''(3), 287-302. Chaudoir, S. R., & Fisher, J. D. (2010). The disclosure processes model: understanding disclosure decision making and post disclosure outcomes among people living with a concealable stigmatized identity. ''Psychological bulletin, 136''(2), 236. Chaudoir, S. R., Fisher, J. D., & Simoni, J. M. (2011). Understanding HIV disclosure: A review and application of the Disclosure Processes Model. ''Social Science & Medicine, 72''(10), 1618-1629. Filipas, H. H., & Ullman, S. E. (2001). Social reactions to sexual assault victims from various support sources. ''Violence and victims, 16''(6), 673. Herek, G. M. (2007). Confronting sexual stigma and prejudice: Theory and practice. ''Journal of Social Issues, 63''(4), 905-925. Klump, M. C. (2006). Posttraumatic stress disorder and sexual assault in women. ''Journal of College Student Psychotherapy, 21''(2), 67-83. Maier, S. F., & Seligman, M. E. (1976). Learned helplessness: Theory and evidence. ''Journal of experimental psychology: general, 105''(1), 3. Major, B., & O'Brien, L. T. (2005). The social psychology of stigma. ''Annu. Rev. Psychol., 56'', 393-421. Mason, G. E., Ullman, S., Long, S. E., Long, L., & Starzynski, L. (2009). Social support and risk of sexual assault revictimization. ''Journal of Community Psychology, 37''(1), 58-72. Miller, A. K., Canales, E. J., Amacker, A. M., Backstrom, T. L., & Gidycz, C. A. (2011). Stigma-Threat Motivated Nondisclosure of Sexual Assault and Sexual Revictimization A Prospective Analysis. ''Psychology of Women Quarterly, 35''(1), 119-128. Sable, M. R., Danis, F., Mauzy, D. L., & Gallagher, S. K. (2006). Barriers to reporting sexual assault for women and men: Perspectives of college students. ''Journal of American College Health, 55''(3), 157-162. Ullman, S. E. (1996). Social Reactions, Coping Strategies, and Self‐blame Attributions in Adjustment to Sexual Assault. ''Psychology of Women Quarterly, 20''(4), 505-526. Values, M. (2015). ''Media's New Mood: Sexual Violence''. United States of America: Center for Media Literacy. Willis, M. (2011). ''Non-disclosure of violence in Australian Indigenous communities''. Retrieved from <nowiki>http://www.aic.gov.au/media_library/publications/tandi_pdf/tandi405.pdf</nowiki>: Wortman, C. B., & Brehm, J. W. (1975). Responses to uncontrollable outcomes: An integration of reactance theory and the learned helplessness model. ''Advances in experimental social psychology, 8'', 277-336. }} == External Links == [https://www.ted.com/talks/jessica_ladd_the_reporting_system_that_sexual_assault_survivors_want?language=en Ted talks. Jessica Lad: The reporting system that sexual assault survivors want] [https://www.ted.com/talks/ione_wells_how_we_talk_about_sexual_assault_online Ted talks. Ione Wells: How we talk about sexual assault online] [https://www.ted.com/talks/melissa_walker_art_can_heal_ptsd_s_invisible_wounds Ted talks: Melissa Walker: Art can heal PTSD's invisible wounds] == External links == ;Where to find help * LifeLine Australia https://www.lifeline.org.au/<nowiki/>13 11 14 * Canberra Rape Crisis Centre http://www.crcc.org.au (02) 6147 2525 * The Nguru Program, counselling for members of the Aboriginal and Torres Strait Islander community who have been victims of sexual assault and their families. http://www.crcc.org.au/support-services/the-nguru-program.aspx (02) 6147 2525 * Service Assisting Male Survivors of Sexual Assault [http://www.samssa.org.au/ www.samssa.org.au] (02) 6287 3935 * NSW Rape Crisis Centre http://www.nswrapecrisis.com.au/ 1800 424 017 [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Forensic/Sex crime‎]] [[Category:Motivation and emotion/Book/Sexual motivation]] fq5xu21cx88f4dullh2nmf6wa6l784e 2801502 2801501 2026-03-30T08:18:00Z Anikmolla786 3007678 Corrected “criteria” (plural) to “criterion” (singular). 2801502 wikitext text/x-wiki {{Title|Sexual assault non-reporting motivation:<br> What motivates non-reporting of sexual assault by victims?}} {{MECR3|1=https://www.youtube.com/watch?v=SbPEjN0lqEE}} __TOC__ == Overview == {{Warning|'''Warning:''' The contents of this page may be a trigger for victims of sexual assault. If the contents of this page are distressing please seek support.}} Sexual assault remains one of the most under-reported serious crime types despite numerous studies that indicate a high prevalence of sexual assault in today's society. It is often referred to as a hidden crime because of the high rate of nondisclosure. The motivations behind non-reporting of sexual assault are complex, and include the influence of societal constructs and the victim's mental and physical well-being. Victims will often report negative experiences of initial disclosure which dissuades them from further disclosure and seeking prosecution of their perpetrator through the justice system. This chapter will outline contributing motivations behind the non-reporting of sexual assault by victims. It will explain relevant theories such as learned helplessness and social stigma theory. The symptoms of post-traumatic stress disorder (PTSD) will be outlined. PTSD is a psychological disorder that affects a large number of sexual assault victims. The disclosure process model will be explained from the perspective of a sexual assault victim and the chapter will conclude with the societal factors that motivate non-reporting of sexual assault. Victims are likely to experience a complex combination of all of the motivations as set out in this chapter that will contribute to their unwillingness to disclose their assault. == Definitions == Sexual assault is as ''"an act of a sexual nature carried out against a person's will through the use of physical force, intimidation or coercion, and includes any attempt to do this. This includes rape, attempted rape, aggravated sexual assault (assault with a weapon), indecent assault, penetration by objects, forced sexual activity that did not end in penetration and attempts to force a person into sexual activity"'' (Australian Bureau of Statistics, 2012). == Sexual assault statistics == * One in five women will experience sexual assault in their lifetime. * One in thirteen men will experience sexual assault in their lifetime. * Females aged between 15 and 19 years were seven times more likely to have been a victim of sexual assault compared to the overall Australian population. * 70 per cent of sexual assaults are committed by a perpetrator known to the victim and are often a family member, friend or acquaintance. * One in ten adult women who are sexually assaulted are victims to their current or past intimate partner. * Most sexual assaults occurred at a residential location and did not involve the use of a weapon. * Victims will wait an average of 11 months before disclosing their assault (Australian Bureau of Statistics, 2012). == Disclosure process model == The Disclosure Process Model (DPM) explains the motivations and timing behind interpersonal verbal disclosure and how it can be beneficial to those who have a concealable identity including victims of sexual assault (Chaudoir & Fisher, 2010). The DPM aims to encapsulate the disclosure process as a single process including the initial decision as to whether disclosure will occur. It also includes the processes involving the outcome of the disclosure and whether it will occur again. The DPM theorises that disclosure can have serious effects on the individual, interpersonal relationships and wider societal relationships, through three mediating processes: * Alleviation of inhibition * Social support * Changes in social information (Chaudoir, Fisher, & Simoni, 2011). The DPM represents one disclosure process in what could be an ongoing number of decisions to disclose by a sexual assault victim over their lifetime (Chaudoir et al., 2011). It is theorised that a victim of sexual assault will go through the DPM whether the disclosure is informal to a friend or family member, or part of the formal process involved in reporting their assault as part of the legal process to prosecution. === Antecedent goals === The first step of the disclosure as purported by the DPM involves the setting by the individual of antecedent goals that are represented by avoidance and approach goals (Chaudoir & Fisher, 2010). Approach goals are made with the outlook to pursue what the victim perceives as positive outcomes, in the case of a victim of sexual assault this may include the following: * Increase understanding by another about the victim's personal situation * Strengthening interpersonal relationships with others through disclosure * Education of others on the nature and prevalence of sexual assault (Chaudoir et al., 2011) Avoidance goals are designed to avoid negative outcomes that the victim might recognise as having the potential to be punishing or have long term effects on their well-being and functioning (Chaudoir et al., 2011). In the case of a sexual assault victim, this may include the following: * Social rejection and/or social stigma * The breakdown of relationships both intimate and social * Reveals the victim as vulnerable to exploitation or revictimisation (Chaudoir et al., 2011) Victims of sexual assault who decide to disclose with approach goals as their motivator are more likely to benefit from the resulting disclosure as opposed to those who decide to disclose despite having avoidance goals in the initial stage (Chaudoir et al., 2011). Avoidance goals can stop the victim from progressing past the first stage of the DPM and the avoidance of negative outcomes may be strong enough to motivate the victim to never disclose their incident of sexual assault. === Disclosure event === The disclosure event involves the verbal communication by the victim to the receiver about their identity of a sexual assault victim. According to the DPM theory, the extent and nature of the disclosure event is characterised by the breadth, depth, duration and emotional content of the disclosure (Chaudoir & Fisher, 2010). The victim will receive and gauge the response of the recipient of the disclosure as being a positive, negative or neutral response. The manner in which a victim communicates during the disclosure can be influenced by the antecedent goal, with victims who had an initial approach goal being more likely to communicate more effectively and receive positive responses from the recipient of the disclosure (Chaudoir et al., 2011). === Mediating processes and disclosure outcomes === The long-term effects of disclosure can be both negative and positive. The negative effects can include psychological distress for the victim, increasingly risky sexual behaviour or increased drug or alcohol use. The positive effects can include increased support from social or official support networks, social and personal awareness of the nature of sexual assault. In the longer term, the disclosure may serve to change stereotypes of sexual assault victims in the victim’s social circle (Chaudoir et al., 2011). === Feedback loop === A single disclosure event can shape future instances of a victim disclosing their sexual assault experience. The victim may take the results of the previous disclosures, particularly if they were negative and decide to conceal the incident and never disclose again. Those who had an overly positive experience are likely to continue to cycle through the DPM for future potential disclosure opportunities (Chaudoir et al., 2011). == Social stigma theory == [[File:A poster produced by the U.S. Navy supports the Sexual Assault Prevention and Response (SAPR) program at Joint Base Pearl Harbor-Hickam, Hawaii, Oct. 30, 2013 131030-N-XD424-001.jpg|thumb|left|''Figure 1.'' A poster produced by the U.S. Navy supports the Sexual Assault Prevention and Response (SAPR) program]] The social stigma theory comes from the work of Erving Goffman who theorised that "stigma is an attribute that extensively discredits an individual, reducing him or her from a whole and usual person to a tainted discounted one" (Major & O'Brien, 2005). In response to his early work, further research into social stigma theory expanded the explanation to include the notion that "stigmatisation occurs when a person possesses (or is believed to possess) some attribute or characteristic that conveys a social identity that is devalued in a particular social context" (Major & O'Brien, 2005). In the context of the non-disclosure of sexual assault by victims, social stigma theory has an influence on the motivations of a victim to avoid disclosure of the incident so as to avoid social stigma. The concept of social stigma is not a phenomenon that exists only within an individual’s concept of self, but is a socially created one with the idea that in order to live successfully in a society, those who possess undesirable attributes are to be avoided (Herek, 2007). Victims of sexual assault are likely to avoid disclosure to escape the social identity stigma that goes with being a victim of sexual assault; these can include being perceived as sexually promiscuous, being a less than ideal partner for social exchange and being vulnerable to exploitation (Miller, Canales, Amacker, Backstrom, & Gidycz, 2011). == Learned helplessness theory == [[File:-The Blackmail.png|thumb|''Figure 2.'' Victims can feel they are trapped]] The learned helplessness theory is used to explain the phenomenon of individuals being exposed to uncontrollable and inescapable situations such as a sexual assault and as a result developing an inability to act or extract themselves from similar future environments or occurrences (Maier & Seligman, 1976). The learned helplessness theory originated from the findings of Maier and Seligman who exposed dogs to inescapable electric shocks and then observed that during the next exposure to the shocks the dogs didn't attempt to escape and were observed to be frozen in their inability to act and extract themselves from the situation (Wortman & Brehm, 1975). The observations and similar research conducted into the behaviour of individuals repeatedly exposed to traumatic situations has demonstrated that learned helplessness has de-motivational consequences on human behaviour (Maier & Seligman, 1976). The theory has serious implications in the context of nondisclosure of sexual assault by victims. Victims of sexual assault, particularly those who experience repeated instances of abuse are likely to develop learned helplessness. The victim may not disclose the assault as a result of experiencing feelings that they have no alternatives or choices to better their situation. Victims reported that during the period of sexual abuse they felt emotionally unsupported, without financial or socially supportive options and a general feeling of being unable to better their situation (Sable, Danis, Mauzy, & Gallagher, 2006). In many instances of repeated sexual assault by a family member, victims did not disclose the instances of assault until a lengthy period of time had passed and the perpetrator had died. The victims expressed that they had felt helpless at the time of the ongoing assault and that it wasn't until significant time had passed that they felt able to disclose the details of the sexual assault to someone (Sable et al., 2006). == Post-traumatic stress disorder == An experience of a traumatic event such as a sexual assault can lead to development of post-traumatic stress disorder (PTSD). The symptoms can become so serious that the victim is unable to disclose the assault. This is due to a number of factors, including fear of flashbacks, emotional detachment from friends and family leading to reduction in social support and a general inability to complete everyday tasks as a result of the PTSD (Klump, 2006). The Diagnostic and Statistical Manual of Mental Disorders V criteria for PTSD (American Psychiatric Association, 2013) are outlined as follows: ;Criteria A: stressor Individual is exposed to: death, threatened death, actual or threatened serious injury, or actual or threatened sexual assault: (one symptom required for diagnosis) * Direct exposure * Witnessing an event * Indirect exposure through learning that a close family member or friend has experienced trauma * Frequent or indirect exposure to traumatic elements of the event, often as part of professional duties. Examples include, emergency first responders, individuals working with details of child abuse (American Psychiatric Association, 2013). ;Criteria B: intrusion symptoms The traumatic event is re-experienced in the following ways: (one symptom required for diagnosis) * Re-occurring, involuntary, and intrusive memories * Disturbing nightmares. Note: Children may have frightening dreams that do not contain elements of the trauma * Flashbacks, which may range from brief episodes to complete loss of consciousness * Intense short or longer term distress after exposure to traumatic reminders * Marked physiological reactivity after exposure to trauma-related stimuli (American Psychiatric Association, 2013). ;Criteria C: avoidance Persistent avoidance of distressing trauma-related stimuli after the event: (one symptom required for diagnosis) * Avoiding thoughts or feelings connected to the traumatic event * Avoiding people or situations connected to the traumatic event (American Psychiatric Association, 2013). [[File:PTSD.png|thumb|''Figure 3.'' PTSD represented visually]] ;Criteria D: negative alterations in mood or cognitions === A recently added criterion that encapsulates symptoms reported by sufferers of PTSD. * Memory problems associated with the event * Negative thoughts about themselves or the world * Distorted sense of blame relating to the event * Ruminate on severe emotions such as shame or horror related to the event * Isolated or detached from others * Loss of interest in pre-event activities (American Psychiatric Association, 2013). ;Criteria E: increased arousal The brain remains alert and hypervigilant: * Difficulty concentrating * Increase in irritability and temper * Difficulty sleeping * Hypervigilance * Easily startled (American Psychiatric Association, 2013). Victims of sexual assault may experience all or a large number of the common symptoms of sexual assault. Disclosure of the sexual assault can be hindered not only by PTSD but also by additional mental health issues. These may have developed as a result of the traumatic event and/or after experiencing the long term effects of PTSD. Common additional mental health issues can include depression and anxiety, and can be emphasised by heavy use of drugs and alcohol (Klump, 2006). == Societal factors == {{expand}} === Informal support networks === Two thirds of sexual assault victims disclose the incident informally to at least one person they trust (Ahrens, 2006; Campbell, Ahrens, Sefl, Wasco, & Barnes, 2001). However, negative reactions and subsequent actions of the recipient of the information can often influence further disclosure of the incident (Ahrens, 2006). The most common reactions that prevent disclosure to additional sources of assistance or in order to begin legal proceedings are blaming victims for the assault occurring or doubting the victim’s disclosure (Ullman, 1996). In addition to this, victims reported that they were encouraged by informal support networks to maintain secrecy and were discouraged from seeking legal assistance (Ahrens, 2006). {{RoundBoxTop}}Case Study: "Karen was abandoned by her boyfriend at a coffee shop during a cross-country road trip. The manager offered her a room at the adjacent motel, but when the night watch-man brought her food, he raped her. When Karen tried to tell her sister about the assault, her sister didn’t seem to identify the experience as rape. This unsupportive interchange caused Karen to question the efficacy of disclosure. Her sister’s response was so unsympathetic that she never spoke to her about the assault again. Believing that there was no-one in her life who would support her, Karen did not disclose again for 19 years." (Filipas & Ullman, 2001).{{RoundBoxBottom}} === Relationship to the perpetrator === Victims were less likely to disclose an incident of sexual assault if the perpetrator was someone that was known to them or someone with whom they were in a close familial or intimate relationship. The following statistics reveal the true extent of the effect of the perpetrator’s relationship to the victim on nondisclosure of the sexual assault (Willis, 2011): * 25 per cent of victims disclosed sexual assault if the perpetrator was a stranger * 17 per cent of victims disclose sexual assault if the perpetrator was a former partner * 1 per cent of victims disclose sexual assault if the perpetrator was a current partner (Willis, 2011). The motivation for non-disclosure of sexual assault committed by a known offender can be severely hindered by the victim’s fear of repercussions from the perpetrator and the family and friends of both parties. The perpetrator may have the ability to exert power over the victim over a long period of time, by threatening financial hardship and/or breakdown of the familial unit. These strong motivators for nondisclosure are likely to encourage the victim to deal with the incidence of sexual assault privately and to avoid pursuing prosecution through the legal system (Willis, 2011). === Revictimisation === A number of studies into victims of sexual assault indicate that following disclosure by victims of assault to a confidant or as part of a public awareness campaign, in some instances the victim was targeted by an additional perpetrator. This perpetrator reportedly used the methods that the victim had described as part of their disclosure (Sable, Danis, Mauzy, & Gallagher, 2006). Victims often reported feeling extremely vulnerable to further attacks after disclosure and that they would be less motivated to report subsequent incidences due to the threat and occurrence of revictimisation (Mason, Ullman, Long, Long, & Starzynski, 2009). Those who had disclosed sexual assault, reported that often they were coping with additional issues such as increased alcohol and drug use and had become increasingly promiscuous as part of coping with the event (Mason et al., 2009). Often because of the additional issues that they were experiencing, victims were vulnerable to revictimisation and experienced further incidences of sexual assault. Victims felt that because of their history of being a sexual assault victim, the drug and alcohol use, and the promiscuity, they were unable to disclose any of their sexual assault incidents (Mason et al., 2009). Revictimisation may additionally come through the disclosure process itself. The victim may have to endure invasive medical procedures, give lengthy and detailed statements of the sexual assault and may come into contact with the perpetrator as part of the prosecution process. The victim may feel that they have just recovered from the ordeal only to be forced to go through it again through revictimisation during the legal process (Willis, 2011). === Knowledge of services === Victims of sexual assault are often unaware of the services available to them after their experience. The threat of a negative response as a result of previous experiences of disclosure and social stigma often prevents the victim from conducting research into what options are available to them. Without an informative support network such as a rape crisis centre or a victim support network, victims are likely to feel that disclosing the assault will only lead to undesirable outcomes. The victim may feel that the prosecution process is too daunting to face on their own (Sable, Danis, Mauzy, & Gallagher, 2006). === Perceptions of police === Perceptions of police, and the justice system as a whole, may severely hinder disclosure of a sexual assault. The victim may not feel assured that they will be believed and that the outcome of any legal action may not reach a desired outcome. Police are often under resourced and face the challenge of gathering enough evidence for the matter to appear before court. The victim may feel that the police do not have enough empathy towards them and are simply stepping through the required process. Sexual assault is often a difficult offence to prosecute. Recent studies have found that in the Australian Capital Territory one third of cases that proceeded to prosecution resulted in a conviction and one in ten apprehended offenders was acquitted. Sexual assault victims may not feel that a disclosure as part of the prosecution process will be beneficial to them (Willis, 2011). Disclosure can additionally be affected by cultural aversions towards police. Disclosure is unlikely to occur if the victim has had negative interactions with the police in their community, or if the victim comes from a cultural background that has an inherent distrust of law enforcement bodies. Victims may be encouraged to deal with the issue within the community and not disclose the incident further (Willis, 2011). == The role of media and popular culture == Media and popular culture have often portrayed sexual assaults as violent crimes involving weapons, and as being committed in public spaces by strangers. Statistics reported by sources such as the Australian Bureau of Statistics show that this is indeed largely a false representation (Australian Bureau of Statistics, 2012). The majority of sexual assault victims will be assaulted by someone that is known to them and the methods used by the perpetrator often do not involve a weapon. The false representation of how sexual assault occurs perpetuates the idea to victims that what they experienced was not a 'true' representation of rape or an attack as portrayed by popular culture. This means that the victim can often decide to not report their assault as they feel that they will not be believed, or they experience self-doubt that their attack is not worth reporting (Dowler, Fleming, & Muzzatti, 2006). In addition to the portrayal of how a sexual assault occurs, media and popular culture can shape societal views on how acceptable such an act is. Through media and popular culture, sexual assault can often be sensationalised to appear alluring and in the extreme desirable or exciting. University of California Los Angeles media researcher Dr Neil Malamuth stated that ''"media portrayals can promote the view that women desire violence; they can transform sensitive individuals' view of rape to make it seem more acceptable and not such an abhorrent act."'' With this in mind, a victim is less likely to report their attack as it may be viewed as just 'something that happens' and a minor incident in everyday life that should be accepted as such (Media Values, 2015). == Conclusion == The motivations behind a victim not reporting their sexual assault can be complex. The circumstances of the assault and the victim's own perceptions and emotional responses mean they can be overwhelmed and indecisive. They may not be aware of support services available to them.  Individuals need access to a positive and supportive audience. They need to be assisted to understand and accept they are not to blame for an incident. This is as an important first step in encouraging disclosure. A victim should be made aware of the support services available and be assisted to understand their options and the processes that may take place if they decide to disclose (Sable, Danis, Mauzy, & Gallagher, 2006). Victims may need encouragement to seek help for peripheral or consequential issues such as serious drug or alcohol use or symptoms of post-traumatic stress disorder. This will help them to recover from their ordeal fully or as part of pursuing a prosecution. Some victims may decide to disclose their sexual assault, but express a desire to keep their details confidential for example, to avoid social stigma. The victim’s wishes should be respected and they should be reassured that every effort will be taken to ensure that this is adhered while they decide how they proceed. == See also == [[Motivation and emotion/Book/2013/Emotional impacts of sexual assault]] [[Motivation and emotion/Book/2014/Sexual harassment and emotion]] == References == {{Hanging indent|1= Ahrens, C. E. (2006). Being Silenced: The Impact of Negative Social Reactions on the Disclosure of Rape. ''American Journal of Community Psychology, 38''(3), 263-274. doi:10.1007/s10464-006-9069-9 American Psychiatric Association. (2013). ''Diagnostic and statistical manual of mental disorders'' (5th ed.). Arlington, VA: American Psychiatric Publishing. Australian Bureau of Statistics. (2012). 4906.0 - Personal Safety, Australia, 2012. Retrieved from http://www.abs.gov.au/AUSSTATS/abs@.nsf/Lookup/4906.0Main+Features12012?OpenDocument. Campbell, R., Ahrens, C. E., Sefl, T., Wasco, S. M., & Barnes, H. E. (2001). Social reactions to rape victims: Healing and hurtful effects on psychological and physical health outcomes. ''Violence and victims, 16''(3), 287-302. Chaudoir, S. R., & Fisher, J. D. (2010). The disclosure processes model: understanding disclosure decision making and post disclosure outcomes among people living with a concealable stigmatized identity. ''Psychological bulletin, 136''(2), 236. Chaudoir, S. R., Fisher, J. D., & Simoni, J. M. (2011). Understanding HIV disclosure: A review and application of the Disclosure Processes Model. ''Social Science & Medicine, 72''(10), 1618-1629. Filipas, H. H., & Ullman, S. E. (2001). Social reactions to sexual assault victims from various support sources. ''Violence and victims, 16''(6), 673. Herek, G. M. (2007). Confronting sexual stigma and prejudice: Theory and practice. ''Journal of Social Issues, 63''(4), 905-925. Klump, M. C. (2006). Posttraumatic stress disorder and sexual assault in women. ''Journal of College Student Psychotherapy, 21''(2), 67-83. Maier, S. F., & Seligman, M. E. (1976). Learned helplessness: Theory and evidence. ''Journal of experimental psychology: general, 105''(1), 3. Major, B., & O'Brien, L. T. (2005). The social psychology of stigma. ''Annu. Rev. Psychol., 56'', 393-421. Mason, G. E., Ullman, S., Long, S. E., Long, L., & Starzynski, L. (2009). Social support and risk of sexual assault revictimization. ''Journal of Community Psychology, 37''(1), 58-72. Miller, A. K., Canales, E. J., Amacker, A. M., Backstrom, T. L., & Gidycz, C. A. (2011). Stigma-Threat Motivated Nondisclosure of Sexual Assault and Sexual Revictimization A Prospective Analysis. ''Psychology of Women Quarterly, 35''(1), 119-128. Sable, M. R., Danis, F., Mauzy, D. L., & Gallagher, S. K. (2006). Barriers to reporting sexual assault for women and men: Perspectives of college students. ''Journal of American College Health, 55''(3), 157-162. Ullman, S. E. (1996). Social Reactions, Coping Strategies, and Self‐blame Attributions in Adjustment to Sexual Assault. ''Psychology of Women Quarterly, 20''(4), 505-526. Values, M. (2015). ''Media's New Mood: Sexual Violence''. United States of America: Center for Media Literacy. Willis, M. (2011). ''Non-disclosure of violence in Australian Indigenous communities''. Retrieved from <nowiki>http://www.aic.gov.au/media_library/publications/tandi_pdf/tandi405.pdf</nowiki>: Wortman, C. B., & Brehm, J. W. (1975). Responses to uncontrollable outcomes: An integration of reactance theory and the learned helplessness model. ''Advances in experimental social psychology, 8'', 277-336. }} == External Links == [https://www.ted.com/talks/jessica_ladd_the_reporting_system_that_sexual_assault_survivors_want?language=en Ted talks. Jessica Lad: The reporting system that sexual assault survivors want] [https://www.ted.com/talks/ione_wells_how_we_talk_about_sexual_assault_online Ted talks. Ione Wells: How we talk about sexual assault online] [https://www.ted.com/talks/melissa_walker_art_can_heal_ptsd_s_invisible_wounds Ted talks: Melissa Walker: Art can heal PTSD's invisible wounds] == External links == ;Where to find help * LifeLine Australia https://www.lifeline.org.au/<nowiki/>13 11 14 * Canberra Rape Crisis Centre http://www.crcc.org.au (02) 6147 2525 * The Nguru Program, counselling for members of the Aboriginal and Torres Strait Islander community who have been victims of sexual assault and their families. http://www.crcc.org.au/support-services/the-nguru-program.aspx (02) 6147 2525 * Service Assisting Male Survivors of Sexual Assault [http://www.samssa.org.au/ www.samssa.org.au] (02) 6287 3935 * NSW Rape Crisis Centre http://www.nswrapecrisis.com.au/ 1800 424 017 [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Forensic/Sex crime‎]] [[Category:Motivation and emotion/Book/Sexual motivation]] 78e4nvg18pcuuumi4a708qo9lipohs6 0 224775 2801585 2474865 2026-03-30T11:45:11Z Bert Niehaus 2387134 2801585 wikitext text/x-wiki [[File:Sustainable Development Goal 04QualityEducation.svg|thumb|SDG4: Quality Education - Learning Resouce supports the SDGs - [https://www.un.org/sustainabledevelopment/news/communications-material/ UN-Guidelines]<ref>UN-Guidelines for Use of SDG logo and the 17 SDG icons (2019/05/10) - https://www.un.org/sustainabledevelopment/news/communications-material/</ref> - see [[Sustainable Development Goals/SDG-Tagging|SDG-Tagging]].]] Green open access<ref>Harnad, S., Brody, T., Vallieres, F. O., Carr, L., Hitchcock, S., Gingras, Y., ... & Hilf, E. R. (2004). The access/impact problem and the green and gold roads to open access. Serials review, 30(4), 310-314.</ref><ref>Harnad, S., Brody, T., Vallieres, F., Carr, L., Hitchcock, S., Gingras, Y., ... & Hilf, E. R. (2008). The access/impact problem and the green and gold roads to open access: An update. Serials review, 34(1), 36-40.</ref> - Green OA, is the practice of placing a version of an author’s manuscript into a repository, making it freely accessible for the scientific community. The primary motivation of Open Access was * to provide Open Access to Knowledge to the READER of Publications and * to allow Open Access to AUTHORS Publication (unbiased publication of Knowledge e.g. with peer reviewing for [[quality assurance]]) == Green Open Access vs. Commercial Open Access Strategies == === Commercial Gold Open Access === Commercial Publishers need to create profit for the company, so it necessary to charge someone in the educational and academic institutions for the publications: * '''(charge reader)''' charge either the Reader of publications, * '''(charge authors)'''the authors of publications by publication fees (e.g in commercial Gold Open Access), * '''(charge institutions)'''or the Institutions, the authors works for. === Publishing Costs for Educational System in Commercial Settings === * Universities or organisations have to pay for the scientist for creating a publication, * Universities have to pay for scientist for peer-reviewing and * when the paper is published the Universities have to pay for Journals in a classical setting or for Gold Standard Open Access Publication. === Green Open Access in Wikiversity === * papers developed in Wikiversity have by default a transparent versioning (i.e. readers of the paper can access the version history and can identify who contributed, when, what, ...) * Green Open Access includes a step of self-archiving of a paper release (e.g. on public accessible repository, where the scientific community has not write-access). * includes an optional step of external quality assurance (if nobody performed an external quality assurance, this is visible for the user), quality assurance could finalize the assessment with ** no errors found, ** errors found and corrected, ** discussion added to paper, where improvements could be performed * The activity on the peer-reviewing side defines the scientific interest in the paper. * release management of authors: the authors can create releases of the paper and make these release accessible to the scientific community == History of Green OA in the Context of AT6FUI == This approach to Green Open Access is based of the [[Open Community Approach]] introduced to Action Team 6 Follow-Up Initiative AT6FUI<ref>Action Team 6 Follow-up Initiative AT6FUI (2011-2015) - URL: http://at6fui.weebly.com/at6fui-background.html (accessed 2020/02/07)</ref> in 2012. The main purpose of Green Open Access is to introduce scientific results as [[Wikipedia:Decision support system|decision support product]]. Image you have a scientific result that is constantly updated with new data. A classical paper must be evaluated by scientist or engineers that include the new results into a [[Wikipedia:Decision support system|Decision Support System]]. A dynamic paper management create constantly statiscal and/or numerical assessment of the data. The green open access creates a dynamic report of the updated input data (e.g. sensor data, financial data, ...). The new data could have an impact the statistical significance of a scientific result (e.g. medical drug has show good success rates for treating a disease, by an constant update of anonymous patient data the benefit of the drug is not clear anymore). A decision support system could be linked to such dynamic digital paper and [[Wikipedia:Decision support system|decision support]] is update as well the new results are quality assured by a digital signature of an institution e.g. WHO. Such a dynamic paper generation could be realized with Knitr<ref>Learn Knitr Basics in 5 min - see YouTube Tutorial https://www.youtube.com/watch?v=DUAaNVlC6FE (replace this tutorial by an Wikiversity internal tutorial)</ref>. The pages on AT6FUI website are published under Creative Commons, so that this documents can be build on those recommendations and results. ==Role of Reviewers== A [[w:Peer_review|peer-reviewing process]] is used for quality assurance. If reviewers provide feedback to the authors they * rate a publication as ready for publication * suggest orthographic or grammar improvements, * highlight missing scientific results, that should be included in the publication, * identify missing information about the methodology so that the results are [[w:Reproducibility|reproducible]] for other scientist, * logical, methodological, ... errors, * ... ==Public-Private-Versioning and Open Access== [[Trust]] in a reviewing process is create by community of reviewers that perform the reviewing. In an [[Open Community Approach]] the strict separation of Reviewers and Authors is not there. Review is regarded as a contribution to the evolution of the paper. A reviewer becomes an author can take over the reviewer role during a quality assurance process. The concept of [[Version Control/Public-Private-Versioning|public-private-versioning]] can be used to create institutional private version in a development branches. Private versions can be regarded as releases (major version) of a document in the private branch. The evolution of a publication is not finished with the publication, it evolves with new scientific results. Who created what, when is transparently documented in a version control system. The reader can track, if the author provided major contributions to paper development or just removed typos and altered the wording. == Learning Tasks == * '''(Cite Wikiversity/Wikipedia)''' Learn how to cite a wikipedia or wikiversity resource by using the ''"Cite this page"'' menu item. * '''([[WikiJournal_of_Medicine|WikiJournal of Medicine]])''' Explore the [[WikiJournal_of_Medicine|WikiJournal of Medicine in Wikiversity]]. ** What are the elements of Green Open Access that you identify in the Journal? ** How is the quality assurance implemented in the journal? ** Compare these concepts with [[Version Control/Public-Private-Versioning|Public-Private-Versioning]] and describe similarities and differences in the approach. * '''([[WikiJournal of Science]])''' Explore the scope of the Journal and identify and analyse the peer-reviewing concept! * '''([[Open Community Approach]])''' Consider the following two options: ** you can regard knowledge as a commercial product, that can be sold or ** you can donate knowledge to the community as common good, on which the build on, access it, modify it and intergrate the common good into the problem solving strategy of the commuity. : Does Green Open Access contribute to that concept and how? * '''([[/Who did when what why/]])''' Transparent evolution of a scientific paper allows authors to publish not only the final result of the paper but also the intermediate steps on transparent way. Open Access combined with evolution of the paper will be visible to other authors as a learning resource. The key principle of transparent levels of paper evolution can be described by open access to information about "[[/Who did when what why/]]". Would you prefer to publish all intermediate steps of just the main steps of the paper development of just the final result of the paper? What can other Master students and/or PhD students learn from an evolutionary process of the paper development and from the access to the feedback of supervisor for the students? == See also == * [[Wikipedia:Peer review]] * [[Risk_management/Spatial_risk_management|Spatial Risk Management]] * [[Version Control/Public-Private-Versioning]] * [[Digital Signature]] for validation of origine of publications. * [[Trust]]: Improve [[Trust|trust in Green Open Access publication]] and peer-reviewing. * [[Space and Global Health|Community of Practice: Space and Global Health]] * [[Open Educational Resources]] * [[Sustainable Development Goals]] * [[Open Community Approach]] == References == [[Category:Open Community Approach]] [[Category:Knowledge management]] [[Category:Open access]] [[Category:Public-Private-Versioning]] <noinclude> [[de:Green Open Access]] </noinclude> dwf4qz4575c5694asews8k824r53uxz 0 231817 2801499 2468894 2026-03-30T08:10:31Z Anikmolla786 3007678 Corrected “graphic” to “graphical” in the first sentence. 2801499 wikitext text/x-wiki <noinclude>{{:{{BASEPAGENAME}}/Sidebar}}</noinclude> This lesson provides a brief introduction to graphical user interface (GUI) programming. GUI options include native frameworks, cross-platform frameworks, and web applications. __TOC__ == Objectives and Skills == Objectives and skills for this lesson include: * Understand the options available for GUI application development * Implement a simple GUI application == Readings == # [[Wikipedia: Graphical user interface]] # [[Wikipedia: List of widget toolkits]] == Multimedia == # [https://www.youtube.com/watch?v=D8-snVfekto&t: YouTube: How to Program a GUI Application] # [https://www.youtube.com/watch?v=RJB1Ek2Ko_Y: YouTube: GUI Tkinter Introduction] # [https://www.youtube.com/watch?v=RHCg3p9aP_I: YouTube: Python GUI Frameworks ] == Examples == <!-- {{colbegin|3}} --> * [[/Python3/]] * [[/Python3-PyQt/]] * [[/Python3-tkInter/]] * [[/Python3/flask/]] <!-- {{colend}} --> == Activities == === GUI Applications === # Using the default GUI framework for your selected programming language, create a graphical user interface for one of the programs developed in this course. == Lesson Summary == * A GUI is a graphical user interface, where actions get funneled through widgets, as opposed to a CLI, a command line interface—where the program is composed exclusively of text.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ** A widget is an element of user-computer interaction that facilitates the direct manipulation of the program by the user (through events).<ref name="Wikipedia: Widget (GUI)">[[Wikipedia: Widget (GUI)]]</ref> * Python's Tkinter library is a binding to the Tk GUI toolkit, which is a framework written in C.<ref name="Wikipedia: Tkinter">[[Wikipedia: Tkinter]]</ref> * A language binding is an API (application programming interface) that allows one language to interface with the code of another language. ** Glue code is a name for this code written solely to inter-operate between libraries, providing no other useful functionality. ** An API is a means to communicate between two or more components of software.<ref name="Wikipedia: Application programming interface">[[Wikipedia: Application programming interface]]</ref> * GUIs are used in a variety of handheld devices like our smartphones, media players, and even in computerized control systems. It aims to enhance efficiency and usability of a program within these devices by allowing users to choose from elements such as windows, menus, and icons.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ** Operating systems still provide some sort of command-line interface along GUI, allowing it to be used easily for beginners. Although CLI may not be preferred, it still does offer greater efficiency and productivity, especially in complex operations, as long as the commands are known by the user. Also, command-line interfaces do not need as much storage and processing power when compared to GUI. <ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> * Large widgets, such as windows, usually provide a frame or container for the main presentation content such as a web page, email message or drawing. Smaller ones usually act as a user-input tool.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> *The visible graphical interface features of an application are sometimes referred to as chrome or GUI (pronounced gooey). Typically, users interact with information by manipulating visual widgets that allow for interactions appropriate to the kind of data they hold.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> * Most modern operating systems provide both a GUI and some level of a CLI, although the GUIs usually receive more attention.<ref name="Wikipedia: Graphical user interface" /> * A GUI uses a combination of technologies and devices to provide a platform that users can interact with, for the tasks of gathering and producing information.<ref name="Wikipedia: Graphical user interface" /> * Good user interface design relates to users more, and to system architecture less.<ref name="Wikipedia: Graphical user interface" /> *Designing the visual composition and temporal behavior of a GUI is an important part of software application programming in the area of human–computer interaction. Its goal is to enhance the efficiency and ease of use for the underlying logical design of a stored program, a design discipline named usability.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> *Operating systems often have low level widget toolkits integrated within them. Some examples of these are:<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> **OS X uses Cocoa. Mac OS9 and OS X use Carbon for 32-bit applications.<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> **The Windows API used in Microsoft Windows. Microsoft had the graphics functions integrated in the kernel until 2006.<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> == Key Terms == ;API :An API is a means to communicate between two or more components of software. API interfaces make it easier for developers to use certain technologies in building applications.<ref>{{Cite journal|date=2018-07-19|title=Application programming interface|url=https://en.wikipedia.org/w/index.php?title=Application_programming_interface&oldid=850975267|journal=Wikipedia|language=en}}</ref> ;CLI :Command-line interface, the program is composed purely of textual content.<ref name="Wikipedia: Command-line interface">[[Wikipedia: Command-line interface]]</ref> ;Client : A piece of computer hardware or software that accesses a service made available by a server. Web application clients vary greatly and this warrants extended forms of testing.<ref>{{Cite journal|date=2018-06-03|title=Client (computing)|url=https://en.wikipedia.org/w/index.php?title=Client_(computing)&oldid=844165126|journal=Wikipedia|language=en}}</ref> ;GUI :Graphical user interface, the program is composed of interactive widgets.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;GUI wrapper :Graphical user interface wrappers circumvent the command-line interface versions of (typically) Linux and Unix-like software applications. Applications may also provide both interfaces, and when they do the GUI is usually a WIMP wrapper around the command-line version.<ref name="Wikipedia: Graphical user interface" /> ;integration testing :A phase in software testing in which individual software modules are combined and tested as a group.<ref name="Wikipedia: Integration testing">[[Wikipedia: Integration testing]]</ref> ;post-WIMP interface :As of 2011, some touchscreen-based operating systems such as Apple's iOS and Android use the class of GUIs named post-WIMP. These support styles of interaction using more than one finger in contact with a display, which allows actions such as pinching and rotating, which are unsupported by one pointer and mouse.<ref name="Wikipedia: Graphical user interface" /> ;skin :Refers to the appearance of a program's interface.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;system testing :Testing conducted on a complete, integrated system to evaluate the system's compliance with its specified requirements.<ref name="Wikipedia: System testing">[[Wikipedia: System testing]]</ref> ;unit testing :A phase of software testing where individual units of a software are tested to determine whether they are fit for use.<ref name="Wikipedia: Unit testing">[[Wikipedia: Unit testing]]</ref> ;widget :A vessel for human-computer interaction that is characteristic of a GUI;a widget could be a button, a check box, a text box, etc.<ref name="Wikipedia: Widget (GUI)">[[Wikipedia: Widget (GUI)]]</ref> ;WIMP :Stands for "Windows, Icons, Menus, Pointer." which describes the graphical user interface (GUI) of personal computers.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;window :A graphical control element which consists of a visual area containing some of the graphical user interface of the program it belongs to and is framed by window decoration.<ref>[[Wikipedia: Window (computing)]]</ref> ; Workbench : provides the user with a graphical interface to work with file systems and launch applications. It uses a workbench metaphor (in place of the more common desktop metaphor) for representing file system organisation.<ref>[[Wikipedia: Workbench (AmigaOS)]]</ref> ;web framework :A structure support of web applications, libraries, APIs, and compilers to streamline the creation of new software.<ref name="Wikipedia: Web framework">[[Wikipedia: Web framework]]</ref> ;web application :A client-server program that runs in a web browser.<ref name="Wikipedia: Web application">[[Wikipedia: Web application]]</ref> ;web application development :The process and practice of developing web applications.<ref name="Wikipedia: Web application development">[[Wikipedia: Web application development]]</ref> == See Also == * [[Internet Fundamentals]] *https://www.geeksforgeeks.org/python-gui-tkinter/ * [[HTML]] * [https://www.tutorialspoint.com/python/python_gui_programming.htm TutorialsPoint: Tkinter Tutorial ] * [https://www.webdesignerdepot.com/2009/03/operating-system-interface-design-between-1981-2009/ Web Designer Depot - Operating System Interface Design Between 1981-2009] * [https://blog.hubspot.com/marketing/look-back-20-years-website-design HubSpot - A Look Back at 20+ Years of Website Design ] * [https://archive.org/web/web.php Wayback Machine ] == References == {{reflist}} {{subpage navbar}} [[Category:Lessons]] [[Category:Completed resources]] {{CourseCat}} gvwv2ou96bmcj02nlvfap8njvdpk4sh 2801500 2801499 2026-03-30T08:13:04Z Anikmolla786 3007678 Removed inaccurate statement about graphics functions in the Windows kernel. 2801500 wikitext text/x-wiki <noinclude>{{:{{BASEPAGENAME}}/Sidebar}}</noinclude> This lesson provides a brief introduction to graphical user interface (GUI) programming. GUI options include native frameworks, cross-platform frameworks, and web applications. __TOC__ == Objectives and Skills == Objectives and skills for this lesson include: * Understand the options available for GUI application development * Implement a simple GUI application == Readings == # [[Wikipedia: Graphical user interface]] # [[Wikipedia: List of widget toolkits]] == Multimedia == # [https://www.youtube.com/watch?v=D8-snVfekto&t: YouTube: How to Program a GUI Application] # [https://www.youtube.com/watch?v=RJB1Ek2Ko_Y: YouTube: GUI Tkinter Introduction] # [https://www.youtube.com/watch?v=RHCg3p9aP_I: YouTube: Python GUI Frameworks ] == Examples == <!-- {{colbegin|3}} --> * [[/Python3/]] * [[/Python3-PyQt/]] * [[/Python3-tkInter/]] * [[/Python3/flask/]] <!-- {{colend}} --> == Activities == === GUI Applications === # Using the default GUI framework for your selected programming language, create a graphical user interface for one of the programs developed in this course. == Lesson Summary == * A GUI is a graphical user interface, where actions get funneled through widgets, as opposed to a CLI, a command line interface—where the program is composed exclusively of text.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ** A widget is an element of user-computer interaction that facilitates the direct manipulation of the program by the user (through events).<ref name="Wikipedia: Widget (GUI)">[[Wikipedia: Widget (GUI)]]</ref> * Python's Tkinter library is a binding to the Tk GUI toolkit, which is a framework written in C.<ref name="Wikipedia: Tkinter">[[Wikipedia: Tkinter]]</ref> * A language binding is an API (application programming interface) that allows one language to interface with the code of another language. ** Glue code is a name for this code written solely to inter-operate between libraries, providing no other useful functionality. ** An API is a means to communicate between two or more components of software.<ref name="Wikipedia: Application programming interface">[[Wikipedia: Application programming interface]]</ref> * GUIs are used in a variety of handheld devices like our smartphones, media players, and even in computerized control systems. It aims to enhance efficiency and usability of a program within these devices by allowing users to choose from elements such as windows, menus, and icons.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ** Operating systems still provide some sort of command-line interface along GUI, allowing it to be used easily for beginners. Although CLI may not be preferred, it still does offer greater efficiency and productivity, especially in complex operations, as long as the commands are known by the user. Also, command-line interfaces do not need as much storage and processing power when compared to GUI. <ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> * Large widgets, such as windows, usually provide a frame or container for the main presentation content such as a web page, email message or drawing. Smaller ones usually act as a user-input tool.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> *The visible graphical interface features of an application are sometimes referred to as chrome or GUI (pronounced gooey). Typically, users interact with information by manipulating visual widgets that allow for interactions appropriate to the kind of data they hold.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> * Most modern operating systems provide both a GUI and some level of a CLI, although the GUIs usually receive more attention.<ref name="Wikipedia: Graphical user interface" /> * A GUI uses a combination of technologies and devices to provide a platform that users can interact with, for the tasks of gathering and producing information.<ref name="Wikipedia: Graphical user interface" /> * Good user interface design relates to users more, and to system architecture less.<ref name="Wikipedia: Graphical user interface" /> *Designing the visual composition and temporal behavior of a GUI is an important part of software application programming in the area of human–computer interaction. Its goal is to enhance the efficiency and ease of use for the underlying logical design of a stored program, a design discipline named usability.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> *Operating systems often have low level widget toolkits integrated within them. Some examples of these are:<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> **OS X uses Cocoa. Mac OS9 and OS X use Carbon for 32-bit applications.<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> **The Windows API used in Microsoft Windows.<ref name="Wikipedia: List of widget toolkits">[[Wikipedia: List of widget toolkits]]</ref> == Key Terms == ;API :An API is a means to communicate between two or more components of software. API interfaces make it easier for developers to use certain technologies in building applications.<ref>{{Cite journal|date=2018-07-19|title=Application programming interface|url=https://en.wikipedia.org/w/index.php?title=Application_programming_interface&oldid=850975267|journal=Wikipedia|language=en}}</ref> ;CLI :Command-line interface, the program is composed purely of textual content.<ref name="Wikipedia: Command-line interface">[[Wikipedia: Command-line interface]]</ref> ;Client : A piece of computer hardware or software that accesses a service made available by a server. Web application clients vary greatly and this warrants extended forms of testing.<ref>{{Cite journal|date=2018-06-03|title=Client (computing)|url=https://en.wikipedia.org/w/index.php?title=Client_(computing)&oldid=844165126|journal=Wikipedia|language=en}}</ref> ;GUI :Graphical user interface, the program is composed of interactive widgets.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;GUI wrapper :Graphical user interface wrappers circumvent the command-line interface versions of (typically) Linux and Unix-like software applications. Applications may also provide both interfaces, and when they do the GUI is usually a WIMP wrapper around the command-line version.<ref name="Wikipedia: Graphical user interface" /> ;integration testing :A phase in software testing in which individual software modules are combined and tested as a group.<ref name="Wikipedia: Integration testing">[[Wikipedia: Integration testing]]</ref> ;post-WIMP interface :As of 2011, some touchscreen-based operating systems such as Apple's iOS and Android use the class of GUIs named post-WIMP. These support styles of interaction using more than one finger in contact with a display, which allows actions such as pinching and rotating, which are unsupported by one pointer and mouse.<ref name="Wikipedia: Graphical user interface" /> ;skin :Refers to the appearance of a program's interface.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;system testing :Testing conducted on a complete, integrated system to evaluate the system's compliance with its specified requirements.<ref name="Wikipedia: System testing">[[Wikipedia: System testing]]</ref> ;unit testing :A phase of software testing where individual units of a software are tested to determine whether they are fit for use.<ref name="Wikipedia: Unit testing">[[Wikipedia: Unit testing]]</ref> ;widget :A vessel for human-computer interaction that is characteristic of a GUI;a widget could be a button, a check box, a text box, etc.<ref name="Wikipedia: Widget (GUI)">[[Wikipedia: Widget (GUI)]]</ref> ;WIMP :Stands for "Windows, Icons, Menus, Pointer." which describes the graphical user interface (GUI) of personal computers.<ref name="Wikipedia: Graphical user interface">[[Wikipedia: Graphical user interface]]</ref> ;window :A graphical control element which consists of a visual area containing some of the graphical user interface of the program it belongs to and is framed by window decoration.<ref>[[Wikipedia: Window (computing)]]</ref> ; Workbench : provides the user with a graphical interface to work with file systems and launch applications. It uses a workbench metaphor (in place of the more common desktop metaphor) for representing file system organisation.<ref>[[Wikipedia: Workbench (AmigaOS)]]</ref> ;web framework :A structure support of web applications, libraries, APIs, and compilers to streamline the creation of new software.<ref name="Wikipedia: Web framework">[[Wikipedia: Web framework]]</ref> ;web application :A client-server program that runs in a web browser.<ref name="Wikipedia: Web application">[[Wikipedia: Web application]]</ref> ;web application development :The process and practice of developing web applications.<ref name="Wikipedia: Web application development">[[Wikipedia: Web application development]]</ref> == See Also == * [[Internet Fundamentals]] *https://www.geeksforgeeks.org/python-gui-tkinter/ * [[HTML]] * [https://www.tutorialspoint.com/python/python_gui_programming.htm TutorialsPoint: Tkinter Tutorial ] * [https://www.webdesignerdepot.com/2009/03/operating-system-interface-design-between-1981-2009/ Web Designer Depot - Operating System Interface Design Between 1981-2009] * [https://blog.hubspot.com/marketing/look-back-20-years-website-design HubSpot - A Look Back at 20+ Years of Website Design ] * [https://archive.org/web/web.php Wayback Machine ] == References == {{reflist}} {{subpage navbar}} [[Category:Lessons]] [[Category:Completed resources]] {{CourseCat}} 6ltf3vbzou1hzpr9ej3ypoouy3qfu9d 0 247196 2801435 2698717 2026-03-30T05:41:45Z Anikmolla786 3007678 Corrected spelling of “Vasintoniēnsis” to “Vasingtoniēnsis” (Latin for Washington). 2801435 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Salvēte omnēs! Welcome back to Latin for Wikiversity. Here you can peruse a new lesson in Latin, in a simple format. If you would like to catch up, you can find a directory of lessons, a classified vocabulary list, and Memrise courses at the links on the right. In the last lesson we looked at the locative case, a relatively uncommon case used primarily for cities, towns, and small islands. Today we’ll continue an informal study of geography and place-names. ==New grammar== Place-names are nouns, but often we use an adjectival form to describe people or things from that place. For example, :''Rōma'' = Rome. ::''Rōmānus, a, um'' is the adjectival form, and can be used substantively (as a noun) in masc. or fem. to describe a person from Rome: :::''Rōmānus'' is a Roman man. ''Rōmāna'' is a Roman woman. But you can also have ''populus Rōmānus'' (the Roman people), ''virtūtēs Rōmānae'' (Roman virtues), etc., as an adjective. ''[[wiktionary:germanicus#Latin|Germānicus]], [[wiktionary:italicus#Latin|Italicus]], [[wiktionary:britannicus#Latin|Britannicus]]'' would usually be used to describe things, not people, although ''Germānicus'' and ''Britannicus'' were names of famous Romans, as was ''[[wiktionary:Tiberius#Latin|Tiberius]]'' (named after the Tiber river). A German might be ''[[wiktionary:germanus#Latin|Germānus]]'', a UK citizen might be ''[[wiktionary:britannus#Latin|Britannus]]'' or ''[[wiktionary:brito#Latin|Britō]]''. Some adjectival forms are created by adding ''–(i)ēnsis'' to the noun describing a city or country, e.g. ''[[wiktionary:branta#Latin|Branta]] [[wiktionary:canadensis#Latin|Canadēnsis]]'' (Canadian goose), ''[[wiktionary:Gallia#Latin|Gallia]] [[wiktionary:Narbonensis#Latin|Narbōnēnsis]]'' (the Roman province in Gaul around the city ''[[wiktionary:Narbo#Latin|Narbō]],'' now Narbonne), ''[[wiktionary:philosophia#Latin|philosophia]] [[wiktionary:atheniensis#Latin|Athēniēnsis]]'' (Athenian philosophy). This is all made more complicated by the fact that Latin has been changing and adapting for thousands of years and there is no one definitive stylebook of modern usage. We therefore include some basic vocabulary and sentences, some of them taken from Vicipaedia. Also, we’ll look at some of the Latin terms from higher education, which would not exist if not for Latin. It’s a bit of a digression, but not if we see the development of the human intellect as the most important journey of all. And we wish hearty congratulations to all of you reading this who are celebrating a completion of one part of your intellectual journey! ==New Vocabulary== {| class="wikitable" ! | Latin ! | English ! | Audio (Classical) ! | Notes |- |+ Nouns |- | [[wiktionary:caelum#Latin|caelum]], ī || sky, heaven || [[File:la-cls-caelum.ogg|thumb|]] || |- | [[wiktionary:vexillum#Latin|vexillum]], ī || flag || <!--[[File:la-cls-vexillum.ogg|thumb|]]--> || |- | [[wiktionary:continens#Latin|continēns]], continentis, f. || continent, mainland || <!--[[File:la-cls-continens.ogg|thumb|]]--> || From contineō = hold in, enclose |- | [[wiktionary:universitas#Latin|ūniversitās]], ūniversitātis, f. (ūniversitās magistrōrum et scholārium) || university (community of teachers and scholars) || <!--[[File:la-cls-universitas.ogg|thumb|]]--> || |- | [[wiktionary:diploma#Latin|dīplōma]], dīplōmatis, n. || letter of recommendation, letter of authority || <!--[[File:la-cls-diploma.ogg|thumb|]]--> || |- | [[wiktionary:gradus#Latin|gradus]], ūs || step, degree, position, rung of a ladder || <!--[[File:la-cls-gradus.ogg|thumb|]]--> || |} {| class="wikitable" ! | Latin ! | English ! | Audio (Classical) ! | Notes |- |+ Adjectives |- | [[wiktionary:almus#Latin|almus]], a, um || nurturing, nourishing, kind || <!--[[File:la-cls-almus.ogg|thumb|]]--> || |- | [[wiktionary:universus#Latin|ūniversus]], a, um || all together, all in one, whole, universal || <!--[[File:la-cls-universus.ogg|thumb|]]--> || |- | [[wiktionary:australis#Latin|austrālis]], e (also meridionalis, e or meridianus, a, um) || southern, of the south || <!--[[File:la-cls-australis.ogg|thumb|]]--> || from auster, the south wind, and meridies, midday or noon) |- | [[wiktionary:occidentalis#Latin|occidentālis]], e || western, of the west || <!--[[File:la-cls-occidentalis.ogg|thumb|]]--> || From occidens, or occasus, the falling or setting of the sun |- | [[wiktionary:orientalis#Latin|orientālis]], e || eastern, of the east || <!--[[File:la-cls-orientalis.ogg|thumb|]]--> || From oriens, referring to the rising of the sun |- | [[wiktionary:septentrionalis#Latin|septentriōnālis]], e (also borealis, e) || northern, of the north || <!--[[File:la-cls-septentrionalis.ogg|thumb|]]--> || From the seven stars, septentrio, the big or little dipper, or boreas, the north wind |} ==New Sentences== {| class="wikitable" style="width: 70%;" ! style="width: 30%;" | Latin ! style="width: 30%;" | English ! style="width: 40%;" | Notes |- | Septem continentēs in mundō sunt; Eurōpa, Asia, Africa, Oceania, America septentriōnālis, America austrālis, et Antarctica. || There are seven continents in the world; Europe, Asia, Africa, Oceania, North America, South America, and Antarctica.  || |- | Ex oriente lūx, ex occidente lēx. || Light from the east, law from the west. || A phrase sometimes used in the study of civilization; the great religions of humanity arise in the East, bringing metaphorical light, and its legal and political traditions arise in the West, bringing, well, whatever it is they bring. |- | Ad occidentālem navigāvērunt. || They sailed to the west. || |- | In terram austrālem iter fēcit. || He made a journey into the south (to the southern land). || |- | Ursa Minor constellātiō in caelō septentriōnālī sita est. || Ursa Minor (the smaller bear) is a constellation located in the northern sky. || |- | Scīpiō Africānus Hannibalem et Carthāginiēnsēs vīcit. || Scipio Africanus conquered Hannibal and the Carthaginians. || Africanus was a name given him in honor of his victory in Africa, not because he was African. |- | Britanniārum Rēgnum est lībera Eurōpae Occidentālis cīvitās, quae ex cīvitātibus Anglia, Scotia, Cambria, in Britannia Maiore īnsula, et Hibernia Septentriōnālī, in Hibernia, cōnstat. || The United Kingdom (kingdom of the British states) is an independent country in western Europe, which consists of England, Scotland, Wales on the island of Great Britain, and Northern Ireland, on Ireland. || Check out the [[w:la:Britanniarum regnum|Vicipaedia]] article |- | Cīvitātēs Foederātae Americae (CFA) || The United States of America (USA) || There is some dispute about this designation, and the [[w:la:Civitates Foederatae Americae|Vicipaedia article]] is much less detailed than others. |- | Sententiae Franciae est “lībertās, aequālitās, frāternitās.” || The motto of France is “liberty, equality, fraternity.” || Check out the [[w:la:Liberté, égalité, fraternité|Vicipaedia article]] |- | Caput et urbs maxima Germāniae est Berōlinum. || The capital and largest city of Germany is Berlin. || Check out the [[w:la:Berolinum|Vicipaedia article]]. Note that “nōmen incolārum – Germānus” (the name of inhabitants is “German” but the adjective is “Germānicus.” |- | Folium acernum in vexillō Canadae est. || There is a maple leaf on the flag of Canada.  || |- | Crēdisne solum ūnum ūniversum esse? || Do you believe that there is only one universe? || |- | In ūniversā terrā. || In the whole earth.  || |- | Ūniversitās Bonōniēnsis || University of Bologna (the oldest university) || From ''[[wiktionary:Bononia#Latin|Bonōnia]]'' |- | Alma māter || nurturing mother || |- | Ūniversitās Oxoniēnsis || University of Oxford || |- | Ūniversitās Cantabrigiēnsis || University of Cambridge || |- | Ūniversitās Harvardiana || Harvard University || |- | Nuntiī Latīnī Occidentālis Studiōrum Ūniversitās Vasingtoniēnsis Bellinghamiae || The [https://nuntiilatini.com Latin News podcast] of Western Washington University at Bellingham || not a particularly old university, but they put on an enjoyable podcast with a transcript, and you should definitely check it out if you want to improve your Latin. |- | Alma mater || nurturing mother || |- | Magnā cum laude || with great praise || |- | Summā cum laude || with greatest praise || |- | Artium Baccalaureus / Magister|| Bachelor / Master of Arts || |- | Scientiae Baccalaureus / Magister || Bachelor of Science || |- | Philosophiae Doctor || Ph.D., Doctor of Philosophy || |- | Gradum / dīplōma suscēpit. || He received his degree/diploma  || ie, graduated |- | Honōris causā || for the reason of honor  || an honorary degree |} ==Practice== {| class="wikitable style="width: 70%;" ! colspan="2" | Practice and learn the words and phrases in this lesson |- | '''Step one''' || First learn the words using this lesson: * [https://community-courses.memrise.com/community/course/748509/carpe-lanams-latin/39/ Locative and Geography vocabulary] |- | '''Step two''' || Next try learning and writing the sentencing using this: * [https://community-courses.memrise.com/community/course/906792/carpe-lanams-latin-sentences/40/ Locative and Geography Sentences] |- | colspan="2" | Note that the Memrise stage covers the content for all lessons in each stage. <br/>If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |} ''Valēte et bonam fortūnam!'' {{subpage navbar}} {{CourseCat}} gh2350a9hsupn7w56lbaz3vhq0e0dca 0 267737 2801327 2799336 2026-03-29T16:56:31Z Saltrabook 1417466 2801327 wikitext text/x-wiki ==Introduction == The Maritime Health research institutes that was created in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes. Due to these changes, the objectives in the "Maritime Health Research and Education-NET" has ben transferred to the John Snow and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. <nowiki>https://doi.org/10.1186/s13063-021-05500-5</nowiki>. # Screening for Prediabetes, T2 DM, and Hypertension in all health examinations # Health promotion program integrated with the Prediabetes, T2DM & HTN screening program # Systematic Literature Reviews # Standardized health questionnaires research The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.<ref>‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals</ref><ref>[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]</ref> We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''<ref> [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]</ref>,<ref>https://pubmed.ncbi.nlm.nih.gov/24737131/</ref> <ref>https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y</ref>''' The research program includes monitoring of the main topics of the EU-Occupational Health strategy ==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] == == [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] == ==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] == ==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]== ==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4: Effectiveness of training in prevention for type 2 diabetes]]== ==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] == ==[[/Systematic Reviews/|Systematic Review Studies]] == ==[[/Organisation / |Organisation]] == ==[[/Presentations pptx /|Presentations]] == ==[[/Invitations for collaboration/]] == ==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]] == == Objectives == # International prospective exposure and health risk cohort studies with maritime students and workers # All other industries are invited to promote the similar research and education goals # Using the standard protocols with different themes # Harmonise exposure and outcome information by using standard questionnaires # Objective and subjective assessments of workplace hazards exposures # Develop and validate Job-exposure Matrices # Systematic reviews and pooled studies from the cohort rounds # Cohort-Linkage to pre-entry and follow-up health exams and other health registers # Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension] # International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing VCT@WORK'''] # Make training materials based on the cohort study and the clinical study outcomes and other scientific sources. .... # OHS training to maritime doctors, seafarers, fishermen, students, and others # Integrate research methodology in the supervision of student’s thesis work # Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data # Keep the Excel data file copies safely (producing country and supervisor) # Disseminate the knowledge in publications and organize seminars/webinars/symposia ==Phases of the preventive program for single industry== # Epidemiological standardized questionnaire studies # Occupational therapists investigate troubled job positions # Occupational Medical Doctors supply with their patients from the workplace # Statistics of work accidents in the specific areas of the workplace years # Dialogue with companies to improve safety and ergonomics positions # Improve and continue if they have done well ==Ethical requirements == The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect "personal data" as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version''] Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials<ref>https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse</ref> ==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] == The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section. ==[[/Contribution to UNs 17 Sustainable Development Goals/]] == Goal 3: Good health and well-being for all workers <br> Goal 4: Quality Education<br> Goal 5: Gender Equity<br> Goal 8: Decent Work and Economic Growth<br> Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)<br> Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)<br> Goal 14: Life underwater observations on compliance with good waste management <br> Goal 17: Partnerships to achieve the Goals<br> ==Integrated Mental Health and Work Policy OECD's recommendation == To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement; Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities. Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace <ref>https://legalinstruments.oecd.org/public/doc/334/334.en.pdf</ref> ==Contribution to quality education == The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases. ===== For the medical doctors doing health examinations===== Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. <br> https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases<br> ===== For the students ===== Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data. They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research. ==Links to relevant organizations, documents, and funds == [http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health] [http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health] [[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]] [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST] [http://dimopex.eu/about/ DiMoPEx (CA 15129)] [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications] [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda] [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values] [https://www.europeansurveyresearch.org The European Survey Research Association] [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)] [https://ec.europa.eu/esf/main.jsp?catId=67&langId=en&newsId=9691 The European Social Fund] == https://www.fi-compass.eu/esif/emff<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] <br> [https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] <br> [https://www.seafarerstrust.org/about/ ITF Seafarers Trust]<br> [https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation] ==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]== Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more). ==Subpages== {{Subpages/List}} '''Bibliographic References.''' ==References== {{reflist}} {{CourseCat}} cuxp2y4mg9tz3pbywaxnte8tcu064c0 0 267893 2801304 2214476 2026-03-29T12:34:59Z Anikmolla786 3007678 Added missing apostrophe for plural possessive (“suitors’ plot”). 2801304 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ships route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelopes sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helens actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In todays world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." as9sgh1ikchuetxspzx9t5ql3ejctsr 2801305 2801304 2026-03-29T12:35:55Z Anikmolla786 3007678 Added missing apostrophe (“Penelope’s sister”). 2801305 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ships route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helens actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In todays world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." c88tpibwkvgumxeim7mss7od07jr07m 2801306 2801305 2026-03-29T12:36:46Z Anikmolla786 3007678 Added apostrophe for possessive (“ship’s route”). 2801306 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helens actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In todays world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." ax1ioyn0isigdm4odtm0hew2j770t8q 2801307 2801306 2026-03-29T12:37:41Z Anikmolla786 3007678 Added possessive apostrophe (“Odysseus’s wealth”). 2801307 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helens actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In todays world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." ociwkuu3oi3fnh9bfp8eqhsze1nncwh 2801308 2801307 2026-03-29T12:38:30Z Anikmolla786 3007678 Added missing apostrophe (“Helen’s actions”). 2801308 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helen’s actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In todays world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." kd581js53hymdtt1cpczc0032yhic66 2801309 2801308 2026-03-29T12:39:18Z Anikmolla786 3007678 Added apostrophe (“today’s world”). 2801309 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helen’s actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In today’s world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the upmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." odiah8mgnctj2wsd0lom2xyem5hh53h 2801310 2801309 2026-03-29T12:40:04Z Anikmolla786 3007678 Changed “upmost” to “utmost” (standard idiom). 2801310 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not know even know who his guest were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helen’s actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In today’s world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the utmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." mz3aso0mgbctvgon4r807liax1oe4t6 2801311 2801310 2026-03-29T12:41:12Z Anikmolla786 3007678 Removed duplicate “know”; changed “guest” to “guests” and “were” to “were” (already). 2801311 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope-The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not even know who his guests were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helen’s actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In today’s world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the utmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." 5qzcrwsmyqqr2qf43kzhwyypzyac472 2801312 2801311 2026-03-29T12:42:28Z Anikmolla786 3007678 Added spaces around dash for readability. 2801312 wikitext text/x-wiki == Summary of Book 4 == Telemachus and Pisistratus arrive in Sparta where a celebration of a wedding is being held at Menelaus's palace. They are invited inside to eat with Menelaus and his wife Helen, when they recognize Telemachus to be the son of Odysseus and begin recounting stories of Odysseus' heroic acts in the battle with Troy including the Trojan Horse. After the stories of Odysseus are told, Menelaus begins to talk about his own journey home from Troy and how difficult it was to get back. He had been stranded on an island on the coast on Egypt and the gods would not send any wind to help guide his ship back home. Menelaus then comes in contact with the goddess Eidothea, who pities him and explains that if he can catch the sea god Proteus and hold him down until he becomes too tired to fight back, Proteus will tell him how he can appease the gods in order to return home. Menelaus is instructed to kill 100 cows at the river of Egypt and after he does so he is able to return back home. While Menelaus had Proteus captured, Proteus revealed to him the fate of some of the other people who also had difficulty returning home from Troy. Menelaus then begins to tell the story of Agamemnon and Ajax, who both met deadly fates on their journey home. Ajax's ship struck a rock and he drowned in the ocean, while Agamemnon was able to make it all the way home despite minor struggles. However, Agamemnon was killed by Aegisthus over dinner, and all of Agamemnon's men were also killed. After hearing about the deaths of Agamemnon and Ajax, Menelaus then reveals that he knows where Odysseus is, and that he is trapped on an island with Calypso. After hearing about his fathers whereabouts, Telemachus thanks Menelaus and leaves to sail back to Ithaca. It is then revealed that while Telemachus was gone, the suitors at his house have devised a plot to murder Telemachus as he returns to Ithaca, saying they will cut off his ship’s route and carry out the murder there. Medon overhears the suitors’ plot and informs Penelope of their plot. Penelope breaks down as she now fears losing both her husband and her son, but she is visited by a phantom in the form of Iphtime, Penelope’s sister, who informs her that the gods will keep Telemachus safe on his journey home. == Characters == Telemachus- He is the son of Odysseus, and is on a quest to find out what happened to his father. Pisistratus- He is the son of Nestor, and is accompanying Telemachus on his journey. Menelaus- He is the King of Sparta who also fought alongside Odysseus in the city of Troy. Helen- She is the Queen of Sparta, and is married to Menelaus. She is also a child of Zeus, who is a powerful god in Greek mythology. Asphalion- A house slave of Menelaus. Eidothea- The goddess who pitied Menelaus and told him how he could work to get home. Proteus- He is described as a "deathless old sea god" who holds the answers to how Menelaus could get home. He had to be pinned down by Menelaus until he got tired and stopped shapeshifting. Calypso- A goddess who kept Odysseus imprisoned on her island. The suitors- A group of single men who are competing to marry Penelope and gain Odysseus’s wealth. Penelope – The wife of Odysseus and mother of Telemachus. == Ancient Worldview == === Host and party etiquette: === The larger portion of book 4 takes place at Menelaus' palace, where Telemachus and Pisistratus are invited in to feast. Throughout this encounter, the importance of being a gracious and generous host is revealed as servants wash their guests hands with "water in a golden jug" (154) and get them whatever food or specific thing they request. Before Menelaus even knows who his guests are, he tells them that they can eat whatever they want and share the meal with them. Although it is subtle, the fact that Menelaus did not even know who his guests were but still showed generous hospitality towards them reinforces just how important being a gracious host was in this ancient worldview. Further on in the book, when everyone is crying at dinner over their lost friends and family, Helen steps in to "mix the wine with drugs to take all the pain and rage away"(159). This is an unnecessary but very gracious thing that Helen did especially to Telemachus who has been devastated about his father Odysseus for a long time. Helen’s actions show us that in this period of time it was very important to meet the needs of your guests and take care of them as well as you can, even if that means going out of your way to help them. By the end of the book Telemachus is eager to leave after learning that his father is still alive, and still Menelaus continues his gracious hosting by offering to shower Telemachus with expensive gifts of horses, but when Telemachus declines he still gives him a "finely crafted bowl of purest silver" (171). In today’s world we do not say goodbye to our guests by giving them gifts, but the ancient worldview of host and party etiquette made this a necessity in order to be considered a hospitable host. === Family legacy: === In the entire book of the Odyssey, a lot of characters are identified by who they are related to, including in book 4 where Pisistratus is referred to as "Nestors son" more than he is called by his real name. The way that these characters are identified show how important family legacy was in these ancient times. Even when Menelaus brings Telemachus and Pisistratus into his home, he was even more ecstatic when he realized who his guests were and that one of them was the son of the great Odysseus. Menelaus says he "always thought that I would greet that friend with warmth beyond all other"(157), which shows that even though he has never met Telemachus, he is greeting him with the utmost respect because of who his father is. Another story in this book that shows how important family legacy is in ancient world-views is the story that Menelaus tells of Aegisthus killing Agamemnon after taking his wife. After Aegisthus killed Agamemnon, Agamemnons son Orestes then took revenge on Aegisthus and killed him, to which Menelaus says his "heart was warmed inside" (169) after hearing that news. Menelaus and all of the gods were proud of Orestes because he had avenged his fathers death and essentially brought honor back to his family. Even in murder and death, family legacy is still a huge factor of the ancient worldview that effects how you and your actions are perceived. === Diminution of women in the Odyssey: === Throughout the Odyssey, the role of women is often subservient to the men around them. Most striking, is the moment when Helen describes her reason for running off to Troy with her lover, Paris. In the translation we are reading by Emily Wilson, Helen describes meeting Odysseus in the streets of Troy. After their meeting, Odysseus leaves, slaughtering Trojans as he goes. Helen says that "The Trojan women keened in grief, but I was glad - by then I wanted to go home. I wished that Aphrodite had not ''made me go crazy'', when she took me from my country [...]" (italics added for emphasis) In an older translation, (E.V. Rieu 1946, revised Peter V. Jones 1991) Helen describes herself as 'suffering a change of heart' and "repenting the blindness Aphrodite sent me." Both translations see Helen shirking responsibility for the destruction of Troy by blaming Aphrodite. However, the more recent translation paints Helen's tryst as an act committed by a trophy wife who had been elevated beyond her station. The idea that the gods made her "go crazy" sounds like the excuse of a tertiary character in a daytime soap opera. Throughout many translations, Helen never seems to be worthy of the epithet: "the face that launched a thousand ships." 00vtr0drmn6mbmn9zfqkhxynbm9j20i 0 285380 2801388 2800971 2026-03-30T01:30:59Z Young1lim 21186 /* Applications */ 2801388 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260330.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> p3j300s03oesryt3329j7m3d8g2ciuz 0 285908 2801451 2800272 2026-03-30T06:16:23Z IronGargoyle 23939 /* What is disappointment? */ remove AI nonsense 2801451 wikitext text/x-wiki {{title|Disappointment:<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://youtu.be/BVUPkwnCYao}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features.]] Have you ever received a lower grade than you expected? Have you ever been let down by someone? If so, you may have experienced disappointment. Disappointment is one of the most common and frequently experienced negative [[wikipedia:Emotion|emotions]] (Van Dijk & Zeelenberg, 2002). Emotions are complex, coordinated, [[wiktionary:multidimensional|multidimensional]] responses that help us in our everyday lives (Izard, 2010); they motivate us to cope, communicate, and [[wiktionary:adapt|adapt]] to the world around us (Izard, 2010). Although there is no official definition for the word 'emotion' (see Mulligan & Scherer, 2012), it is agreed that emotions also involve feelings, bodily arousal, purpose, and expression (see Figure 1) (Izard, 2010). This chapter describes disappointment, explores the causes of disappointment, and discusses what can be done to manage disappointment. {{RoundBoxTop|theme=13}} [[File:Crystal Clear app ktip.svg|right|90px|]] '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== Disappointment is an emotion that occurs when you compare the actual outcome to the perceived better outcome that did not occur, or when your expectations are not met (Zeelenberg et al., 1998a, b). In the context of disappointment, outcomes could be anything, for example, a friend forgetting to do the task you asked them to do, receiving a lower mark on an assignment than you expected, or listening to the new album of your favourite artist and discovering that you don't like any of the songs. Disappointment is all about expectations, and reflecting on what ''could'' have happened (Zeleenberg et al., 1998a, b). Disappointment is a decision-making emotion and has historically been researched using forced choice tasks, where participants are forced to choose between two options, or asking participants to recall moments when they have experienced disappointment (Zeleenberg et al., 1998a; see [[The Regret and Disappointment Scale|the regret and disappointment scale]] for a way to measure disappointment). Researchers generally use choice tasks when researching how disappointment works and various aspects of disappointment, and recall tasks are generally used to define or gain insight on everyday disappointment. One downside to researching disappointment in this way is that disappointment has been shown to increase in forced choice tasks (Matarazzo et al., 2021). However, Matarazzo et al. (2021) found that the thinking, action tendencies, and feelings of disappointment in forced choice tasks are possibly due to the nature of forced choice tasks. Like [[wikipedia:Envy|envy]] or [[wikipedia:Empathy|empathy]], disappointment is a cognitively complex emotion (Ramachandran & Jalal, 2017). Disappointment typically involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999). In some cases, disappointment can look like [[wikipedia:Depression_(mood)|depression]], [[wikipedia:Sadness|sadness]], [[wikipedia:Embarrassment|embarrassment]], or [[wikipedia:Extraversion_and_introversion|introversion]]; as the disappointed individual may withdraw from social situations, feel as if they have experienced a loss, try to avoid similar situations, or not want to participate in general. Disappointment can be paralysing, especially experiencing a string of disappointing events back-to-back, however, people are less likely to hold on to their disappointment and are more likely to move on from the experience in a relatively short amount of time (Zeleenberg et al., 1998a, b). See Table 1 for examples of emotions similar to disappointment. Table 1 ''Emotions Similar to Disappointment'' {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:100%;" |- | style="width:10%;" | '''Emotion''' | style="width:90%;" | '''Definition''' |- |[[w:Regret|Regret]] |A cognitively complex negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a, p.222). |- |[[w:Anger|Anger]] |A simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al.,1999). |- |[[wiktionary:disillusionment|Disillusionment]] |A complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. |} {{Robelbox|theme={{{theme|2}}}|title=Spotlight: The history of disappointment}} The history of disappointment research begins with regret. Many researchers, including David Bell, [[wikipedia:Graham_Loomes|Graham Loomes]], and [[wikipedia:Robert_Sugden_(economist)|Robert Sugden]], were exploring decision making under uncertainty and the emotions that accompany these decisions. After simultaneously publishing their regret theories in 1982, Bell (1985), and Loomes and Sugden (1986) developed their theories of disappointment. A key assumption these theories make is that decision makers anticipate emotions and take them into account when making a decision (Zeleenberg et al., 1998b, 2000). According to Bell (1985), disappointment "is a psychological reaction to an outcome that does not match up with expectations" (p. 1). Bell (1985) believed that perceived disappointment changes the desirability of the outcome and influences how people will act. According to Loomes and Sugden (1986), "when considering any uncertain prospect, an individual forms some ''prior expectation'' ... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (p.271). Loomes and Sugden (1986) have acknowledged that they share the same basic intuition about disappointment as Bell (1985). {{Robelbox/close}} === Types of disappointment === There are two widely recognised types of disappointment. These are outcome-related disappointment [ORD] and person-related disappointment [PRD] (van Dijk & Zeelenberg, 2002). ORD occurs when the expected pleasurable outcome does not occur (van Dijk & Zeelenberg, 2002). This type of disappointment is often researched using forced choice tasks. People who experience ORD may feel [[wiktionary:hopeless|hopeless]] or empty, want a second chance, or try harder to change the outcome next time (van Dijk & Zeelenberg, 2002). PRD occurs when you attribute the undesirable outcome to another person (van Dijk & Zeelenberg, 2002). This type of disappointment is not often focused upon, however, it is probably the most commonly experienced type. People who experience PRD may feel abandoned or distanced from the other person, disapprove of them, and ignore or avoid them (van Dijk & Zeelenberg, 2002). One important consideration is that van Dijk and Zeelenberg (2002) assume that PRD is cause by another person, however, one can be disappointed in themselves. While there has not been research into dimensions of PRD, it would be useful to refine the idea of PRD or research self-disappointment and determine if it should be included in PRD or if it should be considered self-related disappointment. === Test yourself === <quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + PRD - ORD {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - PRD + ORD </quiz> == What causes disappointment? == [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Brain image highlighting the posterior, mid, and anterior insula.]] Disappointment is caused by thoughts and [[wikipedia:Cognition|mental processes]] that originate in the [[wikipedia:Cerebral_cortex|cerebral cortex]]. Multiple brain regions have been shown to be active during disappointment or to contribute to the process of disappointment, namely the [[wikipedia:Insular_cortex|insula]] (see Figure 3), and various regions of the [[wikipedia:Prefrontal_cortex|prefrontal cortex]] (see Figure 4) (Chua et al., 2009; Kalat, 2019; Mohr et al., 2010). Due to the complexity of disappointment, some brain regions work together to produce disappointment. === Insula === The insula is the brain region responsible for knowing what actions are caused by the self and what actions are not, as well as learning and processing risk and uncertainty (Farrer & Frith, 2002). The [[wikipedia:Anatomical_terms_of_location#Anterior_and_posterior|anterior]] insula monitors, evaluates, and consciously represents emotions and feelings that arise from bodily states monitored by the [[wikipedia:Anatomical_terms_of_location#Anterior_and_posterior\|posterior]] insula, including risk (Craig, 2009). When individuals experience disappointment their anterior insula becomes active (Chua et al., 2009; Mohr et al., 2010); it is also active in the presence of potential loss (Mohr et al., 2010). This could be because individuals can predict that they will feel disappointed if loss was to occur. === Prefrontal cortex === [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 4.'' Brain image highlighting the prefrontal cortex (PFC).]] The prefrontal cortex [PFC] is a large section of the brain that is involved various processes, including decision making, working memory, emotional reactions, and movement (Kalat, 2019). It has been shown that the anterior regions of the PFC are responsible for decision making, evaluating which course will provide the best outcome, and determining the probability of achieving a good outcome (Kalat, 2019). This is why [[wikipedia:Lateralization_of_brain_function|hemispherical differences]], the [[wikipedia:Ventromedial_prefrontal_cortex|ventromedial PFC]] (see Figure 5), [[wikipedia:Orbitofrontal_cortex|orbitofrontal cortex]] (see Figure 6), and [[wikipedia:Dorsomedial_prefrontal_cortex|dorsomedial PFC]] (see Figure 5) are considered to be contributing factors to the experience of disappointment (Chua et al., 2009; Davidson, 2004; Kalat, 2019). ==== Hemispherical differences ==== The right PFC is sensitive to punishment and controls impulsive behaviour, and the left is associated with coping, resilience, and psychological wellbeing (Davidson, 2004). When an individual experiences damage to their right PFC, cues that would normally signal danger are no longer received and the individual acts impulsively (Davidson, 2004). Therefore, when an individual encounters a risky or potentially disappointing situation, the right PFC activates and sends a "no-go" message to avoid the situation and perceived disappointment. ==== Ventromedial prefrontal cortex ==== [[File:Cortical midline structures.png|thumb|''Figure 5.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), and the dorsomedial prefrontal cortex (DMPFC).]] The ventromedial PFC [VMPFC] learns what choices are beneficial and what choices are not, adjusting decision making accordingly (Kalat, 2019). The VMPFC also monitors confidence in one's decisions (Kalat, 2019). As the VMPFC is connected to the insula, it is able to attach emotions to choices and other stimuli that is being considered (Craig, 2009). For example, if you feel confident that you have made the right decision and will achieve a good outcome, you will feel more disappointed than you would have felt if you were less confident that you will achieve a good outcome. Damage to the VMPFC has been shown to cause impairments in the ability to make considered decisions. Individuals with VMPFC damage tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). This can lead to constant or [[wiktionary:chronic|chronic]] disappointment as the VMPFC cannot adjust decision making based on previous experience. Below is an example of how the VMPFC works. {{Robelbox|theme=6|title= Case study|width=1000px}}<div style="{{Robelbox/pad}}"> You are playing [[w:Uno|Uno]]. You have 5 cards to play, and so do your four opponents. You only have number cards, and based off of the cards that have already been played, your opponents must have at least one draw 4 card. You strategically match the number that was last played so that you change the colour of the deck; this makes it more likely that a draw 4 card will be played after your turn, and not used on you. If the draw 4 card is used on you, you will feel more disappointed as your strategy did not work. If the draw 4 card is not used on you, you will feel good about your strategy and continue to use it in the future. </div> {{Robelbox-close}} ==== Orbitofrontal cortex ==== [[File:MRI of orbitofrontal cortex.jpg|alt=Orbitofrontal cortex highlighted on brain MRI|thumb|''Figure 6.'' Approximate location of the orbitofrontal cortex (OFC) on an MRI.]] The orbitofrontal cortex [OFC] responds to information from the VMPFC. It is the part of the brain that changes and updates expected outcomes of our actions based on current circumstances (Kalat, 2019). The OFC actively differentiates between 'disappointing' (not good) and 'not disappointing' (good) options or outcomes, and chooses the option that is most likely to lead to a 'not disappointing' outcome (O'Doherty, 2004). Damage or inactivity of the OFC is associated with impulsive and otherwise poor decision making, leading to disappointing outcomes (Kalat, 2019). Below is an example of how the OFC decides what to do. {{Robelbox|theme=6|title= Case study|width=1000px}}<div style="{{Robelbox/pad}}"> You decide to go to 54 Benjamin for breakfast. When you arrive, you remember that the last time you went there you didn't like the drink you ordered. You also remember that when you were there your friend ordered a drink that you liked the look of, and your friend said it was quite good. This time you order what your friend had last time and you are not disappointed. </div> {{Robelbox-close}} ==== Dorsomedial prefrontal cortex ==== The dorsomedial PFC [DMPFC] plays a role in both cognition and emotion (Eickhoff et al., 2016). The DMPFC is responsible for anticipating rewards, monitoring performance, selecting actions, and signalling errors and [[wiktionary:adverse|adverse]] outcomes (Taren et al., 2011); and is activated when individuals experience disappointment (Chua et al., 2009). The DMPFC regulates responses to unpredictable negative [[w:Stimulus (psychology)|stimuli]] and regulates [[wiktionary:reappraisal|reappraisal]] and [[wikipedia:Distraction|distraction]] (Helion et al., 2019). Once an emotion is identified, the DMPFC shapes the intensity of the emotion based on the individual's goals (Helion, Krueger, & Ochsner, 2019). The more invested or important the outcome is, and the more adverse the opposite outcome is, the more disappointment is experienced. Below is an example of how the DMPFC works. {{Robelbox|theme=6|title= Case study}}<div style="{{Robelbox/pad}}"> Your assignment is finally graded and you receive a lower grade than you expected. You feel disappointed in yourself for achieving a lower grade than usual. However, your disappointment starts to disappear when you think about how important the class is to you. You know that it is important to do well, but this particular class is a major, not a core class, so you know that as long as you pass the class you are doing well. </div> {{Robelbox-close}} === Test yourself === <quiz display="simple"> {Which brain region monitors who causes what action? |type="()"} + Insula - OFC {Which brain region modifies emotion intensity? |type="()"} - PFC + DMPFC {Damage to which brain region causes people to confidently make poor decisions? |type="()"} - OFC + VMPFC </quiz> ==How can disappointment be managed?== After learning about the mental processes that contribute to disappointment, it may feel as if disappointment is inevitable. After all, if you were to take a minute to think of the last time you felt disappointed, you would probably be able to think of an event that occurred in the last month. Because disappointment is so unpleasant, researchers have found different ways to manage disappointment. The main three strategies to manage disappointment are lowering expectations, living up to expectations, and avoiding risk-taking (van Dijk et al., 2003; Zeleenberg et al., 1998a, 2000). But should disappointment be managed? === Lowering expectations === When an unfavourable outcome occurs, so does disappointment. One way to combat disappointment is to lower expectations (van Dijk et al., 2003). In general, people tend to lower their expectations when feedback or the outcome is anticipated in the near future. For example, as a patient gets closer to their surgery, their expectations of a positive outcome could reduce until the patient no longer wants surgery (van Dijk et al., 2003). de Meza and Dawson (2021) have found that people with mistaken expectations or unrealistic expectations (i.e., unrealistic optimism) experience lower levels of [[wikipedia:Well-being|well-being]]. In the long-run, realists (people who have a realistic world view) have significantly higher wellbeing than both [[wikipedia:Pessimism|pessimists]] and [[wikipedia:Optimism|optimists]] (de Meza & Dawson, 2021). Overall, lowering expectations leads to a lower chance of experiencing disappointment, however, it is important to keep in mind that this can lower your overall wellbeing if you gain a pessimistic outlook (de Meza & Dawson, 2021; van Dijk et al., 2003). === Living up to expectations === Disappointment, like many other emotions, can be anticipated. If disappointment is anticipated, people attempt to avoid it by living up to expectations (Zeleenberg et al., 2000). In this instance, disappointment is a motivator, either to decrease the likelihood of disappointment, or to increase the likelihood of a desired outcome (Zeleenberg et al., 2000). To live up to expectations, the amount of effort that an individual puts in must be able to increase the likelihood of a good outcome. Therefore, this method is only useful when an individual's effort is able to decrease the probability of disappointment and is only appropriate when effort or something controlled by the individual can lead to obtaining the desired outcome (van Dijk et al., 2003). For example, extra study and preparation can result in a better chance at passing a test which would decrease disappointment, but extra time studying a dice will not result in a better chance at predicting which number it will land on. === Avoid risk-taking === A more proactive approach to managing disappointment is avoiding it. Choosing safe alternatives that lead to known outcomes do not risk disappointment (Zeleenberg et al., 1998a, b, 2000). This approach could be called [[wikipedia:Risk_aversion|risk aversion]] (Zeleenberg et al., 1998b, 2000). Below is an example of how risk-taking can be avoided, however, disappointment is not always avoidable. This begs the question, should disappointment be managed or avoided? {{Robelbox|theme=6|title= Case study}}<div style="{{Robelbox/pad}}"> You are picking ice cream at a new restaurant and you have three options, vanilla, chocolate, and strawberry. You like all three options, however, you find that some chocolate ice creams are disgusting, and you only like specific strawberry ice creams. Vanilla is not your favourite flavour but you find it edible even if you do not completely like it. You pick the vanilla ice cream. </div> {{Robelbox-close}} === Should disappointment be managed? === Disappointment helps us improve our circumstances, improve ourselves, and alerts us to our own expectations and from this we readjust our expectations or adapt to avoid similar disappointing experiences in the future. For example, if someone continually disappoints us we then decide to distance ourselves from that person, or if we are disappointed in the feedback we receive we then work to achieve an acceptable standard, and if we travel with a specific company and their service is disappointing the next time we travel we will most likely try a different company. If disappointment is interpreted as a message that needs to be heard and acted upon, then disappointment occurs less and is perceived as less detrimental (Grainger, 1991). {{RoundBoxTop|theme=5}} '''Questions to consider:''' * What do you think about disappointment? * Is disappointment good or bad? Why? {{RoundBoxBottom}} ==Conclusion== Disappointment is a cognitively complex emotion that occurs when your expectations are not met (Zeleenberg et al., 1998a). Whether you experience ORD or PRD, the insula, VMPFC, OFC, and DMPFC work together to choose the most beneficial choice, determine how likely the beneficial option is, and signal when adverse outcomes occur (Craig, 2009; Kalat, 2019; Taren et al., 2011). Sometimes disappointment is unexpected, however, when it is anticipated, techniques such as lowering expectations, living up to expectations, and avoiding risk-taking are effective in reducing disappointment (van Dijk et al., 2003; Zeleenberg et al., 1998a, 2000). Although disappointment is a negative emotion, it helps us to adapt, avoid negative outcomes, and improve ourselves (Grainger, 1991). Overall, successfully managing expectations is a difficult task, but when done well, reduces disappointment. == See also == * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2022/Resentment|Resentment]] (Book chapter, 2022) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in decision making under uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, H. F., Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Craig, A. D. (2009). How do you feel - now? The anterior insula and human awareness. ''Nature Reviews: Neuroscience, 10''(1), 59–70. https://doi.org/10.1038/nrn2555 Davidson, R. J. (2004). 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Not having what you want versus having what you do not want: The impact of type of negative outcome on the experience of disappointment and related emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The experience of regret and disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. R., van Empelen, P., & Reinderman, D. (1998b). Emotional reactions to the outcomes of decisions: The role of counterfactual thought in the experience of regret and disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 }} ==External links== * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) * [https://mensline.org.au/how-to-deal-with-anger/how-to-deal-with-disappointment/ How to deal with disappointment] (MensLine.org) * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.ted.com/talks/dan_gilbert_why_we_make_bad_decisions Why we make bad decisions] (TED.com) * [https://www.youtube.com/watch?v=0gks6ceq4eQ You aren't at the mercy of your emotions -- your brain creates them | Lisa Feldman Barrett] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}/Top]] [[Category:Motivation and emotion/Book/Disappointment]] e78kt81u34v6mqr6y652rd7c85xsf3q 0 301040 2801316 2800407 2026-03-29T13:23:35Z D.H 52339 /* Perspective of the traveler */ Link 2801316 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 2 March 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. k) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). ==Human beings in 1911== {| class="wikitable" style="background-color:white;" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text=Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |valign=top|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |valign=top|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If we would let remain the second clock in the motion which was given to it at the start, and if we send after it the first clock after some time by a greater velocity, then at the encounter the first one would be retarded with respect to the second one; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |valign=top| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". He pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround. A similar explanation was subsequently given by Langevin (1922).<ref name=morand />}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Paradoxical?== {| class=wikitable style="background-color:white;" ! width=50% | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, [[:w:Paul Gruner|Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger|Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;" ! width=50% padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |- ! width=50% | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent and anisotropic. This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[..] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' | style="padding: 0px 20px 0px 20px;" | [..] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! width=50% | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.)}} |- ! width=50% | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=Dialog über Einwände gegen die Relativitätstheorie|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|url=https://archive.org/details/sim_naturwissenschaften_1918-11-29_6_48|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. [[Henry Brose|Brose]] (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. [[Henry Brose|Brose]] (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914|orig-date=Translation of Dutch lecture in 1913| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] 5qot36k2zbe9u7o9zn1mkw203gj3cch 2801317 2801316 2026-03-29T13:41:12Z D.H 52339 /* Perspective of the traveler */ 2801317 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 2 March 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. k) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). ==Human beings in 1911== {| class="wikitable" style="background-color:white;" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text=Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |valign=top|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |valign=top|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |valign=top|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If we would let remain the second clock in the motion which was given to it at the start, and if we send after it the first clock after some time by a greater velocity, then at the encounter the first one would be retarded with respect to the second one; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |valign=top| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround. A similar explanation was subsequently given by Langevin (1922).<ref name=morand />}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Paradoxical?== {| class=wikitable style="background-color:white;" ! width=50% | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, [[:w:Paul Gruner|Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger|Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;" ! width=50% padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |- ! width=50% | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent and anisotropic. This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[..] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' | style="padding: 0px 20px 0px 20px;" | [..] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! width=50% | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.)}} |- ! width=50% | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=Dialog über Einwände gegen die Relativitätstheorie|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|url=https://archive.org/details/sim_naturwissenschaften_1918-11-29_6_48|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. [[Henry Brose|Brose]] (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. [[Henry Brose|Brose]] (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914|orig-date=Translation of Dutch lecture in 1913| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] hq08l1sogpqla59abmjduwejl7gx4tk 0 305363 2801366 2800455 2026-03-29T21:41:35Z Dc.samizdat 2856930 /* Octahedral dipyramid */ 2801366 wikitext text/x-wiki {{Short description|Four-dimensional analog of the octahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=16-cell<br />(4-orthoplex)| Image_File=Schlegel wireframe 16-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br />(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]<br />4-[[W:Orthoplex|orthoplex]]<br />4-[[W:Demihypercube|demicube]]| Last=[[W:Rectified tesseract|11]]| Index=12| Next=[[W:Truncated tesseract|13]]| Schläfli={3,3,4}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} | Cell_List=16 [[W:Tetrahedron|{3,3}]] [[File:3-simplex t0.svg|25px]]| Face_List=32 [[W:Triangle|{3}]] [[File:2-simplex t0.svg|25px]]| Edge_Count= 24| Vertex_Count= 8| Petrie_Polygon=[[W:Octagon|octagon]]| Coxeter_Group=B₄, [3,3,4], order 384<br />D₄, order 192| Vertex_Figure=[[File:16-cell verf.svg|80px]]<br />[[W:Octahedron|Octahedron]]| Dual=[[W:Tesseract|Tesseract]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]], [[W:Regular polytope|regular]], [[W:Hanner polytope|Hanner polytope]] }} In [[W:Geometry|geometry]], the '''16-cell''' is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician [[W:Ludwig Schläfli|Ludwig Schläfli]] in the mid-19th century.{{Sfn|Coxeter|1973|p=141|loc=§&nbsp;7-x. Historical remarks}} It is also called '''C₁₆''', '''hexadecachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> or '''hexdecahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is the 4-dimesional member of an infinite family of polytopes called [[W:Cross-polytope|cross-polytope]]s, ''orthoplexes'', or ''hyperoctahedrons'' which are analogous to the [[W:Cctahedron|octahedron]] in three dimensions. It is Coxeter's <math>\beta_4</math> polytope.{{Sfn|Coxeter|1973|pp=120=121|loc=§&nbsp;7.2. See illustration Fig 7.2<small>B</small>}} The [[W:Dual polytope|dual polytope]] is the [[W:Tesseract|tesseract]] (4-[[W:Hypercube|cube]]), which it can be combined with to form [[W:Compound of tesseract and 16-cell|a compound figure]]. The cells of the 16-cell are dual to the 16 vertices of the tesseract. == Geometry == The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=polytopes ordered by size and complexity|group=}} Each of its 4 successor convex regular 4-polytopes can be constructed as the [[W:Convex hull|convex hull]] of a [[W:Polytope compound|polytope compound]] of multiple 16-cells: the 16-vertex [[W:Tesseract|tesseract]] as a compound of two 16-cells, the 24-vertex [[24-cell]] as a compound of three 16-cells, the 120-vertex [[600-cell]] as a compound of fifteen 16-cells, and the 600-vertex [[120-cell]] as a compound of seventy-five 16-cells.{{Efn|There are 2 and only 2 16-cells inscribed in the 8-cell (tesseract), 3 and only 3 16-cells inscribed in the 24-cell, 75 distinct 16-cells (but only 15 disjoint 16-cells) inscribed in the 600-cell, and 675 distinct 16-cells (but only 75 disjoint 16-cells) inscribed in the 120-cell.}} {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === {| class="wikitable floatright" !colspan=2|Disjoint squares |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''xy'' plane |- |( 0, 1, 0, 0)||( 0, 0,-1, 0) |- |( 0, 0, 1, 0)||( 0,-1, 0, 0) |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''wz'' plane |- |( 1, 0, 0, 0)||( 0, 0, 0,-1) |- |( 0, 0, 0, 1)||(-1, 0, 0, 0) |} |}The 16-cell is the 4-dimensional [[W:Cross polytope|cross polytope (4-orthoplex)]], which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in the 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} The 16-cell constitutes an [[W:Orthonormal basis|orthonormal ''basis'']] for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes. === Structure === The [[W:Schläfli symbol|Schläfli symbol]] of the 16-cell is {3,3,4}, indicating that its cells are [[W:Regular tetrahedron|regular tetrahedra]] {3,3} and its [[W:Vertex figure|vertex figure]] is a [[W:Regular octahedron|regular octahedron]] {3,4}. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its [[W:Edge figure|edge figure]] is a square. There are 4 tetrahedra and 4 triangles meeting at every edge. The 16-cell is [[W:Totally bounded|bounded]] by 16 [[W:Cell (mathematics)|cells]], all of which are regular [[W:Tetrahedron|tetrahedra]].{{Efn|The boundary surface of a 16-cell is a finite 3-dimensional space consisting of 16 tetrahedra arranged face-to-face (four around one). It is a closed, tightly curved (non-Euclidean) 3-space, within which we can move straight through 4 tetrahedra in any direction and arrive back in the tetrahedron where we started. We can visualize moving around inside this tetrahedral [[W:Jungle gym|jungle gym]], climbing from one tetrahedron into another on its 24 struts (its edges), and never being able to get out (or see out) of the 16 tetrahedra no matter what direction we go (or look). We are always on (or in) the ''surface'' of the 16-cell, never inside the 16-cell itself (nor outside it). We can see that the 6 edges around each vertex radiate symmetrically in 3 dimensions and form an orthogonal 3-axis cross, just as the radii of an octahedron do (so we say the vertex figure of the 16-cell is the octahedron).{{Efn|name=octahedral pyramid}}}} It has 32 [[W:Triangle (geometry)|triangular]] [[W:Face (geometry)|faces]], 24 [[W:Edge (geometry)|edges]], and 8 [[W:Vertex (geometry)|vertices]]. The 24 edges bound 6 [[W:Orthogonal|orthogonal]] central squares lying on [[W:Great circle|great circles]] in the 6 coordinate planes (3 pairs of completely orthogonal great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the [[W:Apex (geometry)|apex]] of a canonical [[W:Octahedral pyramid|octahedral pyramid]].{{Efn|Each vertex in the 16-cell is the apex of an [[W:Octahedral pyramid|octahedral pyramid]], the base of which is the octahedron formed by the 6 other vertices to which the apex is connected by edges. The 16-cell can be deconstructed (four different ways) into two octahedral pyramids by cutting it in half through one of its four octahedral central hyperplanes. Looked at from inside the curved 3 dimensional volume of its boundary surface of 16 face-bonded tetrahedra, the 16-cell's vertex figure is an octahedron. In 4 dimensions, the vertex octahedron is actually an octahedral pyramid. The apex of the octahedral pyramid (the vertex where the 6 edges meet) is not actually at the center of the octahedron: it is displaced radially outwards in the fourth dimension, out of the hyperplane defined by the octahedron's 6 vertices. The 6 edges around the vertex make an orthogonal 3-axis cross in 3 dimensions (and in the [[W:Octahedral pyramid|3-dimensional projection of the 4-pyramid]]), but the 3 lines are actually bent 90 degrees in the fourth dimension where they meet in an apex.|name=octahedral pyramid}} The 6 orthogonal central planes of the 16-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:Octahedron|octahedron]] with 3 orthogonal great squares. === Rotations === {| class="wikitable" width=480 |- align=center valign=top |rowspan=2|[[File:16-cell.gif]]<br />A 3D projection of a 16-cell performing a [[W:SO(4)#Simple rotations|simple rotation]] |[[File:16-cell-orig.gif]]<br />A 3D projection of a 16-cell performing a [[W:SO(4)#Double rotations|double rotation]] |} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in [[w:Completely_orthogonal|completely orthogonal]] planes.{{Sfn|Kim|Rote|2016|p=6|loc=§&nbsp;5. Four-Dimensional Rotations}} The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).{{Efn|name=Six orthogonal planes of the Cartesian basis}} Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the ''xy'' plane) and another angle of rotation in the completely orthogonal great square plane (the ''wz'' plane).{{Efn|Each great square vertex is {{radic|2}} distant from two of the square's other vertices, and {{radic|4}} distant from its opposite vertex. The other four vertices of the 16-cell (also {{radic|2}} distant) are the vertices of the square's completely orthogonal square.{{Efn|name=Clifford parallel great squares}} Each 16-cell vertex is a vertex of ''three'' orthogonal great squares which intersect there. Each of them has a different ''completely'' orthogonal square. Thus there are three great squares completely orthogonal to each vertex: squares that the vertex is not part of.{{Efn|The three ''incompletely'' orthogonal great squares which intersect at each vertex of the 16-cell form the vertex's octahedral [[W:Vertex figure|vertex figure]].{{Efn|name=octahedral pyramid}} Any two of them, together with the completely orthogonal square of the third, also form an octahedron: a central octahedral hyperplane.{{Efn|Three great squares meet at each vertex (and at its opposite vertex) in the 16-cell. Each of them has a different completely orthogonal square. Thus there are three great squares completely orthogonal to each vertex and its opposite vertex (each axis). They form an octahedron (a central hyperplane). Every axis line in the 16-cell is completely orthogonal to a central octahedron hyperplane, as every great square plane is completely orthogonal to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} The axis and the octahedron intersect only at one point (the center of the 16-cell), as each pair of completely orthogonal great squares intersects only at one point (the center of the 16-cell). Each central octahedron is also the octahedral vertex figure of two of the eight vertices: the two on its completely orthogonal axis.|name=octahedral hyperplanes}} In the 16-cell, each octahedral vertex figure is also a central octahedral hyperplane.|name=completely orthogonal great squares}}|name=vertex and central octahedra}} Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.{{Efn|Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are [[W:Clifford parallel|Clifford parallel]].{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§&nbsp;3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=7-10|loc=§&nbsp;6. Angles between two Planes in 4-Space}} In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all {{radic|2}} apart, so these squares are Clifford parallel polygons.{{Efn|name=completely orthogonal Clifford parallels are special}} Note that only the vertices of the great squares (the points on the great circle) are {{radic|2}} apart; points on the edges of the squares (on chords of the circle) are closer together.|name=Clifford parallels}} They are {{radic|2}} apart at each pair of nearest vertices (and in the 16-cell ''all'' the pairs except antipodal pairs are nearest). The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Because they are perpendicular and share a common center, the two squares are obviously not parallel and separate in the usual way of parallel squares in 3 dimensions; rather they are connected like adjacent square links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallel great squares}} In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]], in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.) In a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] takes place.{{Efn|In an isoclinic rotation, all 6 orthogonal planes are displaced in two orthogonal directions at once: they are rotated by the angle, and at the same time they are tilted ''sideways'' by that same angle. An isoclinic displacement (also known as a [[W:William Kingdon Clifford|Clifford]] displacement) is 4-dimensionally diagonal. Points are displaced an equal distance in four orthogonal directions at once, and displaced a total [[W:Pythagorean distance#Higher dimensions|Pythagorean distance]] equal to the square root of four times the square of that distance (which is two times that distance). All vertices of a regular 4-polytope are displaced to a vertex at least two edge lengths away. For example, when the unit-radius 16-cell rotates isoclinically 90° in a great square invariant plane, it also rotates 90° in the completely orthogonal great square invariant plane.{{Efn||name=Six orthogonal planes of the Cartesian basis}} The great square plane tilts sideways 90° to occupy its completely orthogonal plane. (By isoclinic symmetry, ''every'' great square plane rotates 90° ''and'' tilts sideways 90° into its completely orthogonal plane.) Each vertex (in every great square) is displaced to its antipodal vertex, at a distance of {{radic|1}} in each of four orthogonal directions, a total distance of {{radic|4}}. The original and displaced vertex are two edge lengths apart by three{{Efn|There are six different two-edge paths connecting a pair of antipodal vertices along the edges of a great square. The left isoclinic rotation runs diagonally between three of them, and the right isoclinic rotation runs diagonally between the other three. These diagonals are the straight lines (geodesics) connecting opposite vertices of face-bonded tetrahedral cells in the left-handed [[#Helical construction|eight-cell ring]] and the right-handed eight-cell ring, respectively.}} different paths along two edges of a great square. But the ''isocline'' (the helical arc the vertex follows during the isoclinic rotation) does not run along edges: it runs ''between'' these different edge-paths diagonally, on a geodesic (shortest arc) between the original and displaced vertices.{{Efn|name=isocline}} This isoclinic geodesic arc is not a segment of an ordinary great circle; it does not lie in the plane of any great square. It is a helical 180° arc that bends in a circle in two completely orthogonal planes at once. This [[W:Möbius loop|Möbius circle]] does not lie in any one great circle plane, or intersect any vertices of the 16-cell between the original and the displaced vertex.{{Efn|name=Möbius circle}}|name=isoclinic rotation}} In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 16-cell, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel){{Efn|name=Clifford parallels}} plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great squares are {{radic|4}} (180°) apart; the great squares (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them. If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great square returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 16-cell (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}} === Constructions === ==== Octahedral dipyramid ==== {|class="wikitable floatright" !Octahedron <math>\beta_3</math> !16-cell <math>\beta_4</math> |- |[[File:3-cube t2.svg|160px]] |[[File:4-demicube t0 D4.svg|160px]] |- |colspan=2|Orthogonal projections to skew hexagon hyperplane |} The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the [[W:Octahedron|octahedron]]. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its [[W:Petrie polygon|Petrie polygon]] is the [[W:Hexagon|hexagon]]). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two [[W:Octahedral pyramid|octahedral pyramid]]s on a shared octahedron base that lies in the 16-cell's central hyperplane.{{Sfn|Coxeter|1973|p=121|loc=§&nbsp;7.21. See illustration Fig 7.2<small>B</small>|ps=: "<math>\beta_4</math> is a four-dimensional dipyramid based on <math>\beta_3</math> (with its two apices in opposite directions along the fourth dimension)."}} [[File:stereographic_polytope_16cell_colour.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 16-cell's 6 orthogonal central squares onto their great circles. Each circle is divided into 4 arc-edges at the intersections where 3 circles cross perpendicularly. Notice that each circle has one Clifford parallel circle that it does ''not'' intersect. Those two circles pass through each other like adjacent links in a chain.]]The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with ''two'' of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and ''three more squares'' (which appear edge-on as the 3 ''diameters'' of the hexagon in the projection), and three more octahedra.{{Efn|name=octahedral hyperplanes}} Something unprecedented has also been created. Notice that each square no longer intersects with ''all'' of the other squares: it does intersect with four of them (with ''three'' of the squares crossing at each vertex now), but each square has ''one'' other square with which it shares ''no'' vertices: it is not directly connected to that square at all. These two ''separate'' perpendicular squares (there are three pairs of them) are like the opposite edges of a [[W:Tetrahedron|tetrahedron]]: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of '''''Clifford parallels''''', and the 16-cell is the simplest regular polytope in which they occur. [[W:William Kingdon Clifford|Clifford]] parallelism{{Efn|name=Clifford parallels}} of objects of more than one dimension (more than just curved ''lines'') emerges here and occurs in all the subsequent 4-dimensional regular polytopes, where it can be seen as the defining relationship ''among'' disjoint concentric regular 4-polytopes and their corresponding parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.{{Sfn|Tyrrell & Semple|1971}} For example, as noted [[#Geometry|above]] all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]]. ==== Tetrahedral constructions ==== {| class="wikitable" width=480 |- align=center valign=top |[[File:16-cell net.png|180px|]] |[[File:16-cell nets.png|180px]] |} The 16-cell has two [[W:Wythoff construction|Wythoff construction]]s from regular tetrahedra, a regular form and alternated form, shown here as [[W:Net (polyhedron)|nets]], the second represented by tetrahedral cells of two alternating colors. The alternated form is a [[#Symmetry constructions|lower symmetry construction]] of the 16-cell called the [[W:Demitesseract|demitesseract]]. Wythoff's construction replicates the 16-cell's [[5-cell#Orthoschemes|characteristic 5-cell]] in a [[W:Kaleidoscope|kaleidoscope]] of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facets]] (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} There are three regular 4-polytopes with tetrahedral cells: the [[5-cell]], the 16-cell, and the [[600-cell]]. Although all are bounded by ''regular'' tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different [[5-cell#Isometries|tetrahedral pyramids]], all based on the same characteristic ''irregular'' tetrahedron. They share the same [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron]] (3-orthoscheme) and characteristic [[W:Right triangle|right triangle]] (2-orthoscheme) because they have the same kind of cell.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§&nbsp;7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} {| class="wikitable floatright" !colspan=6|Characteristics of the 16-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "16-cell, 𝛽₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§&nbsp;7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{2} \approx 1.414</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center|<small>60″</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45″</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30″</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center| |align=center| |align=center| |align=center| |} The '''characteristic 5-cell of the regular 16-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|4|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center. The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 16-cell has unit radius edge and edge length 𝒍 = <small><math>\sqrt{2}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small> (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center. ==== Helical construction ==== [[File:Eight face-bonded tetrahedra.jpg|thumb|A 4-dimensional ring of 8 face-bonded tetrahedra, seen in the [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]], bounded by three eight-edge circular paths of different colors, cut and laid out flat in 3-dimensional space. It contains an ''isocline'' axis (not shown), a helical circle of circumference 4𝝅 that twists through all four dimensions and visits all 8 vertices.{{Efn|name=isocline}} The two blue-blue-yellow triangles at either end of the cut ring are the same object.]] [[File:16-cell 8-ring net4.png|thumb|Net and orthogonal projection]] A 16-cell can be constructed (three different ways) from two [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]es of eight chained tetrahedra, each bent in the fourth dimension into a ring.{{Sfn|Coxeter|1970|loc=Table 2: Reflexible honeycombs and their groups|p=45|ps=; Honeycomb [3,3,4]₄ is a tiling of the 3-sphere by 2 rings of 8 tetrahedral cells.}}{{Sfn|Banchoff|2013}} The two circular helixes spiral around each other, nest into each other and pass through each other forming a [[W:Hopf link|Hopf link]]. The 16 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex. The purple edges represent the [[W:Petrie polygon|Petrie polygon]] of the 16-cell. The eight-cell ring of tetrahedra contains three [[W:Octagram|octagram]]s of different colors, eight-edge circular paths that wind twice around the 16-cell on every third vertex of the octagram. The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a [[W:Möbius strip|Möbius strip]]. Thus the 16-cell can be decomposed into two cell-disjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The left-handed and right-handed cell rings fit together, nesting into each other and entirely filling the 16-cell, even though they are of opposite chirality. This decomposition can be seen in a 4-4 [[W:Duoantiprism|duoantiprism]] construction of the 16-cell: {{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} or {{Coxeter–Dynkin diagram|node|4|node_h|2x|node_h|4|node}}, [[W:Schläfli symbol|Schläfli symbol]] {2}⨂{2} or s{2}s{2}, [[W:Coxeter notation|symmetry]] [4,2⁺,4], order 64. Three eight-edge paths (of different colors) spiral along each eight-cell ring, making 90° angles at each vertex. (In the Boerdijk–Coxeter helix before it is bent into a ring, the angles in different paths vary, but are not 90°.) Three paths (with three different colors and apparent angles) pass through each vertex. When the helix is bent into a ring, the segments of each eight-edge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 4𝝅, and one edge wide.{{Efn|name=Möbius circle}} The six four-edge halves of the three eight-edge paths each make four 90° angles, but they are ''not'' the six orthogonal great squares: they are open-ended squares, four-edge 360° helices whose open ends are [[W:Antipodal point|antipodal]] vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined end-to-end in pairs of the same [[W:Chirality|chirality]], the six four-edge paths make three eight-edge Möbius loops, [[W:Helix|helical]] octagrams. Each octagram is both a [[W:Petrie polygon|Petrie polygon]] of the 16-cell, and the helical track along which all eight vertices rotate together, in one of the 16-cell's distinct isoclinic [[#Rotations|rotations]].{{Efn|The 16-cell can be constructed from two cell-disjoint eight-cell rings in three different ways; it has three orientations of its pair of rings. Each orientation "contains" a distinct left-right pair of isoclinic rotations, and also a pair of completely orthogonal great squares (Clifford parallel fibers), so each orientation is a discrete [[W:Hopf fibration|fibration]] of the 16-cell. Each eight-cell ring contains three axial octagrams which have different orientations (they exchange roles) in the three discrete fibrations and six distinct isoclinic rotations (three left and three right) through the cell rings. Three octagrams (of different colors) can be seen in the illustration of a single cell ring, one in the role of Petrie polygon, one as the right isocline, and one as the left isocline. Because each octagram plays three roles, there are exactly six distinct isoclines in the 16-cell, not 18.|name=only one disjoint pair of eight-cell rings}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at the same [[W:Skew polygon|skew]] [[W:Octagram|octagram]]{{Efn|All five views are the same orthogonal projection of the 16-cell into the same plane (a circular cross-section of the eight-cell ring cylinder), looking along the central axis of the cut ring cylinder pictured above, from one end of the cylinder. The only difference is which {{radic|2}} edges and {{radic|4}} chords are ''omitted'' for focus. The different colors of {{radic|2}} edges appear to be of different lengths because they are oblique to the viewer at different angles. Vertices are numbered 1 (top) through 8 in counterclockwise order.}} |- ![[#Rotations|Edge path]] ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]]{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁''}} !16-cell ![[W:Hopf fibration|Discrete fibration]] ![[#Coordinates|Diameter chords]] |- ![[W:Octagram|Octagram]]_{8/3}{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₂''}} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Octagram]]_{8/1} ![[W:Coxeter element#Coxeter plane|Coxeter plane]] [[W:B4 polytope|B₄]] ![[W:Octagram#Star polygon compounds|Octagram]]_{8/2}=2{4} ![[W:Octagram#Star polygon compounds|Octagram]]_{8/4}=4{2} |- |align=center|[[File:16-cell skew octagram (8-3).png|120px]] |align=center|[[File:16-cell skew octagram (8).png|120px]] |align=center|[[File:16-cell skew octagram.png|120px]] |align=center|[[File:16-cell skew octagram 2(4).png|120px]] |align=center|[[File:16-cell skew octagram 4(2).png|120px]] |- |The eight {{radic|2}} chords of the edge-path of an isocline.{{Efn|name=isocline curve}} |Skew [[W:Octagon|octagon]] of eight {{radic|2}} edges. The 16-cell has 3 of these 8-vertex circuits. |All 24 {{radic|2}} edges and the four {{radic|4}} orthogonal axes. |Two completely orthogonal (disjoint) great squares of {{radic|2}} edges.{{Efn|name=Clifford parallel great squares}} |The four {{radic|4}} chords of an isocline. Every fourth isocline vertex is joined to its antipodal vertex by a 16-cell axis.{{Efn|Each isocline has the eight continuous {{radic|2}} chords of its octagram_{8/3} edge-path, and also four discontinuous {{radic|4}} diameter chords that connect every ''fourth'' vertex on the octagram but do not connect to each other. Antipodal vertices also have a twisted continuous path of four mutually orthogonal {{radic|2}} edges connecting them. Between antipodal vertices, the isocline curves smoothly around in a helix over the four {{radic|2}} chords of its edge-path, hitting the three intervening vertices. Each {{radic|2}} edge is an edge of a great square, that is completely orthogonal to another great square, in which the {{radic|4}} chord is a diagonal.|name=isocline curve}} |} Each eight-edge helix is a [[W:Skew polygon|skew]] [[W:Octagram|octagram]]_{8/3} that [[W:Winding number|winds three times]] around the 16-cell and visits every vertex before closing into a loop. Its eight {{radic|2}} edges are chords of an ''isocline'', a helical arc on which the 8 vertices circle during an isoclinic rotation.{{Efn|An isocline is a circle of special kind corresponding to a pair of [[W:Villarceau circle|Villarceau circle]]s linked in a [[W:Möbius loop|Möbius loop]]. It curves through four dimensions instead of just two. All ordinary circles have a 2𝝅 circumference, but the 16-cell's isocline is a circle with an 4𝝅 circumference (over eight 90° chords). An isocline is a circle that does not lie in a plane, but to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''circle'' for an ordinary circle in the plane.|name=Möbius circle}} All eight 16-cell vertices are {{radic|2}} apart except for opposite (antipodal) vertices, which are {{radic|4}} apart. A vertex moving on the isocline visits three other vertices that are {{radic|2}} apart before reaching the fourth vertex that is {{radic|4}} away.{{Efn|In the 16-cell, two antipodal vertices are opposite vertices of two face-bonded tetrahedral cells. The two antipodal vertices are connected by (three different) two-edge great circle paths along edges of the tetrahedral cells, by various three-edge paths, and by four-edge paths on isoclines and Petrie polygons. {{Efn|name=Möbius circle}}|name=isocline}} The eight-cell ring is [[W:Chiral|chiral]]: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eight-cell ring twists. Each isocline visits all eight vertices of the 16-cell.{{Efn|In the 16-cell each ''single'' isocline winds through all 8 vertices: an entire [[W:Hopf fibration|fibration]] of two completely orthogonal great squares.{{Efn|name=completely orthogonal Clifford parallels are special}} The 5-cell and the 16-cell are the only regular 4-polytopes where each discrete fibration has just one isocline fiber.{{Efn|Except in the 5-cell and 16-cell,{{Efn|name=two special cases}} a pair of left and right isocline circles have disjoint vertices: the left and right isocline helices are non-intersecting parallels but counter-rotating, forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect).|name=counter-rotating double helix}}|name=each 16-cell isocline reaches all 8 vertices}} Each eight-cell ring contains half of the 16 cells, but all 8 vertices; the two rings share the vertices, as they nest into each other and fit together. They also share the 24 edges, though left and right octagram helices are different eight-edge paths.{{Efn|The left and right isoclines intersect each other at every vertex. They are different sequences of the same set of 8 vertices. With respect only to the set of 4 vertex pairs which are {{radic|2}} apart, they can be considered to be Clifford parallel. With respect only to the set of 4 vertex pairs which are {{radic|4}} apart, they can be considered to be completely orthogonal.{{Efn|name=completely orthogonal Clifford parallels are special}}}} Because there are three pairs of completely orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} there are three congruent ways to compose a 16-cell from two eight-cell rings. The 16-cell contains three left-right pairs of eight-cell rings in different orientations, with each cell ring containing its axial isocline.{{Efn|name=only one disjoint pair of eight-cell rings}} Each left-right pair of isoclines is the track of a left-right pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation.{{Efn|name=Clifford parallel great squares}} At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16-cell axis chord.{{Efn|This is atypical for isoclinic rotations generally; normally both the left and right isoclines do not occur at the same vertex: there are two disjoint sets of vertices reachable only by the left or right rotation respectively.{{Efn|name=counter-rotating double helix}} The left and right isoclines of the 16-cell form a very special double helix: unusual not just because it is circular, but because its different left and right helices twist around each other through the ''same set'' of antipodal vertices,{{Efn|name=each 16-cell isocline reaches all 8 vertices}} not through the two ''disjoint subsets'' of antipodal vertices, as the isocline pairs do in most isoclinic rotations found in nature.{{Efn|For another example of the left and right isoclines of a rotation visiting the same set of vertices, see the [[5-cell#Geodesics and rotations|characteristic isoclinic rotation of the 5-cell]]. Although in these two special cases left and right isoclines of the same rotation visit the same set of vertices, they still take very different rotational paths because they visit the same vertices in different sequences.|name=two special cases}} Isoclinic rotations in completely orthogonal invariant planes are special.{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal. There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]]. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'' unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|pp=7-8|loc=§&nbsp;6 Angles between two Planes in 4-Space|ps=; Left and Right Pairs of Isoclinic Planes.}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. Because planes separated by a 90° isoclinic rotation are 180° apart, the plane to the ''left'' and the plane to the ''right'' are the same plane.{{Efn|name=exchange of completely orthogonal planes}}|name=completely orthogonal Clifford parallels are special}} To see ''how'' and ''why'' they are special, visualize two completely orthogonal invariant planes of rotation, each rotating by some rotation angle ''and'' tilting sideways by the same rotation angle into a different plane entirely.{{Efn|name=isoclinic rotation}} ''Only when the rotation angle is 90°,'' that different plane in which the tilting invariant plane lands will be the completely orthogonal invariant plane itself. The destination plane of the rotation ''is'' the completely orthogonal invariant plane. The 90° isoclinic rotation is the only rotation which takes the completely orthogonal invariant planes to each other.{{Efn|name=exchange of completely orthogonal planes}} This reciprocity is the reason both left and right rotations go to the same place.}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}8 & 6 & 12 & 8 \\ 2 & 24 & 4 & 4 \\ 3 & 3 & 32 & 2 \\ 4 & 6 & 4 & 16 \end{matrix}\end{bmatrix}</math> == Tessellations == One can [[W:Tessellation|tessellate]] 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 16-cells. This is called the [[W:16-cell honeycomb|16-cell honeycomb]] and has [[W:Schläfli symbol|Schläfli symbol]] {3,3,4,3}. Hence, the 16-cell has a [[W:Dihedral angle|dihedral angle]] of 120°.{{sfn|Coxeter|1973|p=293}} Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation. The dual tessellation, the [[W:24-cell honeycomb|24-cell honeycomb]], {3,4,3,3}, is made of regular [[24-cell]]s. Together with the [[W:Tesseractic honeycomb|tesseractic honeycomb]] {4,3,3,4} these are the only three [[W:List of regular polytopes#Tessellations of Euclidean 4-space|regular tessellations]] of '''R'''⁴. == Projections == {{B4 Coxeter plane graphs|t3|150}} [[File:Orthogonal projection envelopes 16-cell.png|thumb|Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)]] The cell-first parallel projection of the 16-cell into 3-space has a [[W:cube|cubical]] envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope. The cell-first perspective projection of the 16-cell into 3-space has a [[W:triakis tetrahedron|triakis tetrahedral]] envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection. The vertex-first parallel [[W:Graphical projection|projection]] of the 16-cell into 3-space has an [[W:octahedron|octahedral]] [[W:projection envelope|envelope]]. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron. Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a [[W:hexagonal bipyramid]]al envelope. == 4 sphere Venn diagram == A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a [[W:Venn diagram|Venn diagram]] of 4 sets) are [[W:topology|topologically]] equivalent. {| |- | {{multiple image | align = left | total_width = 700 | image1 = 4 spheres, cell 00, solid.png | image2 = 4 spheres, weight 1, solid.png | image3 = 4 spheres, weight 2, solid.png | image4 = 4 spheres, weight 3, solid.png | image5 = 4 spheres, cell 15, solid.png | footer = The 16 cells ordered by number of intersecting spheres (from 0 to 4) &nbsp;&nbsp;&nbsp; <small>(see all [[commons:Category:Venn diagrams rgby; single cells|cells]] and [[v:Tesseract and 16-cell faces|''k''-faces]])</small> }} | {{multiple image | align = right | total_width = 290 | image1 = 4 spheres as rings, vertical.png | image2 = Stereographic polytope 16cell.png | footer = 4 sphere Venn diagram and 16-cell projection in the same orientation }} |} == Symmetry constructions == The 16-cell's [[W:Coxeter group|symmetry group]] is denoted [[W:B4 polytope|B₄]]. There is a lower symmetry form of the ''16-cell'', called a '''demitesseract''' or '''4-demicube''', a member of the [[W:Demihypercube|demihypercube]] family.{{Sfn|Conway, Burgiel & Goodman-Strauss|2008| loc=Chapter 26. Hemicubes: 1_n1 | p=409 }} It is represented by h{4,3,3} and [[W:Coxeter diagram|Coxeter diagram]]s {{Coxeter–Dynkin diagram|node_h1|4|node|3|node|3|node}} or {{Coxeter–Dynkin diagram|nodes_10ru|split2|node|3|node}}. It can be drawn bicolored with alternating [[W:tetrahedron|tetrahedral]] cells. It can also be seen in lower symmetry form as a '''tetrahedral antiprism''', constructed by 2 parallel [[W:tetrahedron|tetrahedra]] in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: {{Coxeter–Dynkin diagram|node_h|2x|node_h|4|node|3|node}}. It can also be seen as a snub 4-[[W:Orthotope|orthotope]], represented by s{2^1,1,1}, and Coxeter diagram: {{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} or {{Coxeter–Dynkin diagram|node_h|2x|node_h|split1-22|nodes_hh}}. With the [[W:Tesseract|tesseract]] constructed as a 4-4 [[W:Duoprism|duoprism]], the 16-cell can be seen as its dual, a 4-4 [[W:Duopyramid|duopyramid]]. {| class=wikitable !Name ![[W:Coxeter diagram|Coxeter diagram]] ![[W:Schläfli symbol|Schläfli symbol]] ![[W:Coxeter notation|Coxeter notation]] !Order ![[W:Vertex figure|Vertex figure]] |- align=center !Regular 16-cell |{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} |{3,3,4} |[3,3,4]||384 |{{Coxeter–Dynkin diagram|node_1|3|node|4|node}} |- align=center !Demitesseract<br />[[W:Quasiregular polytope|Quasiregular]] 16-cell |{{Coxeter–Dynkin diagram|nodes_10ru|split2|node|3|node}} = {{Coxeter–Dynkin diagram|node_h1|4|node|3|node|3|node}}<br />{{Coxeter–Dynkin diagram|node_1|3|node|split1|nodes}} = {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node_h0}} |h{4,3,3}<br />{3,3^1,1} |[3^1,1,1] = [1⁺,4,3,3]||192 |{{Coxeter–Dynkin diagram|node|3|node_1|3|node}} |- align=center !Alternated 4-4 [[W:Duoprism|duoprism]] |{{Coxeter–Dynkin diagram|label2|branch_hh|4a4b|nodes}} |2s{4,2,4} |[[W:4,2⁺,4|4,2⁺,4]]||64 | |- align=center !Tetrahedral antiprism |{{Coxeter–Dynkin diagram|node_h|2x|node_h|4|node|3|node}} |s{2,4,3} |[2⁺,4,3]||48 | |- align=center !Alternated square prism prism |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|4|node}} |sr{2,2,4} |[(2,2)⁺,4]||16 | |- align=center !Snub 4-[[W:Orthotope|orthotope]] |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} = {{Coxeter–Dynkin diagram|node_h|2x|node_h|split1-22|nodes_hh}} |s{2^1,1,1} |[2,2,2]⁺ = [2^1,1,1]⁺||8 |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h}} |- align=center !rowspan=6|4-[[W:Rhombic fusil|fusil]] |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node|3|node}} |{3,3,4} |[3,3,4]||384 |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1|4|node}} |{4}+{4} or 2{4} |<nowiki>[[W:4,2,4|4,2,4]]</nowiki> = [8,2⁺,8]||128 |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node|2x|node_f1}} |{3,4}+{ } |[4,3,2]||96 |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node}}<br />{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1|2x|node_f1}} |{4}+2{ } |[4,2,2]||32 |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}}<br />{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1|2x|node_f1}} |{ }+{ }+{ }+{ } or 4{ } |[2,2,2]||16 |{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1}} |} == Related complex polygons == The [[W:Möbius–Kantor polygon|Möbius–Kantor polygon]] is a [[W:Regular complex polytope|regular complex polygon]] ₃{3}₃, {{Coxeter–Dynkin diagram|3node_1|3|3node}}, in <math>\mathbb{C}^2</math> shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.{{Sfn|Coxeter|1991|pp=30,47}}{{Sfn|Coxeter & Shephard|1992}} The regular complex polygon, ₂{4}₄, {{Coxeter–Dynkin diagram|node_1|4|4node}}, in <math>\mathbb{C}^2</math> has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is ₄[4]₂, order 32.{{Sfn|Coxeter|1991|p=108}} {| class=wikitable width=320 |+ [[W:Orthographic projection|Orthographic projection]]s of ₂{4}₄ polygon |- valign=top |[[File:Complex polygon 2-4-4.png|160px]]<br />In B₄ [[W:Coxeter plane|Coxeter plane]], ₂{4}₄ has 8 vertices and 16 2-edges, shown here with 4 sets of colors. |[[File:Complex polygon 2-4-4 bipartite graph.png|160px]]<br />The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a [[W:Complete bipartite graph|complete bipartite graph]], K_4,4.{{Sfn|Coxeter|1991|p=114}} |} == Related uniform polytopes and honeycombs == The regular 16-cell and [[W:Tesseract|tesseract]] are the regular members of a set of 15 [[W:B4 polytope|uniform 4-polytopes with the same B₄ symmetry]]. The 16-cell is also one of the [[W:D4 polytope|uniform polytopes of D₄ symmetry]]. The 16-cell is also related to the [[W:Cubic honeycomb|cubic honeycomb]], [[W:Order-4 dodecahedral honeycomb|order-4 dodecahedral honeycomb]], and [[W:Order-4 hexagonal tiling honeycomb|order-4 hexagonal tiling honeycomb]] which all have [[W:Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures|octahedral vertex figures]]. It belongs to the sequence of [[W:Order-6 tetrahedral honeycomb#Related polytopes and honeycombs|{3,3,p} 4-polytopes]] which have tetrahedral cells. The sequence includes three [[W:Regular 4-polytope|regular 4-polytope]]s of Euclidean 4-space, the [[5-cell]] {3,3,3}, 16-cell {3,3,4}, and [[600-cell]] {3,3,5}), and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. It is first in a sequence of [[W:Tetrahedral-octahedral honeycomb#Quasuiregular honeycombs|quasiregular polytopes and honeycombs]] h{4,p,q}, and a [[W:Order-4 hexagonal tiling honeycomb#Quasiregular honeycombs|half symmetry sequence]], for regular forms {p,3,4}. == See also == *[[24-cell]] *[[W:4-polytope|4-polytope]] *[[W:D4 polytope|D4 polytope]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} == External links == * [https://bendwavy.org/klitzing/incmats/hex.htm hex], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Hexadecachoron Hexadecachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Aerochoron Aerochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/uniform Uniform polychora (The tesseract/16-cell family)], at [https://www.qfbox.info/4d/index 4D Euclidean Space] [[Category:Geometry]] [[Category:Polyscheme]] akg0r145x2xqqsweblayvocz2o3skdx 4 305980 2801303 2801257 2026-03-29T12:04:55Z Jtneill 10242 Removed unnecessary word 2801303 wikitext text/x-wiki {{Proposed policy|WV:AI}} The policy sets out requirements for editors who use [[w:artificial intelligence|artificial intelligence]] (AI) when contributing to Wikiversity. At Wikiversity, an AI is a machine learning model which generates text ([[w:language model|language model]]) but may also involve text-to-image models, text-to-video models, text-to-speech models, and so on, as well as software tools with AI support such as [[w:Grammarly|Grammarly]], [[w:Google Translate|Google Translate]], or [[w:Notion (productivity software)|Notion]]. Use of AI-generated text on Wikiversity is acceptable if it follows good [[w:scholarly method|scholarly practice]]s including: * The origin of the text is clearly indicated in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], ideally with a publicly available link to the chatbot conversation * The text is [[w:Fact-checking|fact-checked]] by the contributor and, where [[Wikiversity:Cite sources|citations]] are included, they must be checked for appropriateness and relevance by the contributor * The text should ideally be revised and rewritten by the contributor * The copyright of the text needs to be owned by the contributor (who contributes it to Wikiversity with an [[Wikiversity:Copyrights|appropriate license]]) * Display the {{tl|AI-generated}} template at the top of pages (or below the description template of a file) which have significant AI-generated content ==See also== * [[Is the output of ChatGPT copyrighted?]] (Wikidebate) * [[Should Wikiversity allow editors to post content generated by LLMs?]] (Wikidebate) * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] (Motivation and emotion project guidelines) * [[b:Wikibooks:Artificial intelligence|Wikibooks:Artificial intelligence]] (Wikibooks policy) * [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page) [[Category:Artificial intelligence]] [[Category:Wikiversity policy]] m78np84t4w0ke543qaw05ahylkbwqhy 0 307306 2801417 2677316 2026-03-30T04:28:36Z Dronebogus 3054149 /* Overview */ slop 2801417 wikitext text/x-wiki {{title|Lucid dream facilitation:<br>What techniques can facilitate lucid dreaming?}} {{MECR3|1=https://youtu.be/asB62uH_e60}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario Imagine waking up in the middle of a dream, fully aware that you are dreaming and can control your actions and the dream environment. {{expand}} {{RoundBoxBottom}} This experience of [[wikipedia:Lucid_dream|lucid dreaming]]is a phenomenon that has intrigued and inspired individuals across cultures for centuries. Whether it is to explore creativity, confront fears, or enhance personal growth, pursuing lucidity during sleep presents an array of possibilities{{g}}. Figure 1 shows an AI generated Image of "the Brain During Lucid Dreaming".'''What is Lucid Dreaming?''' Lucid dreaming occurs primarily during [[wikipedia:Rapid_eye_movement_sleep|rapid eye movement (REM) sleep]], a phase when the brain exhibits activity levels similar to those in waking consciousness (Baird et al., 2019). Understanding the significance of dream awareness is crucial, as it opens the door to various techniques that can enhance the likelihood of achieving lucidity. These techniques include [[wikipedia:Reality_testing|reality testing]], mnemonic induction of lucid dreams (MILD), and wake-initiated lucid dreams (WILD), each employing different psychological and neurological mechanisms. The exploration of lucid dreaming extends beyond mere curiosity; it touches on deeper psychological realms, offering a pathway for personal development and emotional healing. As we delve into the intricacies of this phenomenon, we will uncover the profound interplay between motivation, emotion, and cognition that shapes our dream experiences, paving the way for a richer understanding of both lucid dreaming and the human psyche.{{rewrite}} {{RoundBoxTop|theme=3}} ;Focus questions {{ic|Use numbered list per Tutorial 02}} '''1. What are the most effective techniques for facilitating lucid dreaming?''' '''2. How do these techniques operate on a psychological and neurological level?''' '''3. What potential benefits and risks are associated with the practice of lucid dreaming?''' {{RoundBoxBottom}} == Theoretical framework for understanding lucid dreaming techniques == {{expand}} === Reality testing === {{expand}} ==== Definition and mechanism ==== Reality testing involves the practice of questioning one’s environment to determine whether one is dreaming or awake. According to Aspy et al. (2020), this technique enhances dream awareness and is particularly effective when the dreamer is motivated by a strong desire to achieve lucidity. By consistently engaging in reality checks throughout the day, individuals can increase their chances of recognising the dream state when it occurs at night. This proactive approach not only builds awareness but also reinforces the habit of questioning reality, which is crucial for achieving lucidity in dreams. ==== Psychological theory ==== From a psychological standpoint, reality testing can be understood through the lens of [[wikipedia:Cognitive_behavioral_therapy|Cognitive Behavioural Theory (CBT)]]. CBT posits that our thoughts, feelings, and behaviours are interconnected, and modifying one aspect can lead to changes in the others. Regular reality checks enhance dream awareness, particularly when motivated by a strong desire to become lucid. Emotional events can also serve as significant triggers for reality testing within dreams. As noted by Baird et al. (2019), “Research suggests that this cognitive conditioning fosters a greater likelihood of recognising the dream state.” This implies that by training the mind to question reality consistently, individuals can develop a heightened sense of awareness, facilitating the recognition of dreams as they unfold. ==== Case study ==== A compelling illustration of the effectiveness of reality testing can be found in a case study conducted by Forrer (2014). In this study, an individual engaged in reality testing techniques over several weeks, which included frequent questioning of their surroundings and the use of various reality checks, such as looking at their hands or reading text. This participant reported an increased ability to recognise when they were dreaming, leading to a significant number of lucid dreams. The findings from Forrer’s study highlight the practical application of reality testing and its potential to empower individuals in their journey towards achieving lucidity in dreams. === Mnemonic induction of lucid dreams === {{expand}} ==== '''Definition and mechanism''' ==== Mnemonic Induction of Lucid Dreams (MILD) is a technique designed to enhance the likelihood of achieving lucidity during dreaming by setting clear intentions before sleep. This method involves the repetition of affirmations, such as "I will realise I am dreaming," which serves to prime the mind for lucidity (Aspy et al., 2017). The effectiveness of MILD is significantly influenced by the individual's motivation and emotional investment in the process, as these factors contribute to a stronger intention to achieve lucidity{{f}}. ==== '''Psychological theory''' ==== The theoretical foundation for MILD can be understood through the lens of Motivational Theory, particularly Goal-Setting Theory. According to Locke and Latham (2002), specific and challenging goals yield higher performance than vague or easily attainable ones. When individuals set explicit goals, such as the intention to recognise when they are dreaming, they are more likely to achieve those goals{{f}}. This aligns with the MILD technique, which emphasises the importance of clarity and commitment in setting lucid dreaming intentions. === Wake-initiated lucid dreams === {{expand}} ==== Definition and mechanism ==== Wake-Initiated Lucid Dreams (WILD) is a technique that allows individuals to transition directly from wakefulness into a dream state while maintaining consciousness (Baird et al., 2019). This technique requires a calm emotional state and positive motivation, enabling the dreamer to remain aware as they slip into the dream world. ==== Psychological Theory ==== The concept of WILD aligns with [[wikipedia:Mindfulness|Mindfulness Theory]], which emphasises present-moment awareness and emotional regulation. Practising mindfulness helps individuals maintain awareness of their thoughts and feelings as they enter a dream state, facilitating a seamless transition into lucidity (Baird et al., 2019). === Critical analysis of lucid dreaming techniques === While techniques such as Reality Testing, Mnemonic Induction of Lucid Dreams (MILD), and Wake-Initiated Lucid Dreams (WILD) present promising avenues for facilitating lucid dreaming, their effectiveness can vary significantly among individuals{{f}}. A key consideration is the reliance on motivation, which may not be equally effective for all, particularly for those struggling with mental health issues like anxiety or depression. Research indicates that such conditions can hinder emotional regulation, making it challenging for affected individuals to engage effectively with techniques that require a positive emotional state (Menefee et al., 2022). Furthermore, while Reality Testing is a foundational technique in lucid dream facilitation, it may not be sufficient on its own for everyone. Some individuals may find greater success when complemented with strategies like MILD or WILD, which focus on intention-setting and maintaining awareness during the transition into sleep.{{f}} The integration of these methods can potentially enhance the likelihood of achieving lucidity and navigating dream scenarios effectively{{f}}. The emotional outcomes of lucid dreaming are not universally beneficial. Research by Neuhäusler et al. (2018) suggests that not all individuals experience positive emotional benefits from lucid dreaming; instead, some may encounter anxiety or distress, particularly when faced with challenging dream scenarios. This highlights the necessity of considering individual differences and emotional responses when exploring lucid dreaming techniques. The implications of these findings suggest that while techniques such as MILD and WILD can be valuable tools for some individuals, there is a need for tailored approaches that consider individual psychological profiles. Practitioners and researchers alike should remain cognisant of these variances and incorporate complementary strategies and supports that can enhance emotional regulation and overall dream experience. {{robelbox|theme=8|title=Quiz question 1|iconwidth=55px|icon=Search-icon-white-background.png}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> Which lucid dreaming induction technique involves questioning the reality of one's environment? |type="()"} - Wake-initiated lucid dreams (WILD) - Mnemonic induction of lucid dreams (MILD) + Reality testing </quiz> </div>{{RoundBoxBottom}} == Motivation in lucid dreaming == {{expand}} === The importance of motivation in lucid dream success === Motivation plays a pivotal role in the success of lucid dreaming techniques. Research indicates that strong motivation significantly increases the likelihood of practising and effectively utilising these techniques (Konkoly & Burke, 2019). Personal goals, such as overcoming nightmares or exploring creativity, often drive this motivation{{f}}. Individuals who set specific intentions for their dream experiences are more likely to engage consistently in practices that lead to lucidity, making motivation a critical factor in the overall process of achieving and maintaining awareness within dreams{{f}}. For example, a study by Aspy et al. (2020) found that dreamers who were motivated by specific objectives, like problem-solving or self-exploration, reported higher instances of lucid dreams. This demonstrates the importance of aligning personal aspirations with the techniques employed to achieve lucidity, highlighting how motivation can transform the dream experience. === Techniques to enhance motivation for lucid dreaming === Several strategies can be employed to enhance motivation for lucid dreaming. Keeping a dream journal is one effective method; by documenting dreams and tracking progress, individuals can cultivate a deeper connection to their dream life, which reinforces their motivation to become lucid (Baird et al., 2022). Setting clear goals related to dream experiences further strengthens this motivation. For instance, specific intentions, such as “I want to confront my fears in a dream,” can foster a proactive mindset towards achieving lucidity. Additionally, visualising the desired outcome before sleep can enhance motivation. This technique, often used in sports psychology, encourages individuals to picture themselves successfully becoming lucid within their dreams. This mental rehearsal not only solidifies their intentions but also primes the mind for the experience of lucidity (Menefee et al., 2022). By employing these strategies, individuals can create a motivating environment conducive to achieving their lucid dreaming goals. {{robelbox|theme=8|title=Quiz question 2|iconwidth=55px|icon=Search-icon-white-background.png}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 2. What is an example of setting a specific intention to strengthen motivation for lucid dreaming? |type="()"} - Deciding to sleep longer hours each night. + Intending to confront fears within a dream. - Choosing to ignore nightmares altogether. </quiz> </div>{{RoundBoxBottom}} == Emotion in lucid dreaming == {{expand}} === Emotional triggers for lucid dreams === Emotions experienced during the day, particularly intense emotions like fear, excitement, or anxiety, can serve as powerful triggers for becoming lucid in a dream. These emotions often carry over into the dream state, creating unusual or exaggerated emotional experiences that prompt the dreamer to question the reality of the situation, thus enhancing lucidity (Menefee et al., 2022). For example, individuals dealing with anxiety may experience heightened dream awareness when these feelings surface in their dreams, providing an opportunity to become lucid and address underlying issues. === The role of emotion in dream awareness === Within dreams, the intensity of emotions often plays a critical role in fostering self-awareness. Emotions that feel unusually strong or deviate from the dreamer's normal emotional experiences, such as extreme joy or fear, can serve as cues that they are dreaming (Neuhäusler et al., 2018). These emotionally charged moments create a heightened sense of awareness, prompting the dreamer to recognise the dream state and potentially achieve lucidity. Research has demonstrated that dreamers who frequently experience intense emotions, particularly negative emotions like fear, are more likely to become lucid as they learn to recognise these feelings as dream cues (van Agteren et al., 2021{{ic|This article doesn't refer to dreams}}). === Emotion regulation and exploration through lucid dreaming === Lucid dreaming provides a valuable space for emotion regulation and exploration. Within the lucid state, individuals have the ability to consciously engage with difficult emotions, such as fear, grief, or anger, in a controlled and safe environment. This allows for emotional processing and, over time, emotional healing. Dreamers can use lucidity to confront emotional challenges, rewrite the narrative of distressing dreams, or even transform nightmares into empowering experiences (Neuhäusler et al., 2018). Studies suggest that this emotional engagement can lead to improved emotional regulation and psychological well-being, especially for those with frequent nightmares or emotional distress (van Agteren et al., 2021). {{robelbox|theme=8|title=Quiz question 3|iconwidth=55px|icon=Search-icon-white-background.png}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> What role do emotions intense emotions like fear or anxiety, play in lucid dreaming? |type="()"} - They have no effect on lucid dreaming. + They can serve as triggers, prompting the dreamer to question the reality of the dream and become lucid. - Intense emotions prevent the dreamer from becoming lucid. </quiz> </div> {{RoundBoxBottom}} == Psychological and neurological mechanisms == {{expand}} === Rapid eye movement sleep and lucid dreaming === Lucid dreaming typically occurs during rapid eye movement (REM) sleep, a stage when brain activity closely resembles that of waking consciousness. During REM sleep, vivid dreams are common, and the brain's heightened activity provides an optimal environment for lucidity (Baird et al., 2019). This period of sleep is marked by increased neural activity in regions involved in cognition and sensory processing, making it possible for the dreamer to recognise the dream state{{f}}. === Memory and awareness in lucid dreaming === The ability to become aware of and control dreams relies heavily on memory and self-awareness{{f}}. Memory recall, especially related to recent events, can serve as a trigger for recognising that one is dreaming.{{f}} Awareness of the incongruities in the dream environment further facilitates lucidity{{f}}. Lucid dreamers often demonstrate heightened metacognitive awareness—an understanding of their own thinking processes—which allows them to distinguish between dreaming and waking states (Baird et al., 2019; Voss et al., 2009). === Brain regions involved in lucid dreaming === Studies have shown that certain brain regions, particularly the prefrontal cortex, exhibit heightened activity during lucid dreaming{{f}}. The prefrontal cortex is responsible for higher-order cognitive functions, including decision-making, self-reflection, and awareness—skills that are essential for recognising and manipulating the dream state (Dresler et al., 2015). Additionally, the parietal lobe and temporo-parietal junction, which are involved in spatial awareness and self-representation, also show increased activity during lucid dreams, facilitating the dreamer's sense of control (Baird et al., 2019). {{robelbox|theme=8|title=Quiz question 4|iconwidth=55px|icon=Search-icon-white-background.png}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> Which brain region is primarily responsible for the heightened awareness and decision-making necessary for lucid dreaming? |type="()"} + The prefrontal cortex - The brainstem - The cerebellum </quiz> </div>{{RoundBoxBottom}} ==Conclusion== Lucid dreaming presents a fascinating intersection of psychology, motivation, and emotion, offering individuals a unique opportunity to explore their inner worlds while awake in their dreams. The key points outlined throughout this chapter illustrate the significance of understanding lucid dreaming techniques, such as reality testing, mnemonic induction of lucid dreams (MILD), and wake-initiated lucid dreams (WILD). Each of these techniques leverages psychological theories—like Goal-Setting Theory and Mindfulness Theory—demonstrating how motivation and emotional awareness play pivotal roles in achieving lucidity. The exploration of focus questions reveals that while techniques can enhance the likelihood of experiencing lucid dreams, their effectiveness may vary based on individual emotional states and motivations. For instance, personal goals, such as overcoming nightmares or fostering creativity, can greatly influence one's success in achieving lucidity (Konkoly & Burke, 2019). Conversely, challenges like anxiety and depression may hinder emotional regulation, potentially complicating the lucid dreaming experience (Menefee et al., 2022). By engaging with our dreams, we can confront and process difficult emotions, ultimately leading to a deeper understanding of ourselves. As illustrated in the thought experiment at the beginning of this chapter, the power of lucid dreaming allows us not only to manipulate our dream environments but also to navigate our emotional landscapes. As we move forward, embracing the techniques for achieving lucid dreaming can empower individuals to harness the full potential of their dreaming experiences, transforming nightmares into opportunities for resilience and creativity. Ultimately, lucid dreaming invites us to explore the vast expanse of our consciousness, bridging the gap between our waking and dreaming selves. == See also == * [[wikipedia:Lucid_dream|Lucid Dream (Wikipedia)]] * [[Motivation and emotion/Book/2014/Dreams and emotion|Dreams and Emotion (Book chapter, 2014)]] ==References== {{Hanging indent|1= Aspy, D. J. (2020). Findings from the international lucid dream induction study. Frontiers in Psychology, 11. https://doi.org/10.3389/fpsyg.2020.01746 Baird, B., Mota-Rolim, S. A., & Dresler, M. (2019). The cognitive neuroscience of lucid dreaming. Neuroscience & Biobehavioral Reviews, 100, 305–323. https://doi.org/10.1016/j.neubiorev.2019.03.008 Dresler, M., Wehrle, R., Spoormaker, V. I., Steiger, A., Holsboer, F., Czisch, M., & Hobson, J. A. (2015). Neural correlates of insight in dreaming and psychosis. Sleep Medicine Reviews, 20, 92–99. https://doi.org/10.1016/j.smrv.2014.06.004 Forrer, K. (2014). “To test or not to test, that is the question” - is there a way of verifying the validity of the interpretation of our dreams?. International Journal of Dream Research, 7(2), 153–169. https://doi.org/10.11588/ijodr.2014.2.15033 Konkoly, K., & Burke, C. T. (2019). Can learning to lucid dream promote personal growth? Dreaming, 29(2). https://doi.org/10.1037/drm0000101 Locke, E. A., & Latham, G. P. (2002). Building a practically useful theory of goal setting and task motivation: A 35-year odyssey. American Psychologist, 57(9), 705–717. Menefee, D. S., Ledoux, T., Johnston, C. A., & DeBakey, M. E. (2022). The importance of emotional regulation in mental health. American Journal of Lifestyle Medicine, 16(1), 28–31. https://doi.org/10.1177/15598276211049771 Neuhäusler, A., Schredl, M., & Göritz, A. S. (2018). General knowledge about lucid dreaming and lucid dream induction techniques: An online study. International Journal of Dream Research, 179–185. https://doi.org/10.11588/ijodr.2018.2.50491 van Agteren, J., Iasiello, M., Lo, L., Bartholomaeus, J., Kopsaftis, Z., Carey, M., & Kyrios, M. (2021). A systematic review and meta-analysis of psychological interventions to improve mental wellbeing. Nature Human Behaviour, 5(5), 631–652. https://doi.org/10.1038/s41562-021-01093-w Voss, U., Holzmann, R., Tuin, I., & Hobson, A. J. (2009). Lucid dreaming: A state of consciousness with features of both waking and non-lucid dreaming. Sleep, 32(9), 1191–1200. https://doi.org/10.1093/sleep/32.9.1191 }} == External links == * [https://www.sleepfoundation.org/dreams/lucid-dreams How to Lucid Dream: Expert Tips and Tricks] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Dreams]] 4bxt0z7s2alngxtxcxomizy2ew8xml0 0 307751 2801423 2787773 2026-03-30T05:01:04Z Dronebogus 3054149 2801423 wikitext text/x-wiki {{TOC right | limit|limit=2}} {{AI-generated}} The phrase "every ism creates a [[w:Schism|schism]]" suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as "isms") tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.<ref>[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. </ref>^,<ref>“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl</ref> While "isms" can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social "isms" and how they have created [[w:Schism|schisms]] throughout history. == Political Isms: Capitalism and Communism == One of the clearest examples of an "ism" that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership. This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic "isms" had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations. == Religious Isms: Protestantism and Catholicism == In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an "ism" can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation. [[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an "ism" grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious "ism" left a legacy of division that altered European history and shaped global religious dynamics. == Social Isms: Feminism and Patriarchy == '''[[w:Feminism|Feminism]]''', another significant "ism," arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations. However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism. Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social "isms" can divide societies. == Philosophical Isms: Rationalism and Empiricism == In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience. This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual "isms" can divide schools of thought and shape the trajectory of entire fields of inquiry. == Cultural Isms: Nationalism and Globalism == '''[[w:Nationalism|Nationalism]]''' is another "ism" that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism. The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness. == Remedies == We can gain the wisdom to avoid the schisms born of isms in several ways. Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful. It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]]. [[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]]. Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]]. == Conclusion == The phrase "every ism creates a schism" captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While "isms" can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas. {{CourseCat}} [[Category:Essays]] [[Category:Living Wisely]] 9ri3a2tznxhozoe4l4ikb1i7vygfpzr 2801583 2801423 2026-03-30T11:06:48Z Lbeaumont 278565 Reverted edits by [[Special:Contributions/Dronebogus|Dronebogus]] ([[User_talk:Dronebogus|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]] 2787773 wikitext text/x-wiki {{TOC right | limit|limit=2}} {{AI-generated}} The phrase "every ism creates a [[w:Schism|schism]]" suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as "isms") tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.<ref>[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. </ref>^,<ref>“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl</ref> While "isms" can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social "isms" and how they have created [[w:Schism|schisms]] throughout history. [[File:Every_Ism_Creates_a_Schism.png|thumb|Although every ism creates a schism, we can [[Finding Common Ground|find common ground]].]] == Political Isms: Capitalism and Communism == One of the clearest examples of an "ism" that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership. This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic "isms" had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations. == Religious Isms: Protestantism and Catholicism == In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an "ism" can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation. [[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an "ism" grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious "ism" left a legacy of division that altered European history and shaped global religious dynamics. == Social Isms: Feminism and Patriarchy == '''[[w:Feminism|Feminism]]''', another significant "ism," arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations. However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism. Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social "isms" can divide societies. == Philosophical Isms: Rationalism and Empiricism == In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience. This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual "isms" can divide schools of thought and shape the trajectory of entire fields of inquiry. == Cultural Isms: Nationalism and Globalism == '''[[w:Nationalism|Nationalism]]''' is another "ism" that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism. The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness. == Remedies == We can gain the wisdom to avoid the schisms born of isms in several ways. Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful. It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]]. [[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]]. Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]]. == Conclusion == The phrase "every ism creates a schism" captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While "isms" can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas. {{CourseCat}} [[Category:Essays]] [[Category:Living Wisely]] dlap2uhw6v6kut5fhazmdu8k1lfo5ho 1 313564 2801464 2783852 2026-03-30T06:56:57Z Heptapode 3057546 Complete, reorganize 2801464 wikitext text/x-wiki == Student projects == === Aims and outputs === The aims are to learn more about aspects of CFT that are not dealt with in detail in the course, and to share what is learned with fellow students and the wider world. Possible outputs include: * A written report (maximum 5 pages). * An oral presentation to the class (10 minutes). * Contributions to Wikipedia (see [[w:User:Sylvain_Ribault/YRIS2019|tutorial]]), such as: adding references, correcting mistakes, restructuring, translating, adding missing material, etc. === Guidelines === * The outputs can be in English or in French. * Projects can be done in teams or individually. * In a report, there should be a table of contents, all equations should be numbered, references should be cited from the text and not just listed at the end. === Past projects === 2025 projects: * Central extensions in physics and CFT -> added references to [[w:Affine_Lie_algebra]], [[w:Virasoro_algebra]], [[w:Lie_algebra_extension]]. * Defects in CFT. * Affine Lie algebras and CFT. * Quantum groups in CFT -> written [[w:Groupe_quantique]] by translation from the English, and addition of a section on physical applications. * Renormalization group flows in CFT. * Modular invariance of conformal field theories on torus. * A Comparative Study of CFTs: Lagrangian vs Bootstrap Approach. * Generalized symmetries and topological defects in 2d CFT. 2026 projects: * Vertex operator algebras. * Modular invariance. * Wess-Zumino-Witten Models. o2r5cfymeodbk8vepzlrys9vxo83mrz 2 318980 2801530 2796485 2026-03-30T09:10:38Z Ruud Loeffen 2998353 /* 2.5 Implications for Planetary Formation Models */ add Extension to circumplanetary disks and moon formation 2801530 wikitext text/x-wiki [[File:CITbanner via Paint.png|center|1000px]] = Chapter 2: The Role of VRMS in Planetary Structuring = == Introduction == One of the fundamental insights of the '''Cosmic Influx Theory (CIT)''' is that the structure of planetary systems is not random but follows a predictable pattern based on the '''Root Mean Square Velocity (VRMS)''' of the original protoplanetary disk. This chapter explores: * The definition and significance of '''VRMS''' in planetary dynamics. * How VRMS determines the '''Preferred Distance (D_pref)''' [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.2.8|[8.2.8]]]. * The role of the '''Universal Scaling Constant (κ_CIT)'''. * The link between VRMS and observed exoplanetary systems. ---- <span id="2.1"></span> == 2.1 How VRMS is Related to Cosmic Structuring == The '''Root Mean Square Velocity (VRMS)''' is a statistical measure of velocities within a system. In CIT: * The original '''protoplanetary disk''' had a characteristic VRMS. * This velocity defines a '''Preferred Distance''' at which mass concentration tends to occur. * The '''largest planets''' are often found near this distance. The equation for the Preferred Distance is: <math>D_{\text{pref}} = \kappa_{\text{CIT}} \times M_{\text{star}}</math> ........'''(2.1.1)''' === Definition of κ_CIT === CIT introduces a fundamental proportionality constant: <math>\kappa_{\text{CIT}} = \frac{1}{8\pi c^2} = 4.4 \times 10^{-19} \ \text{m/kg}</math> ........'''(2.1.2)''' This can also be expressed as: <math>\kappa_{\text{CIT}} = \frac{D_{\text{pref}}}{M_{\text{star}}}</math> ........'''(2.1.3)''' or: <math>\kappa_{\text{CIT}} = \frac{G}{V_{\text{RMS}}^2}</math> ........'''(2.1.4)''' *Note: This expression arises from relativistic energy formulations and carries units of m/kg when derived from dimensional analysis of G and VRMS.* The precise calculated value is: <math>\kappa_{\text{CIT}} = 4.427093908810190 \times 10^{-19} \ \text{m/kg}</math> This remarkable connection suggests that: # Planetary structuring is governed by fundamental constants, linking gravitational mass distributions with the speed of light. # The Preferred Distance is an intrinsic property of cosmic structuring, rooted in mechanics and electrodynamics. # The scaling factor <math>\frac{1}{8\pi c^2}</math> unites gravitational and electromagnetic principles, reinforcing the internal consistency of CIT. These equations accurately predict the location of giant exoplanets in other star systems, reinforcing CIT’s predictive value. ---- <span id="2.2"></span> == 2.2 The Connection Between CIT and General Relativity == '''CIT does not reject General Relativity, but offers a complementary perspective, proposing that gravitational effects arise from a continuous influx of energy — an external flow into matter that contributes to its mass-energy and gravitational influence.''' This leads to three key extensions of traditional gravitational theory: # The gravitational field is not just a curvature of spacetime, but the observable result of an influx of energy. This influx guides objects along curved paths, producing the effects attributed to spacetime curvature in General Relativity. # The gravitational acceleration (g or a) at a planet’s surface depends on the intensity of the energy influx, which is related to the total mass of the object. # Observed planetary distances are not random, but reflect a cosmic structuring principle derived from the Root Mean Square Velocity (VRMS) and the resulting Preferred Distance (D_pref). A resonant-field curvature formulation that leads to the same effective coupling as the mechanical influx picture used in CIT is presented in Panagis & Loeffen (2025) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.48|[8.4.48]]]. ---- <span id="2.3"></span> == 2.3 The Preferred Distance (D_pref) and Its Calculation == CIT introduces the concept of the '''Preferred Distance (D_pref)''', the location where the most massive planets tend to form. <math>D_{\text{pref}} = \kappa_{\text{CIT}} \times M_{\text{star}}</math> ........'''(2.3.1)''' where: * <math>\kappa_{\text{CIT}}</math> is the '''Universal Scaling Constant for Planetary Structuring''', approximately <math>4.4271 \times 10^{-19} \ \text{m/kg}</math>. * <math>M_{\text{star}}</math> is the mass of the central star. This proportionality helps explain: * Why Jupiter and Saturn formed at their observed distances in our solar system. * Why exoplanets tend to cluster at specific radii from their stars. * Why ring and gap structures in protoplanetary disks exhibit ordered patterns [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.3.2|[8.3.2]]]. ---- <span id="2.4"></span> == 2.4 Empirical Confirmation from Exoplanetary Systems == The predictions of CIT align closely with observed exoplanetary systems: * The distribution of exoplanets shows clustering of dust and gas at specific distances. * Protoplanetary disks exhibit gaps that correspond with predicted values of D_pref. * The TRAPPIST system may potentially host a yet-undiscovered giant planet near <math>D_{\text{pref}} = 7.825 \times 10^{10} \ \text{m}</math>. [[File:2.2A Image Excel file Selection calculated Preferred Distance.png|thumb|Selection of an extended Excel file with calculations for the Preferred Distance in exo star systems]] Future observations from the '''James Webb Space Telescope (JWST)''' may provide additional confirmation. ---- '''Observational Challenges at the Preferred Distance (D_pref)''' Detecting giant planets at the '''Preferred Distance''' predicted by CIT is a significant observational challenge. These planets typically have '''long orbital periods''' — often spanning many years or even decades — meaning they may not have completed a full orbit since their host systems were first monitored. Furthermore, it may take equally long for such a planet to transit again, delaying confirmation. Detection also depends on alignment: for radial velocity or transit methods to succeed, the planet must pass in front of the star from Earth’s point of view. In addition, the '''central star must be sufficiently massive''' to generate a detectable planetary body at D_pref. A promising alternative is to focus on '''protoplanetary disks'''. In these early-stage systems, concentric rings and gaps may indicate emerging planets. CIT suggests that the '''Preferred Distance''' corresponds to a balance point between the '''inward-directed influx stream''' and the '''orbital motion''' of the disk’s material — making this region a prime location for early planet formation. Observing such structures can provide indirect evidence supporting CIT’s planetary structuring model. <span id="2.5"></span> == 2.5 Implications for Planetary Formation Models == The connection between VRMS and planetary structuring suggests that: # '''Planetary migration models''' may need to include VRMS-based structuring principles. # '''The gravitational constant (G)''' may reflect deeper connections with kinetic and relativistic parameters. # '''Galactic structure formation''' might follow similar VRMS-based ordering on larger scales. '''Recent observational evidence supports CIT predictions''' A study published in the Publications of the Astronomical Society of Japan presents super-resolution imaging of 78 protoplanetary disks in the Ophiuchus star-forming region: [https://academic.oup.com/pasj/article/77/3/572/8117664 ''ALMA 2D super-resolution imaging survey of Ophiuchus Class I/flat spectrum/II disks. I. Discovery of new disk substructures''] (Shoshi et al., 2025). This study reveals that ring and spiral substructures already emerge a few hundred thousand years after star birth—much earlier than expected in traditional models. These findings significantly increase the statistical sample size over earlier ALMA projects (DSHARP and eDisk) and provide strong evidence for patterned disk evolution very early in stellar formation. A ChatGPT-assisted review of this publication concludes: ''"Relevance to Cosmic Influx Theory (CIT):'' ''This article supports CIT indirectly. CIT emphasizes structured, early planetary formation within dynamic disk environments. The observed early appearance of ring-like substructures aligns with CIT’s prediction that large planets form in dominant rings at Preferred Distances (Dₚᵣₑf) shortly after star formation. The rapid onset of such structuring offers observational backing to CIT’s premise of early, patterned planetary genesis—before classical accretion or migration theories would expect such development."'' (See full analysis: [https://chatgpt.com/share/686224d8-6390-8012-aa4e-e91f2081f568 ChatGPT Share Link]) These observations reinforce the view that the cosmic influx does not act only over long timescales, but plays a formative role from the earliest phases of stellar and planetary development. A disk is the '''primary environment'''; collisions are then a '''secondary consequence''' of that environment. That ordering matters. If one starts from collisions, one risks treating violent impact as the default creative mechanism. But if one starts from disks, then collisions become only one of several natural processes inside an already existing, evolving orbital system. That is also the more physically economical picture: A disk already contains the ingredients for growth, sorting, resonance, fragmentation, accretion, migration, and occasional impacts. So one does not need a catastrophic collision as the first explanatory step. One first asks how matter organizes in the disk, and only then whether some observed debris requires a major impact. In that sense, Saturn’s rings are a good intuitive example. They show that orbiting systems are not static: particles collide, merge, scatter, fragment, and reorganize continuously. But nobody would say the rings are fundamentally “about catastrophe.” They are fundamentally about a '''disk-like orbital structure''' with ongoing internal interactions. CIT can fully admit that collisions occur, including severe ones, while still maintaining that the broader formation of planets and moons is rooted in protoplanetary and circumplanetary disk dynamics. In that framing, catastrophic collisions are real but not foundational. They are episodes within the larger whirling history of disk evolution. This opens the possibility of extending CIT from planetary systems to broader cosmic evolution. === Extension to circumplanetary disks and moon formation === The Preferred Distance concept may also have implications beyond planetary formation around stars. If ''Dpref'' expresses a general structuring tendency in rotating matter systems, then a similar principle may also be expected in circumplanetary disks around forming planets. In that case, the same relation <math>D_{\mathrm{pref}} = \frac{M}{8\pi c^2}</math> with <math>M</math> representing the mass of the planet, can define a local preferred formation zone for moons. This suggests that moon systems may represent a smaller-scale repetition of the larger protoplanetary process. For planets around stars, the Preferred Distance relation can also be written in the familiar orbital form: <math>D_{\mathrm{pref}} = \frac{GM}{\mathrm{VRMS}^2}</math> since in CIT: <math>G = \frac{\mathrm{VRMS}^2}{8\pi c^2}</math> Applying the same logic to moons orbiting planets gives: <math>D = \frac{GM}{v_p^2}</math> where <math>D</math> is the actual orbital distance of the moon and <math>v_p</math> its measured orbital velocity around the planet. Combining this with the Preferred Distance relation gives: <math>\frac{D_{\mathrm{pref}}}{D} = \frac{v_p^2}{\mathrm{VRMS}^2}</math> and thus: <math>v_p^2 = \mathrm{VRMS}^2 \cdot \frac{D_{\mathrm{pref}}}{D}</math> or equivalently: <math>v_p = \mathrm{VRMS}\sqrt{\frac{D_{\mathrm{pref}}}{D}}</math> These equations show that the orbital velocity of a moon can be interpreted as a local circumplanetary expression of the more general CIT structuring principle. In this view, ''VRMS'' is not the direct orbital velocity of the moon itself, but the reference velocity scale from which the local moon velocity follows through the ratio between the Preferred Distance and the actual orbital distance. At the same time, present orbital distances of moons should not automatically be expected to match the local Preferred Distance exactly. Circumplanetary disks consist of a complex mixture of dust, grains, gas, ions, and growing clumps, in which aggregation, fragmentation, thermal exchange, chemical differentiation, ionisation, turbulence, and resonance effects all influence the final structure. Within the CIT framework, these processes may occur while matter also continues to absorb mass-energy from the PEW background. As a result, moons may originate near a preferred zone but later shift through migration, tidal evolution, capture, or long-term dynamical adjustment. This broader interpretation suggests that Preferred Distance should be understood as a large-scale organizing principle rather than as a strict final orbital rule. Planets in a protoplanetary disk and moons in a circumplanetary disk may both arise within structured rings or whirls, while their present positions reflect the combined outcome of original structuring and later evolution. In this way, CIT proposes a hierarchical model of cosmic formation in which similar ordering principles may operate from star systems down to moon systems. == 2.6 Measuring the VRMS == A useful interactive tool for exploring the concept of the root-mean-square velocity (VRMS) in relation to the '''Cosmic Influx Theory (CIT)''' is provided by '''Gabino Casanova''' through his open-source visualisation project: [https://gabinoc67.github.io/interstellar-star-clock/demos/vrms.html https://gabinoc67.github.io/interstellar-star-clock/demos/vrms.html] ==== Overview of the tool ==== The web-based simulator, titled '''CST-Locked v-RMS Gravity Equivalence – VT ≡ CIT''', enables users to explore the relationship between the VRMS velocity scale and the estimated gravitational constant. It computes a live value: :<math>G_{est} = \frac{v_{RMS}^2}{8 \pi c^2}</math> where <math>c</math> is the speed of light and <math>v_{RMS}</math> is the user-adjusted velocity parameter. The interface includes: * Planetary presets (Earth, Sun, Mercury, etc.) * An '''Auto-Scan''' routine that compares <math>G_{est}</math> to the standard terrestrial constant <math>G_0 = 6.67430 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}</math> * Fine-tuning sliders for density, noise, and influx/vibration blending * A visual feedback mode showing whether the computed <math>G_{est}</math> passes or fails the target precision. The tool is accessible here: [https://gabinoc67.github.io/interstellar-star-clock/demos/vrms.html Gabino Casanova – VRMS Demo] ==== Relevance to Cosmic Influx Theory ==== * The simulator links directly to the CIT postulate that gravitational behaviour emerges from a universal velocity scale, expressed as the '''VRMS ≈ 12 278 m/s'''. * By setting the simulator to match <math>G_{est} = G_0</math>, users can empirically test which VRMS values reproduce the gravitational constant predicted by CIT: ::<math>G = \frac{v_{RMS}^2}{8 \pi c^2}</math> * The '''Auto-Calibrate''' function provides an intuitive way to visualise how different VRMS settings influence the computed gravitational constant, supporting CIT’s claim of a kinetic relationship between the influx field and gravitational strength. * The tool’s planetary presets also allow testing how CIT’s universal VRMS could manifest across planetary systems of varying mass and density. ==== Observational and critical notes ==== Although the simulator provides an excellent educational interface, it is primarily a '''conceptual visualizer''' rather than a precise experimental instrument. Key considerations: # The author labels it as a “concept visualizer; substitute observed modes when available”, reminding users that the displayed data are for illustration. # The mapping <math>G_{est} = v_{RMS}^2 / (8 \pi c^2)</math> should be interpreted in CIT as a simplified expression of the proportionality between influx velocity and gravitational coupling. # Adjustable factors such as “noise mix” or “influx/vibration blending” are useful for sensitivity studies but require clear physical interpretation within CIT’s framework of the '''Primordial Elementary Whirlings (PEWs)''' that transfer energy through the influx. # When using planetary presets, readers should remember that CIT’s VRMS refers to the early protoplanetary disk conditions, not the present orbital velocities of planets. ==== Integration into the CIT workflow ==== Within this chapter, the website may serve as a visual bridge between the theoretical and observable realms of VRMS. Readers are encouraged to: * Use the tool to verify that a VRMS around 12.3 km/s yields a <math>G_{est}</math> value close to the accepted constant. * Observe the live deviation '''Δ vs G₀ %''' shown by the simulator. * Compare the results with CIT’s derived equations for gravitational coupling and energy influx. A screenshot or embedded link to this page may be included to illustrate this relationship in action. ==== Reference ==== * Gabino Casanova (2025). '''CST-Locked v-RMS Gravity Equivalence – VT ≡ CIT'''. [https://gabinoc67.github.io/interstellar-star-clock/demos/vrms.html https://gabinoc67.github.io/interstellar-star-clock/demos/vrms.html] * Loeffen, R. (2025). '''Cosmic Influx Theory – Chapter 2: Lorentz Transformation and Mass-Energy Relations.''' Wikiversity. ---- == Summary == This chapter introduced: * The concept of '''VRMS''' and its role in planetary dynamics. * The equation for the '''Preferred Distance (D_pref)''' and the derivation of the Universal Scaling Constant. * How CIT aligns with observed data from exoplanetary systems. * Implications for planetary formation and gravitational theory. In the next chapter, we will explore '''how the Cosmic Influx relates to the gravitational constant (G)'''. ---- ++ Navigation * [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|← Previous Chapter]] * [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)|Back to Main Page]] * [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3|Next Chapter →]] ---- dal50pudyoj55q6z9jknjv1tyop2hi4 0 321521 2801352 2800909 2026-03-29T19:57:54Z James500 297601 /* Japanese and Japan */Add 2801352 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:Oricon|Oricon]] (オリコン) *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). [[Category:Music]] 7x3vk9uilq6au2m3ii58msrzbqs00em 0 323938 2801558 2795316 2026-03-30T10:33:30Z Shantanu-sg-01 3006856 /* Researchers and Practitioners I learn from */ 2801558 wikitext text/x-wiki =Happy Haskell Hours!= This page documents my daily practice and exploration of Haskell and functional programming. The purpose is consistency, clarity of thought, and long-term growth toward scientific computing in Haskell. == Daily Log == A daily log of my ongoing practice and exploration of Haskell and functional programming. The purpose of maintaining this log is consistency rather than performance — to document what I study, what I implement, and what I understand each day. Over time, this record should reflect gradual progress toward building a strong foundation for scientific computing in Haskell. [[Happy haskell hours/Daily log Feb March|Feb - March 2026]] == Intend to Explore == # Course: Functional Programming by Graham Hutton # Introduction to Haskell Course by "Well Typed". # Haskell For Dilettantes # Books by Richard Bird # Domain Specific Language for Mathematics Course. # Competitive Programming # Learn Physics with Functional Programming book. # Scientific computing projects # Panel methods # FEM code # Optimization algorithms # Quantum computing # linear and nonlinear solvers # Numerical Methods implementation # Numerical Analysis # Algorithms for FP # Science of FP # Functional data structures # Courses by Patrik Jansson # Finite difference solvers of CFD # Finite volume method # Grid generation and adaptation algorithms in haskell == Researchers and Practitioners I learn from == # [https://simon.peytonjones.org/ Simon Peyton Jones.] # [https://chalmersfp.github.io/members/rjmh.html John Hughes.] # [https://www.cse.chalmers.se/~koen/ Koen Classen.] # [https://simonmar.github.io/ Simon Marlow.] # [https://patrikja.owlstown.net/ Patrick Jansson.] # [https://martinescardo.github.io/ Martin Escardo.] # [https://www.gla.ac.uk/schools/computing/staff/philtrinder/ Phil Thrinder.] # [https://chalmersfp.github.io/members/ms.html Mary Sheeran.] # [https://chalmersfp.github.io/members/secarl.html Carl-Johan Seger.] # [https://bartoszmilewski.com/ Bartosz Milewski.] # [https://www.uu.nl/staff/GKKeller Gabriele Kellar.] # [https://justtesting.org/ Manual M T Chakravarty.] # [https://www.ost.ch/en/person/farhad-d-mehta-8699 Farhad Mehta] # [http://ozark.hendrix.edu/~yorgey/forest/index/index.xml Brent Yogrey] # [https://www.andres-loeh.de/ Andres Loh] # [https://sergey-goncharov.org/ Sergey Goncharov] # [https://www.lvc.edu/profiles/dr-scott-n-walck/ Scott N Walck] # [https://omelkonian.github.io/ Orestis Melkonian] # [https://jmchapman.io/index.html James Chapman] # [https://dylanbeattie.net/ Dylan beattie] # [https://www.cs.ox.ac.uk/people/jeremy.gibbons/ Jeremy Gibbsons] # [https://people.epfl.ch/daniel.kressner?lang=en Daniel Kressner] - Professor - EPFL - Numerical Algorithms and High Performance Computing == Websites & Podcasts == # [https://chalmersfp.github.io/ Chalmers Functional Programming Group.] # [https://haskell.foundation/podcast/ The Haskell Interlude.] # [https://youtube.com/playlist?list=PLD8gywOEY4HaG5VSrKVnHxCptlJv2GAn7&si=DKXSKaajeL7c9kNr Haskell Unfolder Podcast] # Serokell youtube == Hackage Libraries == # physics # vector # linear # learn-physics # dimensional # chiphunk # LPFP # science-constants # lin-alg # [[Category:Haskell]] 5y6hnuttoisyd2mdjdj3rf124hs6kkd 2801561 2801558 2026-03-30T10:38:58Z Shantanu-sg-01 3006856 /* Researchers and Practitioners I learn from */ 2801561 wikitext text/x-wiki =Happy Haskell Hours!= This page documents my daily practice and exploration of Haskell and functional programming. The purpose is consistency, clarity of thought, and long-term growth toward scientific computing in Haskell. == Daily Log == A daily log of my ongoing practice and exploration of Haskell and functional programming. The purpose of maintaining this log is consistency rather than performance — to document what I study, what I implement, and what I understand each day. Over time, this record should reflect gradual progress toward building a strong foundation for scientific computing in Haskell. [[Happy haskell hours/Daily log Feb March|Feb - March 2026]] == Intend to Explore == # Course: Functional Programming by Graham Hutton # Introduction to Haskell Course by "Well Typed". # Haskell For Dilettantes # Books by Richard Bird # Domain Specific Language for Mathematics Course. # Competitive Programming # Learn Physics with Functional Programming book. # Scientific computing projects # Panel methods # FEM code # Optimization algorithms # Quantum computing # linear and nonlinear solvers # Numerical Methods implementation # Numerical Analysis # Algorithms for FP # Science of FP # Functional data structures # Courses by Patrik Jansson # Finite difference solvers of CFD # Finite volume method # Grid generation and adaptation algorithms in haskell == Researchers and Practitioners I learn from == # [https://simon.peytonjones.org/ Simon Peyton Jones.] # [https://chalmersfp.github.io/members/rjmh.html John Hughes.] # [https://www.cse.chalmers.se/~koen/ Koen Classen.] # [https://simonmar.github.io/ Simon Marlow.] # [https://patrikja.owlstown.net/ Patrick Jansson.] # [https://martinescardo.github.io/ Martin Escardo.] # [https://www.gla.ac.uk/schools/computing/staff/philtrinder/ Phil Thrinder.] # [https://chalmersfp.github.io/members/ms.html Mary Sheeran.] # [https://chalmersfp.github.io/members/secarl.html Carl-Johan Seger.] # [https://bartoszmilewski.com/ Bartosz Milewski.] # [https://www.uu.nl/staff/GKKeller Gabriele Kellar.] # [https://justtesting.org/ Manual M T Chakravarty.] # [https://www.ost.ch/en/person/farhad-d-mehta-8699 Farhad Mehta] # [http://ozark.hendrix.edu/~yorgey/forest/index/index.xml Brent Yogrey] # [https://www.andres-loeh.de/ Andres Loh] # [https://sergey-goncharov.org/ Sergey Goncharov] # [https://www.lvc.edu/profiles/dr-scott-n-walck/ Scott N Walck] # [https://omelkonian.github.io/ Orestis Melkonian] # [https://jmchapman.io/index.html James Chapman] # [https://dylanbeattie.net/ Dylan beattie] # [https://www.cs.ox.ac.uk/people/jeremy.gibbons/ Jeremy Gibbsons] # [https://people.epfl.ch/daniel.kressner?lang=en Daniel Kressner] - Professor - EPFL - Numerical Algorithms and High Performance Computing # [https://people.epfl.ch/laura.grigori?lang=en Laura Grigori] - Professor - EPFL - High Performance Numerical Algorithm # [https://www.nicolasboumal.net/ Nicolas Boumal] - Professor - EPFL - Optimization == Websites & Podcasts == # [https://chalmersfp.github.io/ Chalmers Functional Programming Group.] # [https://haskell.foundation/podcast/ The Haskell Interlude.] # [https://youtube.com/playlist?list=PLD8gywOEY4HaG5VSrKVnHxCptlJv2GAn7&si=DKXSKaajeL7c9kNr Haskell Unfolder Podcast] # Serokell youtube == Hackage Libraries == # physics # vector # linear # learn-physics # dimensional # chiphunk # LPFP # science-constants # lin-alg # [[Category:Haskell]] d6o45rdao5d6hzwreckf8zasx3mvkij 0 326040 2801368 2792057 2026-03-29T22:12:48Z TMorata 860721 corrected order of authors 2801368 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Morocco (orthographic projection).svg|https://en.wikipedia.org/wiki/Morocco}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Morocco Morocco], officially the Kingdom Morocco is located in North Africa.The nation is bordered to the north by the Strait of Gibraltar and the Mediterranean Sea, to the south by Mauritania, to the east by Algeria, and to the west by the Atlantic Ocean. Arabic and Berber are the official languages, while French is commonly used in education, administration, and healthcare. Its position at the crossroads of Africa, the Mediterranean, and Europe has created a mix of peoples and culture enriching the linguistic and phonetic diversity of Moroccan identity. {{HTitle|History of Audiology}} Hearing healthcare services in Morocco developed within otorhinolaryngology (ENT) departments before expanding into broader clinical and rehabilitation services. Before the 1990s, services focused on pure-tone audiometry and basic hearing-aid dispensing. Major progress since the early 2000s includes the introduction of OAE, ABR, ASSR, and ECochG in university hospitals (Casablanca, Rabat, Marrakech, Fes, Agadir). Cochlear implant programs have expanded across sectors, but rural areas still face major access gaps. Audiology as a profession in Morocco is still a developing field. Until recently, hearing aid specialists dominated hearing care in the private sector. ==== Deaf Education ==== Between 1960 and 2023, around 22 schools and units were established for Deaf children, where oral methods were predominantly employed, while sign language was only allowed for communication among the students.<ref>{{Cite journal|title=Overview of Deaf Education in Morocco|url=https://www.mecs-press.org/ijmecs/ijmecs-v16-n3/v16n3-7.html|journal=International Journal of Modern Education and Computer Science|date=2024-06-08|pages=86–100|volume=16|issue=3|doi=10.5815/ijmecs.2024.03.07|last=Department of Didactics, Faculty of Education Sciences, Mohammed V University, Rabat, Morocco|first2=Abdelaziz|last2=Arssi|first3=Otmane|last3=Omari}}</ref> Additionally, most of these schools are run by voluntary organizations, highlighting the need for improved governmental systems for hearing related services. There continues to be a push for the “mainstreaming” of Deaf and hard of hearing children which has lead to the lack of unified Moroccan sign language in and large dropout rate for this population. As a result, over 78% of the Deaf population in Morocco is illiterate, and very few Deaf individuals have had access to middle school education. {{HTitle|Incidence and Prevalence of Hearing Loss}} In Morocco, national data on the incidence of hearing loss are limited. Estimates were obtained for presbycusis in Casablanca, ranging from 6.3%, 13.1%, 25.3% and 36.7% for 60-64 year olds, 65-69 year olds, 70-74 year olds and over 75 years respectively.<ref>{{Cite journal|title=Prevalence of Presbycusis in Casablanca|url=https://journalajrs.com/index.php/AJRS/article/view/130|journal=Asian Journal of Research in Surgery|date=2023-04-18|pages=48–55|language=en|first=Loudghiri|last=Myriam|first2=Larhrabli|last2=Ibtissam|first3=Oukessou|last3=Youssef|first4=Mahtar|last4=Mohamed|first5=Redalah Larb|last5=Abada|first6=Roubal|last6=Mohamed}}</ref> There is a lack of a universal newborn hearing screening policy. However, a study at Ibn Rochd University Hospital in Casablanca found that 1% of screened newborns exhibited mild to moderate hearing loss, while 0.7% had severe unilateral or bilateral hearing loss. Genetic deafness is higher than average in Morocco due to the high prevalence of consanguinity.<ref>{{Cite journal|title=CONSANGUINEOUS MARRIAGES IN MOROCCO AND THE CONSEQUENCE FOR THE INCIDENCE OF AUTOSOMAL RECESSIVE DISORDERS|url=https://www.cambridge.org/core/product/identifier/S0021932009003393/type/journal_article|journal=Journal of Biosocial Science|date=2009-09|issn=0021-9320|pages=575–581|volume=41|issue=5|doi=10.1017/S0021932009003393|language=en|first=I. Cherkaoui|last=Jaouad|first2=S. Chafaï|last2=Elalaoui|first3=A.|last3=Sbiti|first4=F.|last4=Elkerh|first5=L.|last5=Belmahi|first6=A.|last6=Sefiani}}</ref> Consanguineous marriages are traditional in Arab countries like Morocco with a rate estimated at 15.25%, This increases the risk of having children with autosomal recessive disorders, and in this case the increase in genetic hearing impairment and syndromic hearing loss. '''Urban–Rural Disparities''' Urban regions (Casablanca-Settat, Rabat-Salé-Kénitra) benefit from earlier diagnosis and better access to ENT and audiology services. Rural areas show higher prevalence of untreated otitis media, preventable conductive loss, and late-detected childhood hearing loss. {| class="wikitable" |+ Table 1. Examples of Available Prevalence Data in Morocco {{Citation needed}} |- ! Population Group / Region !! Prevalence / Observed Trend !! Source / Setting |- | Newborns (Casablanca) || ~2% mild–moderate HL, ~1% severe HL || Ibn Rochd University Hospital |- | Newborns (Marrakech) || Similar to Casablanca || CHU Marrakech |- | Children (rural provinces) || Higher otitis media and conductive HL || Regional school health programmes |- | Adults (national) || Rising presbycusis and SNHL || Hospital reports |- | Rural populations || Higher overall prevalence || Ministry of Health rural data |- | Urban populations || Earlier diagnosis, better access || ENT/Audiology centres |} {| class="wikitable" |+ Table 2. Age-Related Hearing Loss Risk in Morocco |- ! Age Group !! Observed / Expected Trend !! Notes |- | 0–14 years || 1–3% congenital or early-onset HL || Often linked to otitis media |- | 15–59 years || Increasing SNHL || Noise exposure and chronic disease |- | 60+ years || >35–40% disabling HL expected || Consistent with population aging |} These findings underline the need for national prevalence studies, universal newborn hearing screening, expanded rural ENT/audiology access, and strengthened rehabilitation services including hearing aids, cochlear implants, and speech therapy. {{HTitle|Information About Audiology}} In the public sector, most audiometric testing is conducted for diagnostic purposes and is fully covered by Morocco’s universal health insurance system. Services include pure-tone audiometry, tympanometry, and, in some centers, auditory brainstem response (ABR) evaluations. Due to a shortage of specialized equipment in many hospitals, early screening remains inadequate in several areas. However, the recent procurement of OAE and ABR equipment in urban University hospitals such as Casablanca, Rabat, Marrakech, and Fes has improved hearing screening and diagnostics in those areas. For treatment such hearing aid fittings and cochlear implantation, patients are often referred to the private sector. Because of this, services are often limited in rural areas where access to services and socioeconomic stability is lower. A 2019 study on adults over 60 in Casablanca found that 6.3% of 60-64 year olds, 13.1% of 65-69 year olds, 25.3% of 70-74 year olds, and 36.7% of over 75 year olds had related hearing loss not attributed to a diagnosis. Of these hearing impaired persons, non of their participants wore hearing aids. The main reason attributed to this finding was cost. Hearing aids remain prohibitively costly in Morocco, even in urban areas. '''Audiological Services''' *Core Audiological Services *Pure-tone audiometry *Speech audiometry (Arabic, Darija, French) *Tympanometry & acoustic reflexes *OAE (screening & diagnostics) *Screening ABR *Behavioral audiometry (Conditioned play, VRA) *Hearing-aid fitting & counseling *Aural rehabilitation '''Advanced Audiological Services''' *Diagnostic ABR *ASSR *Electrocochleography: very limited practice in some private center *Cochlear implant assessment and programming *Vestibular testing (VNG, vHIT) *CAPD assessment : limited *Tinnitus counseling: limited availability {{HTitle|Scope of Practice and Licensing}} In the absence of comprehensive audiology training programs nationally, most Moroccan hearing aid specialists follow a three-year training model similar to that in France. These professionals primarily focus on basic audiometric tests and fitting hearing aids but are not equipped to handle more complex diagnoses or vestibular disorders. Morocco relies on a multidisciplinary team for hearing healthcare: *ENT specialists: diagnostics, surgery, cochlear implantation, vestibular care *Audioprothesists: hearing-aid fitting, verification, maintenance *Speech-language therapists: AVT, speech rehabilitation, CI follow-up *Community workers: basic screening, referral, parental guidance '''ENT Specialists (Otolaryngologists)''' As of March 2025, Morocco counts 650 ENT specialists distributed as follows:{{Citation needed}} * Private sector: 408 (63%) * Public hospitals: 197 (30%) * Military hospitals: 25 (4%) * University hospitals: 20 (3%) Their scope includes diagnostic evaluation, ear surgery, CI oversight, rehabilitation coordination, and management of pediatric and chronic ear disorders. '''Hearing-Aid Specialists (Audioprothésistes)''' Hearing-aid specialists practice exclusively in the private sector. Their scope includes selection, fitting, verification, earmold fabrication, counseling, and device maintenance. A prescription from an ENT specialist is mandatory for all fittings. '''Speech-Language Therapists (Orthophonistes)''' Speech-language therapists contribute to aural rehabilitation, auditory–verbal therapy, speech and language intervention, CI follow-up, parental counseling, and early intervention for infants identified through screening. {{HTitle|Professional and Regulatory Bodies}} Morocco does not yet offer a standalone university degree in audiology. Training occurs through ENT residency, speech-language therapy programs, and audioprothesist programs. New initiatives include inter-university diplomas (DIU) and the SMA Masterclass in Clinical Audiology. Continuous professional development is growing through SMA workshops, national congresses, and emerging DIU programs aimed at expanding national expertise. {{HTitle|Ongoing audiology research}} Current research on hearing care in Morocco covers specific conditions, new digital technologies, education for the Deaf community, psychological impacts of hearing impairment among older adults, and the development of sign language resources. <ref>{{Cite journal|last=Department of Didactics, Faculty of Education Sciences, Mohammed V University, Rabat, Morocco|last2=Arssi|first2=Abdelaziz|last3=Omari|first3=Otmane|date=2024-06-08|title=Overview of Deaf Education in Morocco|url=https://www.mecs-press.org/ijmecs/ijmecs-v16-n3/v16n3-7.html|journal=International Journal of Modern Education and Computer Science|volume=16|issue=3|pages=86–100|doi=10.5815/ijmecs.2024.03.07}}</ref> <ref>{{Cite journal|last=Soudi|first=Abdelhadi|last2=Vinopol|first2=Corinne|last3=Van Laerhoven|first3=Kristof|date=2025-04|title=Toward the Creation of a Large-Scale Moroccan Sign Language Corpus|url=https://muse.jhu.edu/article/970568|journal=Sign Language Studies|language=en|volume=25|issue=4|pages=741–767|doi=10.1353/sls.2025.a970568|issn=1533-6263}}</ref> <ref>{{Cite journal|last=Hajjioui|first=A.|last2=Fourtassi|first2=M.|last3=Nejjari|first3=C.|date=2015-03|title=Le handicap au Maroc : étude d’adéquation entre l’offre de soins et le besoin en rééducation|url=https://linkinghub.elsevier.com/retrieve/pii/S0242648X14001182|journal=Journal de Réadaptation Médicale : Pratique et Formation en Médecine Physique et de Réadaptation|language=fr|volume=35|issue=1|pages=33–37|doi=10.1016/j.jrm.2014.11.001}}</ref> <ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-032-01967-7_27|title=Management of Digital Health Technologies in Morocco: Bibliometric Analysis of Trends and Challenges|last=Lyaakobi|first=Amal|last2=Dadouh|first2=Achraf|last3=Kharmoum|first3=Nassim|last4=El Abbadi|first4=Abderrazak|date=2026|publisher=Springer Nature Switzerland|isbn=978-3-032-01966-0|editor-last=Rhalem|editor-first=Wajih|volume=1586|location=Cham|pages=273–287|language=en|doi=10.1007/978-3-032-01967-7_27|editor-last2=AL Idrissi|editor-first2=Najib|editor-last3=Lazaar|editor-first3=Mohamed}}</ref> It includes case studies that focus on gene linked deafness, congenital syndromes, and rare diagnoses. Audiology research is conducted mainly in university hospitals. Key domains include newborn screening, cochlear implant outcomes, epidemiology of hearing loss, and development of speech audiometry materials in Darija and Amazigh. The expansion of NASMA has stimulated research on screening logistics, CI candidacy, and long-term developmental outcomes. {{HTitle|Challenges, Opportunities and Notes}} Morocco’s healthcare system, though equipped with some modern infrastructure, faces challenges such as a lack of specialized audiology professionals, limited availability of equipment in public hospitals, and the absence of a national neonatal hearing screening program. However, with ongoing reforms and increasing interest in digitizing the healthcare sector, Morocco is well-positioned to improve its audiological services in the coming years. {{HTitle|Audiology Charities}} The NASMA Program, under the patronage of HRH Princess Lalla Asmae, integrates newborn hearing screening, diagnostic services, hearing aids, cochlear implantation, and rehabilitation for children from low-income families. It is now the backbone of Morocco’s national childhood hearing-care strategy. The global hearing aid brand Phonak has played a large roll in the development of several Moroccan Hearing Aid Centers (Centre Marocain de Prothèse Auditive-C.m.p.a ) in key cities around the country helping to improve accessibility to services and treatment. {{HTitle|References}} {{reflist}} {{:Global Audiology/Authors-3|Aderdour Lahcen|Madison Saunders|Gabriella Green|Aderdour Lahcen||https://www.linkedin.com/in/lahcen-aderdour-05914b30/|https://www.linkedin.com/in/mcmsaunders/|https://www.linkedin.com/in/gabriellagreen206/}} </div></div> [[Category:Audiology]] [[Category:Morocco]] dokw2nqkvnb0dyalmmi39rgqxp7czw0 2 326765 2801336 2800952 2026-03-29T18:26:30Z Dc.samizdat 2856930 /* Conclusions */ 2801336 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this equally regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=,r_3=,r_4=</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the building block of all the other regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally.{{Efn|Fontaine and Hurley's formula will reveal the geodesics of arbitrary non-uniform polytopes. It can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to its gcd regular polygon.}} Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley formula in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} a4imxecqrpa4hrwigmk6g7ah6vanb38 2801339 2801336 2026-03-29T18:29:30Z Dc.samizdat 2856930 /* Conclusions */ 2801339 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this equally regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=,r_3=,r_4=</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the building block of all the other regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, after fitting the polytope to its gcd regular polygon. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley formula in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} h5l6twg9wcibrkvyh1bh24j41mxyt17 2801343 2801339 2026-03-29T18:53:33Z Dc.samizdat 2856930 /* Conclusions */ 2801343 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this equally regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=,r_3=,r_4=</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the building block of all the other regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} hwkd1215uhr2igftds2hcl97cdlb6ml 2801344 2801343 2026-03-29T19:02:53Z Dc.samizdat 2856930 /* ... */ 2801344 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=,r_3=,r_4=</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the building block of all the other regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} drcqphwqw1ks00jy9da2skfgcu5mnq7 2801345 2801344 2026-03-29T19:20:15Z Dc.samizdat 2856930 /* ... */ 2801345 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=1.84776`,r_3=2.41421`,r_4=2.61313`</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} nfjwmner3ayo8mj5gz9q09ysl438o8e 2801346 2801345 2026-03-29T19:21:02Z Dc.samizdat 2856930 /* ... */ 2801346 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2=1.84776~,r_3=2.41421~,r_4=2.61313~</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ec3f9uf35hd5jxaebapmb1nakxy0nt9 2801347 2801346 2026-03-29T19:22:11Z Dc.samizdat 2856930 /* ... */ 2801347 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 /approx 1.84776,r_3 /approx 2.41421,r_4 /approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} euj2n96w6lc7yej3ekd71qajhvkemkg 2801348 2801347 2026-03-29T19:22:34Z Dc.samizdat 2856930 /* ... */ 2801348 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge! In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} n4crbrwf3dcjkchxubnpaurn6k6en45 2801349 2801348 2026-03-29T19:24:05Z Dc.samizdat 2856930 /* ... */ 2801349 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 4ji5fr478yyp7lzd0wqds9fvb0nao0i 2801350 2801349 2026-03-29T19:48:22Z Dc.samizdat 2856930 /* ... */ 2801350 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == ... == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1p7buupham6axhqd1hsboll0xkt0pei 2801351 2801350 2026-03-29T19:51:53Z Dc.samizdat 2856930 /* ... */ 2801351 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its Petrie polygon. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the 4-simplex. It is one of the most symmetrical polytopes in nature, the dimensional analog of the square and the octahedron, and the common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} taedsaqdl3u3tvkd7u3ecpjkgai7wm4 2801353 2801351 2026-03-29T20:16:13Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801353 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <math>r_1=\sqrt{2}</math> has chords <math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math>. That same regular octagon embedded symmetrically in 4-space has <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the dimensional analog of the octahedron and the square, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices exactly define four orthogonal axes. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} m30loagnb7jucjmwxqn3zltrme4w8dl 2801354 2801353 2026-03-29T20:17:16Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801354 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the dimensional analog of the octahedron and the square, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices exactly define four orthogonal axes. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 72q7j2ygzqm2hf4yd4keo3vd8ygk5yx 2801355 2801354 2026-03-29T20:20:33Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801355 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the dimensional analog of the octahedron and the square, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} tjbk7sl79g15s94gkermm9wuh8nydkq 2801356 2801355 2026-03-29T20:25:23Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801356 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the dimensional analog of the octahedron and the square, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} tvsz36guc414n1ox7oqohw8e0bnmv5j 2801357 2801356 2026-03-29T20:27:35Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801357 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the four-dimensional analogue of the octahedron. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} hslpfnd39mnavweo78qyiflyd8kenig 2801358 2801357 2026-03-29T20:32:35Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801358 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the four-dimensional analogue of the octahedron. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} sqsvyg25q41umyb4ntdf9d8l50x2lyz 2801359 2801358 2026-03-29T20:35:26Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801359 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the four-dimensional analogue of the octahedron. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} dpcqplggxpisie7hn2gnmrswt44xovn 2801363 2801359 2026-03-29T21:30:55Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801363 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the four-dimensional analogue of the octahedron. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} sq67kghbj7jruhp0xo7rajwszfunm6q 2801364 2801363 2026-03-29T21:34:43Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801364 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the four-dimensional analogue of the octahedron. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 0w423p4rk4j85v7x7hf468h8o7oyprq 2801365 2801364 2026-03-29T21:38:20Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801365 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} rmsnfycntjn6jh00zlqcbsw3wrhkjdu 2801372 2801365 2026-03-29T23:28:22Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801372 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>\sqrt{2}</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: <small><math>r_3-r_1-r_1=1/r_3r_2 \approx 0.41421</math></small> == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} qksdzzyrfx6n3zile3zj2el97g01jqp 2801373 2801372 2026-03-29T23:28:58Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801373 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>\sqrt{2}</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<small><math>r_3-r_1-r_1=1/r_3r_2 \approx 0.41421</math></small> == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 7snc5xo59znd7xl88jwd5uji7186sw4 2801374 2801373 2026-03-29T23:29:38Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801374 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>\sqrt{2}</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<small><math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math></small> == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1fuo4ezf6r5gt0idj9ycf1rdl8l9eis 2801378 2801374 2026-03-29T23:36:53Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801378 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>\sqrt{2}</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1</math></small>. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} le5lylamvogr4mb1cajkw5fvgi53l07 2801394 2801378 2026-03-30T02:59:35Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801394 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron facets. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are the square and the triangle, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=\sqrt{2}</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2},r_4=2</math></small>. All its chords except its long diameter are the same length as its edge. In fact they are its 24 edges, and it is a unit-radius 16-cell. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>\sqrt{2}</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 4pindltyyr0gcyds0cmfvkfrpxligyr 2801408 2801394 2026-03-30T03:32:47Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801408 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=1</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon embedded symmetrically in 4-space has <small><math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math></small>. All its chords except its long diameter are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>r_1=1</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} mt7ah8mgqit5uf01z0a0kqz1fhg3ag6 2801410 2801408 2026-03-30T03:34:25Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801410 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=1</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon with edge <small><math>r_1=1</math></small> embedded symmetrically in 4-space has <small><math>r_2=1,r_3=1,r_4=\sqrt{2}</math></small>. All its chords except its long diameter are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of the unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>r_1=1</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} dmb6hx421lya0azbpevhyyhn8vg9gt4 2801411 2801410 2026-03-30T03:35:54Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801411 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=1</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon with edge <small><math>r_1=1</math></small> embedded symmetrically in 4-space has <small><math>r_2=1,r_3=1,r_4=\sqrt{2}</math></small>. All its chords except its long diameter are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. Applied to the regular octagon in the plane of edge <small><math>r_1=1</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} o0znu4saatgxfa4v9cyufz9x148qe4h 2801412 2801411 2026-03-30T03:37:26Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801412 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=1</math></small> has chords <small><math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math></small>. That same regular octagon with edge <small><math>r_1=1</math></small> embedded symmetrically in 4-space has <small><math>r_2=1,r_3=1,r_4=\sqrt{2}</math></small>. All its chords except its long diameter are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. Applied to the regular octagon in the plane of edge <small><math>r_1=1</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} gfrmt05iuoe3km6e1k1tww8xgmz8s3f 2801413 2801412 2026-03-30T03:53:47Z Dc.samizdat 2856930 /* 8-point regular polytopes */ 2801413 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - March 2026}} <blockquote>Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the 120-cell, the largest four-dimensional regular convex polytope.</blockquote> == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. These chords may be considered the 30 most significant discrete distances in geometry. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == 8-point regular polytopes == The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]]. The only planar regular polygons in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. Consider the diagonals of this regular octagon, which lies skew in 4-space, in comparison to those of a two-dimensional regular octagon in the plane. A planar octagon with edge <small><math>r_1=1</math></small> has chords: :<math>r_2 \approx 1.84776,r_3 \approx 2.41421,r_4 \approx 2.61313</math> That same regular octagon with edge <small><math>r_1=1</math></small> embedded symmetrically in 4-space has: :<math>r_2=1,r_3=1,r_4=\sqrt{2}</math> All its chords except its long diameter are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The regular octad has higher symmetry in four-dimensional space than it does in two-dimensional space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. It is one of the most symmetrical polytopes in nature, and the smallest common building block of all the larger regular 4-polytopes. The regular octagon exhibits this high symmetry only when embedded in four-dimensional space. Applied to the regular octagon in the plane of edge <small><math>r_1=1</math></small> , Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <small><math>r_3=\sqrt{2}+1</math></small> and <small><math>1/r_3=\sqrt{2}-1=r_3-2</math></small>. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal invariant planes. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]], because each of the 16-cell's 6 great square planes has another completely orthogonal great square plane. == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} d2ii3cuydachegarcg5qiz5azbw58ln 3 327012 2801318 2800680 2026-03-29T14:10:06Z Juandev 2651 /* Test pages */ new section 2801318 wikitext text/x-wiki ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Custodianship/PieWriter]] == I've closed [[Wikiversity:Candidates for Custodianship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) 9gtmeoi35uv0z1tks8a02vwu0euqnwl 2801414 2801318 2026-03-30T04:00:12Z PieWriter 3039865 /* Test pages */ Reply 2801414 wikitext text/x-wiki ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Custodianship/PieWriter]] == I've closed [[Wikiversity:Candidates for Custodianship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) :@[[User:Juandev|Juandev]] Oh, I didn’t notice that. I thought it was also a test page. Is it possible to undelete the page? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:00, 30 March 2026 (UTC) h995867npmmh2moupxnp8rp1c7shsbb 2801484 2801414 2026-03-30T07:24:59Z Juandev 2651 /* Test pages */ Reply 2801484 wikitext text/x-wiki ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Custodianship/PieWriter]] == I've closed [[Wikiversity:Candidates for Custodianship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) :@[[User:Juandev|Juandev]] Oh, I didn’t notice that. I thought it was also a test page. Is it possible to undelete the page? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:00, 30 March 2026 (UTC) ::Yes it is, try it @[[User:PieWriter|PieWriter]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:24, 30 March 2026 (UTC) ml54ubfuspr5zj3vz2nzz6waupmuez1 2801573 2801484 2026-03-30T10:55:07Z PieWriter 3039865 /* Test pages */ Reply 2801573 wikitext text/x-wiki ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Custodianship/PieWriter]] == I've closed [[Wikiversity:Candidates for Custodianship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) :@[[User:Juandev|Juandev]] Oh, I didn’t notice that. I thought it was also a test page. Is it possible to undelete the page? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:00, 30 March 2026 (UTC) ::Yes it is, try it @[[User:PieWriter|PieWriter]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:24, 30 March 2026 (UTC) :::@[[User:Juandev|Juandev]] I tried but only custodians can undelete [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 10:55, 30 March 2026 (UTC) j6ok34p4izbnqdxmca4od4rh99vuibm 0 327179 2801547 2793703 2026-03-30T10:06:45Z Strivenword 3045506 Added links to the checklist for 4.2 and 5.1. 2801547 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} = PaulLee = * '''My contributions:''' [[Special:Contributions/PaulLee]] == My pages == {{Special:PrefixIndex/DesignWriteStudio/Course/StudentPages/PaulLee/|hideredirects=1|stripprefix=1}} == My assignment checklist == * 1.1 → [[DesignWriteStudio/Course/StudentPages/PaulLee/1.1 Defining Hypertext]] * 1.2 → [[DesignWriteStudio/Course/StudentPages/PaulLee/1.2 Hypertext in the Wild]] * 2.1 → [[DesignWriteStudio/Course/StudentPages/PaulLee/2.1 Hypertext Feature Encyclopedia Entry]] * 2.2 → [[DesignWriteStudio/Course/StudentPages/PaulLee/2.2 Expanded Hypertext Feature Encyclopedia Entry]] * 2.3 → [[DesignWriteStudio/Course/StudentPages/PaulLee/2.3 Validating an encyclopedia entry written by another author]] * 3.1 → [[DesignWriteStudio/Course/StudentPages/PaulLee/3.1 Hypertext Examples Fiction Gaming]] * 3.2 → [[DesignWriteStudio/Course/StudentPages/PaulLee/3.2 Hypertext Examples Catalogues Datastores]] * 3.3 → [[DesignWriteStudio/Course/StudentPages/PaulLee/3.3 Hypertext Design Challenges Introduction]] * 3.4 → [[DesignWriteStudio/Course/StudentPages/PaulLee/3.4 Hypertext Design Challenges Advanced]] * 4.2 → [[DesignWriteStudio/Course/StudentPages/PaulLee/4.2 Midsemester Portfolio]] * 5.1 → [[DesignWriteStudio/Course/StudentPages/PaulLee/5.1 Data as Hypertextual Objects]] {{:DesignWriteStudio/SiteElements/Footer}} [[Category:DesignWriteStudio-StudentPages]] 3b4e60l08joyu3eem585u1sa2ecwst7 0 328087 2801415 2794863 2026-03-30T04:22:34Z CommonsDelinker 9184 Removing [[:c:File:RISC-V_Address.svg|RISC-V_Address.svg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: No ticket permission since 27 February 2026. 2801415 wikitext text/x-wiki == Objectives == # Learn the basic concepts of virtual memory. # Translate virtual memory addresses into physical addresses using the RISC-V address translation scheme. # Implement the virtual memory subsystem of HAWX.<span id="reading"></span> == Reading == * [https://pages.cs.wisc.edu/~remzi/OSTEP/dialogue-vm.pdf OSTEP Chapter 12 - A Dialogue on Memory Virtualization] * [https://pages.cs.wisc.edu/~remzi/OSTEP/vm-intro.pdf OSTEP Chapter 13 - Address Spaces] * [https://pdos.csail.mit.edu/6.1810/2025/xv6/book-riscv-rev5.pdf xv6 Chapter 3 - Page Tables]<span id="introduction"></span> == Introduction == [[w:Virtual%20memory|Virtual memory]] is simultaneously one of the most useful and most [[Wiktionary:vexing|vexing]] hardware features for operating system authors! Computers with virtual memory features provide robust mechanisms for [[w:Process%20(computing)|process]] separation and for extending the amount of useful memory allowing users to do more with less expensive hardware. Once the exclusive domain of large [[w:mainframe|mainframe]] systems, virtual memory is implemented in pretty much every modern computer save for more basic microcontrollers. Though, in fact, some larger microcontrollers such as the [[w:PIC_microcontrollers#PIC32MZ|PIC32MZ]] do have virtual memory controllers. You would have to search for a very long time to find a currently produced general purpose computer architecture that does not have a built in virtual memory controller. Your desktop computer, every server you use, and even your mobile phone all make extensive use of virtual memory. In order to understand virtual memory, we need to make a slight adjustment to our concept of computer memory. What we have seen so far is that we have a two part physical bus, to which we might attach several devices ([[w:Random%20Access%20Memory|RAM]], [[w:hard%20disk|hard disks]], [[w:UART|UART controllers]], [[w:Undergraduate%20education|undergraduates]], etc.) The lines in this bus are divided into three groups: control, address and data. When we want to interact with a device, we place its ''physical address'' on the address bus. If we are writing, we put data on the data lines, if we are reading we read data from the lines. We use the control lines to indicate which operation we are performing, and the bus controller uses arcane magic to service our request. This is the arrangement you are likely to see in a computer architecture course, or just in forming a mental image of how [[w:Pointer%20(computer%20programming)|''pointers'']] work. The list of addresses we have available to us is called the [[w:Address%20space|''address space'']] of the system. You’ve been taught to view memory this way, as a flat array where the address serves as the index and we get the cell that’s specified. This view of memory is called the [[w:Flat%20memory%20model|''flat memory model'']]. In the previous two lessons, we’ve seen that some of the things in that “array” are in fact memory mapped devices, but the flat model visualization still works if we broaden things a little. An address is an index into an array of things, some of the cells are associated with RAM, some with other things. The first computers, early microcomputers, and many present day microcontrollers use this flat memory model. These devices are simpler, and programs all run close to the metal, and so this arrangement keeps the hardware simple. In fact, this model is so desirable that all of our programs are written as though this were really what is going on! Now, in a modern system we have lots of processes all running with various levels of access to the hardware. If our system really did use a flat memory model, we could never protect processes from each other nor could we prevent processes from gaining raw access to memory mapped device. [[w:Chaos|Chaos]] would reign supreme and the system would fall! (Or, really, buggy processes would clobber each other and our disk file systems would be very brittle.) So how does virtual memory save us from this fate? Simply put, a virtual memory controller is a device which sits on the address bus between the CPU and the bus controller. The controller translates every address which passes through it. This means that from the CPU’s perspective, the CPU always deals in ''virtual addresses''. A virtual address is an address which belongs to a virtual address space and must be translated to a ''physical address'' in order to access a device. This means that every pointer value you’ve seen is a lie! (Or at least, an untranslated virtual address.) By controlling the translation hardware, we can control what physical addresses a program can access. So then at its core, virtual memory is actually fairly simple. We have some scheme where we can map one address space onto another. The physical memory is hidden by a layer of abstraction provided by the virtual memory controller, which is in turn controlled by the kernel. As is often the case in programming, the details of how virtual memory works add a bit of complexity. It also tends to be rather difficult to debug, even in a simulated hardware environment like the one we are using. The reason debugging is tricky is that the debugger works with the CPU. This, in turn, means the debugger works with virtual addresses. If we get virtual memory wrong, then the debugger literally cannot find any of the memory we need to inspect. It’s as though we are manipulating the low level fabric of the computer’s reality. Not to worry though! Through a painstaking and arduous process of hexadecimal and binary translation, we can see what’s going on by accessing the machine monitor through [[w:Qemu|qemu]]. == How Virtual Memory Works == <blockquote>While in proximity of the obelisk, a voice in your head murmurs sweet lies and maddening truths. — [https://www.wowhead.com/item=187104/obelisk-of-dark-tidings Obelisk of Dark Tidings], [[w:World%20of%20Warcraft|World of Warcraft]] </blockquote> ''Virtual memory'' is any memory access scheme where virtual addresses are translated into physical addresses. This can take on several different forms. Early systems, such as the [[w:IBM_System/7|IBM System/7]], used a simple base and limit register setup. This scheme is often referred to as “indexed memory” because the base register stores the beginning of the program’s address space and the program’s addresses are offsets to the memory beginning at the base register. If a program attempted to read beyond the address <math display="inline">base + limit</math>, the system raises a fault. This scheme is fairly simple, but mandates that all programs be stored in contiguous memory. This leads to memory fragmentation, and generally complicates the allocation process. Realizing the allocation issues of indexed memory models, hardware designers moved toward a page based virtual memory system. Under this scheme, physical memory is subdivided into a series of ''pages''. A ''[[w:Page_(computer_memory)|page]]'' is just a fixed sized chunk of the system’s overall memory. One early example of this sort of page table system is the Dynamic Address Translation hardware present in the [[w:IBM%20System/370|IBM System/370]] line of mainframe computers, introduced in 1970. This system used a hierarchical ''page table'' which maps virtual pages onto physical pages. Page-based virtual address translation is the norm, and in the present day, when someone says “virtual memory,” they are almost always referring exclusively to this type of organization. Because pages are all the same size, they form the basic unit of allocation for memory. This means that when we build programs, we build them a page at a time. The pages need not be contiguous either! Memory within the pages is contiguous, but the pages may be physically located anywhere in memory. Generally, hardware manufacturers use powers of 2 as their page size, with the most common page size being 4096 bytes (4kb). 4096 works well as a sort of sweet spot for allocation control and granularity. Some architectures have variable page sizes, where the page is specified by setting mode bits within the CPU’s control registers. Generally, though, a kernel will set one page size and then never deviate from that. So we can treat page sizes as fixed. Now, that begs the question, where do we put the pages? Do they start at arbitrary addresses? The answer is, typically “no”. Pages begin at boundaries that are multiples of their size. So for a 4096 byte page, the first five physical pages would be distributed like this: {| class="wikitable" |- ! Physical Page Number (PPN) ! Physical Address (DEC) ! Physical Address (HEX) |- | 0 | 0 | <code>0x00000</code> |- | 1 | 4096 | <code>0x01000</code> |- | 2 | 8192 | <code>0x02000</code> |- | 3 | 12288 | <code>0x03000</code> |- | 4 | 16384 | <code>0x04000</code> |} [Five physical pages of 4KB page address space] As you can see, counting by 4096s is best done in hexadecimal! This works because <math display="inline">4096=2^{12}</math>which means we have 12 bits of offset into the page, which in turn means that the offset is the last 3 hexadecimal digits. Now, this also implies that we have a definite format for our virtual addresses. A virtual address is subdivided into at least two parts. The first indicates the virtual page, the second the offset into the page. Internally, the CPU has a page table which maps virtual pages to physical pages. Putting all this together, we can build a sort of generalized algorithm for virtual address translation: # Separate the virtual address into virtual page and offset. # Look up the virtual page in the page table to acquire the physical page number. # Multiply the physical page number by the size of the pages to acquire the physical page address. # Add the offset to the physical page address to form the full physical address. For example, suppose we have a page table which maps the virtual page 7 to the physical page 4. Further suppose that we are presented with a virtual address <code>0x0702a</code>. We would separate this address into virtual page <code>0x07</code> and offset <code>0x02a</code>. Looking into our table, we find that we are using physical page number <code>0x04</code>, which we then multiply by 4096 to form the physical page address <code>0x04000</code>. Now we add the offset, giving us a physical address of <code>0x0402a</code>. While the basic concept is fairly easy to understand there are a few details that tend to make things a bit more challenging. Several key questions are: * How do we know whether a virtual page has been mapped? * What does the CPU do when it encounters a non-mapped page? * How do we actually store the page table? * What other control mechanisms are present in the virtual memory mapping process? The answers to all of these questions are dependent on the architecture on which your system runs. If you are on an [[w:i386|i386]] machine, the page table will be quite difficult; possibly even the stuff of nightmares. If you are on a [[w:RISC-V|RISC-V]] machine, it will be much cleaner and more programmer friendly. If you are on anything else, you’ll have to look into what they do. The key point is that this is a fundamental aspect of the architecture. After all, every address lookup runs through this process. You must look to the hardware documentation for the full story!<span id="swapping-pages"></span> == Swapping Pages == When [[w:luser|lusers]] (i.e. non-os programmer types) speak of virtual memory they usually are not referring to anything like what I have described thus far. What they mean is the ability for your system to pretend it has more RAM than it actually has. Under this scheme, the OS can extend RAM by using part of the hard disk. What they are really referred to is [[w:Memory%20paging|memory paging]]. Under this scheme, memory is subdivided into pages. When we run out of pages, the operating system will temporarily remove a page from memory and then write it to disk. Then it uses the newly freed page for something else. Should a disk page be required again, the operating system will swap out one of the memory pages for a disk page. Because memory paging operates on swapping, most systems refer to the area of disk set aside for paging as a “swap file” or a “swap partition.” As you can no doubt guess, most virtual memory systems include features to facilitate disk paging. This is why these terms are often conflated. This means that the most popular form of virtual memory in use today is, in fact, paged virtual memory. Obviously this is a match made in OS heaven! The hardware handles the page translation for in-memory pages, and the OS is responsible for swapping pages in and out of disk. The way that it does this is through a combination of careful resource allocation and page fault handling. If a page is not present, but it is needed, the OS will look to the swap file to see if it is there. If it is not, then the OS will raise a fault with the running process. However, if it finds it, it will swap it back in. Paging can be an expensive process. Disks are slow, and memory is essential. This means if a process needs a page that is on disk, we have to pause that process until the disk can have time to load the page into RAM. If you’ve ever used a computer and opened a lot of applications, you’ve no doubt seen the hard disk light turn on. The system grinds to a halt, as the light flickers. What’s happening is you’re seeing a lot of paging activity swapping in and out of memory. We will leave further discussion of paged virtual memory alone for now. Instead, we are going to focus on the virtual page table.<span id="risc-v-virtual-memory"></span> == RISC-V Virtual Memory == Now that we have the basic ideas down, let’s turn our attention to the actual hardware we are using. The RISC-V processor has a clean and powerful method of handling paging. Operating systems like xv6, and HAWX, use this mechanism to separate processes by creating address spaces for each process and for the kernel. The kernel has a 1:1 page table where physical memory is mapped to identical virtual addresses. This makes working with the rest of the system quite easy as the kernel needs to operate close to the metal. The [https://pdos.csail.mit.edu/6.1810/2025/xv6/book-riscv-rev5.pdf xv6 book ] does a very good job explaining this in Chapter 3 - Page tables. I recommend that you take a moment and read through that chapter and the accompanying code before you continue with this section. When you come back, we’re going to hand translate some addresses to get a good feel for how we might go about debugging virtual memory issues. [[File:RISC-V Pagetable.svg|alt=RISC-V Pagetable Structure|thumb|RISC-V Pagetable Structure]] In RISC-V, each page has a variety of flags that can be set for it. These are described in the xv6 reading, as well as in the [https://github.com/riscv/riscv-isa-manual/releases/download/riscv-isa-release-8cf4af7-2026-02-28/riscv-privileged.pdf RISC-V Privileged ISA Manual]. Most should be pretty self explanatory. RISC-V uses four kilobyte pages, and we can map any virtual address onto any physical address. However, a 1:1 mapping table would be impossibly large. The RISC-V has a 64 bit address space. This means that it can address a total of <math display="inline">2^{64} = 18,446,744,073,709,551,616</math> or 16 exabytes of memory. Now, we subdivide this into 4kb pages, of <math display="inline">2^{12} = 4,096</math> byte pages. <math display="inline">2^{64} \div 2^{12} = 2^{52}= 4,503,599,627,370,496</math> possible pages to map. Now, consider that each of these pages have associated with them 10 bits of flag data, and they themselves require 52 bits for physical page numbers. This means that each of these entries must be at least 62 bits long. Rounding to the nearest byte boundary and we find that we need 8 bytes to store these 62 bits. So then the total size of a page table mapping all possible pages for RISC-V would be <math display="inline">8 * 2^{52} = 36,028,797,018,963,968</math>or 32 petabytes per page table. Now, consider that we need a page table for each process, and we need one for the kernel as well. Clearly, we can’t spare that kind of memory! In fact, we probably cannot buy that much memory at all. Very few (if any) real systems will ever approach exhausting the full address space, and so we need a way to sort of “store what we need.” The way this is done in RISC-V is to subdivide the page table into 3 chunks. The L2, L1, and L0 segments of the table. Each of these segments points to an index within a ''page directory''. Each page directory occupies a page of data, and each ''page table entry'' is 8 bytes long. This means we can store 512 page table entries in each page directory, giving us the ability to reference <math display="inline">512^3 = 2^{27}= 134,217,728</math> pages. This is enough to address 512GB in each page table. That should be plenty for our processes! If you observe the address format , you will find that the upper 25 bits of the addresses are labeled “EXT” for “Extended”. Should the need arise for more address space, external hardware could be added to provide additional paging support. This will likely be unnecessary for quite a while though. So now, what we want to do is have a look at some page tables and translate some addresses. For those of you that have been itching to do some binary, you’re in for a treat! As you can tell by the address format and the page table entry format, the fields do not neatly align with [[w:Nibble|nibble]] boundaries. The process for the translation is actually pretty easy, though it can be error prone: # Separate the address into L2, L1, L0, and offset. # Examine the SATP (Supervisor Address Translation and Protection) register to determine if paging is turned on and to find the address of the L2 page directory. # Extract the page table entry from the L2 page directory. # Separate the L2 page table entry into the L1 Physical Page Number and Flags. # Extract the page table entry from the L1 page directory. # Separate the L1 page table entry into the L0 Physical Page Number and Flags. # Extract the page table entry from the L0 page table entry. # Separate the L0 page table entry into the Physical Page Number and Flags. # Calculate the physical address as <code>PA = PPN * 4096 + OFFSET</code> Keep in mind that the RISC-V CPU does this with every byte that it retrieves. It can do this billions of times per second. You’re not going to let a little piece of silicon beat you are you? Ok, yes. Yes you are definitely going to do that. The real question you are probably asking yourself is why you would want to do this by hand? Well, the answer is that if something goes wrong with your page tables, your only hope of debugging it is to examine them manually. (I also rather like asking exam questions where you are presented with a bunch of RAM and asked to translate addresses.) So now, let’s go through that process! I recommend that you follow along, and maybe even try running this experiment yourself. That way you understand how to use the commands that are needed to take a look at the page tables. First, let’s make a program that might do something interesting with virtual memory under xv6. I created the file <code>user/vmfork.c</code> with the following code: <pre>#include &quot;kernel/types.h&quot; #include &quot;kernel/fcntl.h&quot; #include &quot;user/user.h&quot; int main() { int pid; // fork the parent offsets itself slightly and lets the child go first. if(fork()) pause(50); //get the process pid and keep stating it over and over again pid = getpid(); for(;;) { printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); pause(100); } }</pre> To compile and run this, we make the necessary <code>Makefile</code> changes and then execute it: <pre>$ vmfork &amp;pid: 0x0000000000004F9C, pid: 4 &amp;pid: 0x0000000000004F9C, pid: 3 &amp;pid: 0x0000000000004F9C, pid: 4 &amp;pid: 0x0000000000004F9C, pid: 3 &amp;pid: 0x0000000000004F9C, pid: 4 ...</pre> This back and forth between pid 4 and 3 will continue with a little time passing between each one. If you look closely, you will note that both processes have the same address, but different values at that address. That’s virtual memory in action! So now, let’s pry under the hood. The first thing to do is run xv6 in single processor mode. So quit out of [[w:qemu|qemu]], make a set of split terminals, and then run this: <pre>$ CPUS=1 make qemu-gdb</pre> This will start xv6 in gdb mode, but more importantly it will create only one CPU core. This is important because stopping and examining processes is tricky when you are competing among several cores! (You won’t have this problem in HAWX since it runs with only one core active.) Now, in the other terminal start up gdb. <pre>$ gdb-multiarch</pre> We need to find a breakpoint. Stopping inside of a user process can be tricky because the debugger cannot maintain source context easily. So we have to be ready to look at underlying instructions in the process. Open up <code>user/vmfork.asm</code> and look for the main function: <pre>int main() { 0: 1101 addi sp,sp,-32 2: ec06 sd ra,24(sp) 4: e822 sd s0,16(sp) 6: 1000 addi s0,sp,32 int pid; // fork the parent offsets itself slightly and lets the child go first. if(fork()) pause(50); 8: 530000ef jal 538 &lt;fork&gt; c: 87aa mv a5,a0 e: c789 beqz a5,18 &lt;main+0x18&gt; 10: 03200513 li a0,50 14: 5bc000ef jal 5d0 &lt;pause&gt; //get the process pid and keep stating it over and over again pid = getpid(); 18: 5a8000ef jal 5c0 &lt;getpid&gt; 1c: 87aa mv a5,a0 1e: fef42623 sw a5,-20(s0) for(;;) { printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); 22: fec42703 lw a4,-20(s0) 26: fec40793 addi a5,s0,-20 2a: 863a mv a2,a4 2c: 85be mv a1,a5 2e: 00001517 auipc a0,0x1 32: f5250513 addi a0,a0,-174 # f80 &lt;malloc+0x144&gt; 36: 421000ef jal c56 &lt;printf&gt; pause(100); 3a: 06400513 li a0,100 3e: 592000ef jal 5d0 &lt;pause&gt; printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); 42: 0001 nop 44: bff9 j 22 &lt;main+0x22&gt;</pre> I am going to arbitrarily select address <code>0x2c</code> because that is on an instruction boundary and it is within the for loop. Let’s set this as a breakpoint in [[w:Gnu%20Debugger|gdb]] and continue: <pre>(gdb) break *0x2c Breakpoint 1 at 0x2c (gdb) c</pre> This will likely result in xv6 fully starting the shell as it probably doesn’t have an instruction at the same address. However, you may need to run multiple continues to get to where you need to be. Now, once we are there, we are going to want to take a look at physical memory. The only problem is, we can’t do that from the debugger. The debugger only accesses virtual memory, and it is currently inside a user process. That’s where the qemu machine monitor comes in. Click on the qemu terminal and press ''Ctrl-a'' followed by ''c''. That is, press Ctrl-a, release, and then press ''c''. You should see the <code>(qemu)</code> prompt. It’s a lot like gdb, only not quite so beginner friendly. The first thing we need to look at to do this translation is this: <pre>(qemu) info registers CPU#0 V = 0 pc 000000000000002c mhartid 0000000000000000 mstatus 0000000a000000a2 hstatus 0000000200000000 vsstatus 0000000a00000000 mip 0000000000000080 mie 0000000000000220 mideleg 0000000000003666 hideleg 0000000000000000 medeleg 000000000000bfff hedeleg 0000000000000000 mtvec 0000000000000000 stvec 0000003ffffff000 vstvec 0000000000000000 mepc 0000000080001834 sepc 00000000000005c6 vsepc 0000000000000000 mcause 0000000000000000 scause 0000000000000008 vscause 0000000000000000 mtval 0000000000000000 stval 0000000000000000 htval 0000000000000000 mtval2 0000000000000000 mscratch 0000000000000000 sscratch 0000000000000000 satp 8000000000087f54 x0/zero 0000000000000000 x1/ra 000000000000001c x2/sp 0000000000004f90 x3/gp 0505050505050505 x4/tp 0505050505050505 x5/t0 0505050505050505 x6/t1 0505050505050505 x7/t2 0505050505050505 x8/s0 0000000000004fb0 x9/s1 0505050505050505 x10/a0 0000000000000004 x11/a1 0000000000004fe0 x12/a2 0000000000000004 x13/a3 0000000000000001 x14/a4 0000000000000004 x15/a5 0000000000004f9c x16/a6 0505050505050505 x17/a7 000000000000000b x18/s2 0505050505050505 x19/s3 0505050505050505 x20/s4 0505050505050505 x21/s5 0505050505050505 x22/s6 0505050505050505 x23/s7 0505050505050505 x24/s8 0505050505050505 x25/s9 0505050505050505 x26/s10 0505050505050505 x27/s11 0505050505050505 x28/t3 0505050505050505 x29/t4 0505050505050505 x30/t5 0505050505050505 x31/t6 0505050505050505 f0/ft0 0000000000000000 f1/ft1 0000000000000000 f2/ft2 0000000000000000 f3/ft3 0000000000000000 f4/ft4 0000000000000000 f5/ft5 0000000000000000 f6/ft6 0000000000000000 f7/ft7 0000000000000000 f8/fs0 0000000000000000 f9/fs1 0000000000000000 f10/fa0 0000000000000000 f11/fa1 0000000000000000 f12/fa2 0000000000000000 f13/fa3 0000000000000000 f14/fa4 0000000000000000 f15/fa5 0000000000000000 f16/fa6 0000000000000000 f17/fa7 0000000000000000 f18/fs2 0000000000000000 f19/fs3 0000000000000000 f20/fs4 0000000000000000 f21/fs5 0000000000000000 f22/fs6 0000000000000000 f23/fs7 0000000000000000 f24/fs8 0000000000000000 f25/fs9 0000000000000000 f26/fs10 0000000000000000 f27/fs11 0000000000000000 f28/ft8 0000000000000000 f29/ft9 0000000000000000 f30/ft10 0000000000000000 f31/ft11 0000000000000000</pre> This rather impressive looking display is showing all the registers in the CPU, including the privileged ones. (gdb will only ever show you the unprivileged user registers.) The one we are interested in is satp, which has a value of <code>8000000000087f54</code>. (Note: All numbers displayed in the monitor are hexadecimal. Also note that you can use the qemu monitor while the CPU is stopped in the debugger. This is probably the best way to do it.) The satp value has an 8 in the leading [[w:nibble|nibble]], which means that it has a 1 in its most significant bit. This is a flag which means that page protection and translation is active. On the lower end of the satp word, we have <code>87f54</code>. This is the physical page number where we will find the L2 page directory. The address of this page is given by multiplying this number by 4,096 (0x1000 hex): <code>0x87f54000</code>. Before we do that though, we need to choose an address to translate! This program is rather small, so we’ll just use the address that we used for our breakpoint. I’ll carry this process out in a preformatted text section: <pre>1.) Convert 0x2c to binary 0010 1100 2.) The lower 12 bits form the offset, we'll need to pad a bit: 0000 0010 1100 0 2 c offset: 0x02c 3.) The L2, L1, and L0 are given by the next 18 bits, which are all 0: 000000000 000000000 000000000 l2=0 l1=0 l0=0</pre> Ok, so this address is not the most interesting one to work with! But it will make the next parts easy. L2, L1, and L0 are index values into the page directory. Let’s have a look at the page directory by examining physical memory in the qemu monitor: <pre>(qemu) xp/512gx 0x87f54000 0000000087f54000: 0x0000000021fd0401 0x0000000000000000 0000000087f54010: 0x0000000000000000 0x0000000000000000 0000000087f54020: 0x0000000000000000 0x0000000000000000 0000000087f54030: 0x0000000000000000 0x0000000000000000 0000000087f54040: 0x0000000000000000 0x0000000000000000 0000000087f54050: 0x0000000000000000 0x0000000000000000 0000000087f54060: 0x0000000000000000 0x0000000000000000 0000000087f54070: 0x0000000000000000 0x0000000000000000 0000000087f54080: 0x0000000000000000 0x0000000000000000 0000000087f54090: 0x0000000000000000 0x0000000000000000 ... 0000000087f547c0: 0x0000000000000000 0x0000000000000000 0000000087f547d0: 0x0000000000000000 0x0000000000000000 0000000087f547e0: 0x0000000000000000 0x0000000000000000 0000000087f547f0: 0x0000000000000000 0x0000000021fd4c01 0000000087f54800: 0x0000000000000000 0x0000000000000000 0000000087f54810: 0x0000000000000000 0x0000000000000000 ... 0000000087f54f80: 0x0000000000000000 0x0000000000000000 0000000087f54f90: 0x0000000000000000 0x0000000000000000 0000000087f54fa0: 0x0000000000000000 0x0000000000000000 0000000087f54fb0: 0x0000000000000000 0x0000000000000000 0000000087f54fc0: 0x0000000000000000 0x0000000000000000 0000000087f54fd0: 0x0000000000000000 0x0000000000000000 0000000087f54fe0: 0x0000000000000000 0x0000000000000000 0000000087f54ff0: 0x0000000000000000 0x0000000000000000</pre> The regions I skipped were all filled with zeroes. Also, here's what that <code>xp/512gx</code> command means: * xp - Examine Physical Memory * / - begin the formatter (The general format here is <code>xp/fmt physical_address</code> with no spaces) * 512 - This is the number that comes after 511. It is one more than 511 and one less than 513. This is how many units I want the monitor to dump to the screen. * g - Tells the monitor to display "huGe words", which are 64 bit numbers. This is the size of the page table entries. * x - We want the numbers in hexadecimal for ease of use Putting it together, this command will display 512 64 bit numbers, 16 bytes to a row, with row address headings at the beginning. This is an entire 4096 byte page, and therefore the size of the page directory. Now, we need to find the 0th entry, which is at address <code>0x87540000</code> This is the L2 page table entry. It reads <code>0x21fd0401</code>. We need to convert it to binary and examine its flags and PPN. <pre>1.) Convert to binary: 0x21fd0401 0010 0001 1111 1101 0000 0100 0000 0001 2.) Subdivide the number 0010 0001 1111 1101 0000 01 | 00 0000 0001 PPN | FLAGS 3. Convert PPN to hex (Group from right to left in 4 digit increments for greater clarity) 00 1000 0111 1111 0100 0001 0 8 7 f 4 1 4. Thus L2_PPN=0x87f41 and the address of the L1 page directory is 0x87f41000</pre> As a quick sanity check, on this simulated system all physical RAM addresses start with <code>0x8</code> as the first non-zero digit. If you get something else out there, something has gone wrong! Also, note that only the valid flag is set. This is normal for page tables as programs should not be reading or writing them! Now we continue the process with the L1 and L0 side of things (being a little less greedy with our data dumps since we know we only need index 0.): <pre>(qemu) xp/8gx 0x87f41000 0000000087f41000: 0x0000000021fc9001 0x0000000000000000 0000000087f41010: 0x0000000000000000 0x0000000000000000 0000000087f41020: 0x0000000000000000 0x0000000000000000 0000000087f41030: 0x0000000000000000 0x0000000000000000 The L1 index is 0, so we look at the first entry in the table: L1_PTE=0x21fc9001 0010 0001 1111 1100 1001 00 | 00 0000 0001 PPN FLAGS L1_PPN=00 1000 0111 1111 0010 0100 8 7 f 2 4 L0 page directory is at 0x87f24000 (qemu) xp/8gx 0x87f24000 0000000087f24000: 0x0000000021fd145b 0x0000000021fc941b 0000000087f24010: 0x0000000021fc9817 0x0000000021fc9c07 0000000087f24020: 0x0000000021fca0d7 0x0000000000000000 0000000087f24030: 0x0000000000000000 0x0000000000000000 L0 index is 0, so we take the first entry. L0_PTE= 0x21fd145b 0010 0001 1111 1101 0001 01 | 00 0101 1011 PPN FLAGS L0_PPN= 00 1000 0111 1111 0100 0101 8 7 f 4 5 PA = 0x87f45000 + 0x2c = 0x87f4502c</pre> Thus we have shown that the virtual address <code>0x2c</code> translates to physical address <code>0x87f4502c</code>. Also, it’s worth noting the flags that are set: * <code>V</code> - Valid * <code>R</code> - Readable * <code>X</code> - Executable * <code>U</code> - User * <code>G</code> - Global * <code>D</code> - Dirty Valid means the page is present. We can read and execute the code on this page, but not write to it. (That’s what you want for program text.) It is user accessible meaning that the user program can use this memory. I’ll leave off global for now, as well as dirty. (You can look them up if you are curious.) Now, as a final puzzle for you, try translating this address: <pre>0x3ffffff000 0011 1111 1111 1111 1111 1111 1111 0000 0000 0000 L2 L1 L0 0 1111 1111 | 1 1111 1111 | 1 1111 1111 | 0000 0000 0000 0 f f | 1 f f | 1 f f | 0 0 0</pre> This one has different index values. The L2 index is <code>0xff</code>. This is an index into the page directory. Every entry is 8 bytes long, so the total offset into the page directory is <code>0xff * 0x8 = 0x7f8</code>. That’s the only other non-zero entry in our L2 page table! Do you see where it is in my RAM dump above? Can you find it in your own? We’ll find out when you do this week’s written assignment!<span id="conclusion"></span> == Conclusion == Virtual memory allows us to create a custom address space for every program. We can arbitrarily map RAM to any address we like, and we can control access while providing process separation. The details are fairly complex, but the concept is quite simple. Programming it can be a challenge, but if you’ve come this far you should be fine. Good luck!<span id="activities-assignments"></span> == Activities / Assignments == # Continue the address translation of <code>0x3ffffff000</code>. Show your work along with all relevant memory dumps. Answer the following: ## What is the physical address? ## What flags are set here? ## This is the TRAMPOLINE page, used to switch back to the kernel. Why is this mapped here and why are these particular flags set? ## Is there anything surprising about this setup? Why or why not? # [[Operating Systems (Hands-On)/Virtual Memory/HAWX Virtual Memory/|HAWX Virtual Memory]] # [[Operating Systems (Hands-On)/Virtual Memory/Review Quiz/|Review Quiz]] [[Category:Virtual memory]] if5c0adgloglgd3g0hqw24qsd8r7ux3 2801416 2801415 2026-03-30T04:22:38Z CommonsDelinker 9184 Removing [[:c:File:RISC-V_Pagetable.svg|RISC-V_Pagetable.svg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: No ticket permission since 27 February 2026. 2801416 wikitext text/x-wiki == Objectives == # Learn the basic concepts of virtual memory. # Translate virtual memory addresses into physical addresses using the RISC-V address translation scheme. # Implement the virtual memory subsystem of HAWX.<span id="reading"></span> == Reading == * [https://pages.cs.wisc.edu/~remzi/OSTEP/dialogue-vm.pdf OSTEP Chapter 12 - A Dialogue on Memory Virtualization] * [https://pages.cs.wisc.edu/~remzi/OSTEP/vm-intro.pdf OSTEP Chapter 13 - Address Spaces] * [https://pdos.csail.mit.edu/6.1810/2025/xv6/book-riscv-rev5.pdf xv6 Chapter 3 - Page Tables]<span id="introduction"></span> == Introduction == [[w:Virtual%20memory|Virtual memory]] is simultaneously one of the most useful and most [[Wiktionary:vexing|vexing]] hardware features for operating system authors! Computers with virtual memory features provide robust mechanisms for [[w:Process%20(computing)|process]] separation and for extending the amount of useful memory allowing users to do more with less expensive hardware. Once the exclusive domain of large [[w:mainframe|mainframe]] systems, virtual memory is implemented in pretty much every modern computer save for more basic microcontrollers. Though, in fact, some larger microcontrollers such as the [[w:PIC_microcontrollers#PIC32MZ|PIC32MZ]] do have virtual memory controllers. You would have to search for a very long time to find a currently produced general purpose computer architecture that does not have a built in virtual memory controller. Your desktop computer, every server you use, and even your mobile phone all make extensive use of virtual memory. In order to understand virtual memory, we need to make a slight adjustment to our concept of computer memory. What we have seen so far is that we have a two part physical bus, to which we might attach several devices ([[w:Random%20Access%20Memory|RAM]], [[w:hard%20disk|hard disks]], [[w:UART|UART controllers]], [[w:Undergraduate%20education|undergraduates]], etc.) The lines in this bus are divided into three groups: control, address and data. When we want to interact with a device, we place its ''physical address'' on the address bus. If we are writing, we put data on the data lines, if we are reading we read data from the lines. We use the control lines to indicate which operation we are performing, and the bus controller uses arcane magic to service our request. This is the arrangement you are likely to see in a computer architecture course, or just in forming a mental image of how [[w:Pointer%20(computer%20programming)|''pointers'']] work. The list of addresses we have available to us is called the [[w:Address%20space|''address space'']] of the system. You’ve been taught to view memory this way, as a flat array where the address serves as the index and we get the cell that’s specified. This view of memory is called the [[w:Flat%20memory%20model|''flat memory model'']]. In the previous two lessons, we’ve seen that some of the things in that “array” are in fact memory mapped devices, but the flat model visualization still works if we broaden things a little. An address is an index into an array of things, some of the cells are associated with RAM, some with other things. The first computers, early microcomputers, and many present day microcontrollers use this flat memory model. These devices are simpler, and programs all run close to the metal, and so this arrangement keeps the hardware simple. In fact, this model is so desirable that all of our programs are written as though this were really what is going on! Now, in a modern system we have lots of processes all running with various levels of access to the hardware. If our system really did use a flat memory model, we could never protect processes from each other nor could we prevent processes from gaining raw access to memory mapped device. [[w:Chaos|Chaos]] would reign supreme and the system would fall! (Or, really, buggy processes would clobber each other and our disk file systems would be very brittle.) So how does virtual memory save us from this fate? Simply put, a virtual memory controller is a device which sits on the address bus between the CPU and the bus controller. The controller translates every address which passes through it. This means that from the CPU’s perspective, the CPU always deals in ''virtual addresses''. A virtual address is an address which belongs to a virtual address space and must be translated to a ''physical address'' in order to access a device. This means that every pointer value you’ve seen is a lie! (Or at least, an untranslated virtual address.) By controlling the translation hardware, we can control what physical addresses a program can access. So then at its core, virtual memory is actually fairly simple. We have some scheme where we can map one address space onto another. The physical memory is hidden by a layer of abstraction provided by the virtual memory controller, which is in turn controlled by the kernel. As is often the case in programming, the details of how virtual memory works add a bit of complexity. It also tends to be rather difficult to debug, even in a simulated hardware environment like the one we are using. The reason debugging is tricky is that the debugger works with the CPU. This, in turn, means the debugger works with virtual addresses. If we get virtual memory wrong, then the debugger literally cannot find any of the memory we need to inspect. It’s as though we are manipulating the low level fabric of the computer’s reality. Not to worry though! Through a painstaking and arduous process of hexadecimal and binary translation, we can see what’s going on by accessing the machine monitor through [[w:Qemu|qemu]]. == How Virtual Memory Works == <blockquote>While in proximity of the obelisk, a voice in your head murmurs sweet lies and maddening truths. — [https://www.wowhead.com/item=187104/obelisk-of-dark-tidings Obelisk of Dark Tidings], [[w:World%20of%20Warcraft|World of Warcraft]] </blockquote> ''Virtual memory'' is any memory access scheme where virtual addresses are translated into physical addresses. This can take on several different forms. Early systems, such as the [[w:IBM_System/7|IBM System/7]], used a simple base and limit register setup. This scheme is often referred to as “indexed memory” because the base register stores the beginning of the program’s address space and the program’s addresses are offsets to the memory beginning at the base register. If a program attempted to read beyond the address <math display="inline">base + limit</math>, the system raises a fault. This scheme is fairly simple, but mandates that all programs be stored in contiguous memory. This leads to memory fragmentation, and generally complicates the allocation process. Realizing the allocation issues of indexed memory models, hardware designers moved toward a page based virtual memory system. Under this scheme, physical memory is subdivided into a series of ''pages''. A ''[[w:Page_(computer_memory)|page]]'' is just a fixed sized chunk of the system’s overall memory. One early example of this sort of page table system is the Dynamic Address Translation hardware present in the [[w:IBM%20System/370|IBM System/370]] line of mainframe computers, introduced in 1970. This system used a hierarchical ''page table'' which maps virtual pages onto physical pages. Page-based virtual address translation is the norm, and in the present day, when someone says “virtual memory,” they are almost always referring exclusively to this type of organization. Because pages are all the same size, they form the basic unit of allocation for memory. This means that when we build programs, we build them a page at a time. The pages need not be contiguous either! Memory within the pages is contiguous, but the pages may be physically located anywhere in memory. Generally, hardware manufacturers use powers of 2 as their page size, with the most common page size being 4096 bytes (4kb). 4096 works well as a sort of sweet spot for allocation control and granularity. Some architectures have variable page sizes, where the page is specified by setting mode bits within the CPU’s control registers. Generally, though, a kernel will set one page size and then never deviate from that. So we can treat page sizes as fixed. Now, that begs the question, where do we put the pages? Do they start at arbitrary addresses? The answer is, typically “no”. Pages begin at boundaries that are multiples of their size. So for a 4096 byte page, the first five physical pages would be distributed like this: {| class="wikitable" |- ! Physical Page Number (PPN) ! Physical Address (DEC) ! Physical Address (HEX) |- | 0 | 0 | <code>0x00000</code> |- | 1 | 4096 | <code>0x01000</code> |- | 2 | 8192 | <code>0x02000</code> |- | 3 | 12288 | <code>0x03000</code> |- | 4 | 16384 | <code>0x04000</code> |} [Five physical pages of 4KB page address space] As you can see, counting by 4096s is best done in hexadecimal! This works because <math display="inline">4096=2^{12}</math>which means we have 12 bits of offset into the page, which in turn means that the offset is the last 3 hexadecimal digits. Now, this also implies that we have a definite format for our virtual addresses. A virtual address is subdivided into at least two parts. The first indicates the virtual page, the second the offset into the page. Internally, the CPU has a page table which maps virtual pages to physical pages. Putting all this together, we can build a sort of generalized algorithm for virtual address translation: # Separate the virtual address into virtual page and offset. # Look up the virtual page in the page table to acquire the physical page number. # Multiply the physical page number by the size of the pages to acquire the physical page address. # Add the offset to the physical page address to form the full physical address. For example, suppose we have a page table which maps the virtual page 7 to the physical page 4. Further suppose that we are presented with a virtual address <code>0x0702a</code>. We would separate this address into virtual page <code>0x07</code> and offset <code>0x02a</code>. Looking into our table, we find that we are using physical page number <code>0x04</code>, which we then multiply by 4096 to form the physical page address <code>0x04000</code>. Now we add the offset, giving us a physical address of <code>0x0402a</code>. While the basic concept is fairly easy to understand there are a few details that tend to make things a bit more challenging. Several key questions are: * How do we know whether a virtual page has been mapped? * What does the CPU do when it encounters a non-mapped page? * How do we actually store the page table? * What other control mechanisms are present in the virtual memory mapping process? The answers to all of these questions are dependent on the architecture on which your system runs. If you are on an [[w:i386|i386]] machine, the page table will be quite difficult; possibly even the stuff of nightmares. If you are on a [[w:RISC-V|RISC-V]] machine, it will be much cleaner and more programmer friendly. If you are on anything else, you’ll have to look into what they do. The key point is that this is a fundamental aspect of the architecture. After all, every address lookup runs through this process. You must look to the hardware documentation for the full story!<span id="swapping-pages"></span> == Swapping Pages == When [[w:luser|lusers]] (i.e. non-os programmer types) speak of virtual memory they usually are not referring to anything like what I have described thus far. What they mean is the ability for your system to pretend it has more RAM than it actually has. Under this scheme, the OS can extend RAM by using part of the hard disk. What they are really referred to is [[w:Memory%20paging|memory paging]]. Under this scheme, memory is subdivided into pages. When we run out of pages, the operating system will temporarily remove a page from memory and then write it to disk. Then it uses the newly freed page for something else. Should a disk page be required again, the operating system will swap out one of the memory pages for a disk page. Because memory paging operates on swapping, most systems refer to the area of disk set aside for paging as a “swap file” or a “swap partition.” As you can no doubt guess, most virtual memory systems include features to facilitate disk paging. This is why these terms are often conflated. This means that the most popular form of virtual memory in use today is, in fact, paged virtual memory. Obviously this is a match made in OS heaven! The hardware handles the page translation for in-memory pages, and the OS is responsible for swapping pages in and out of disk. The way that it does this is through a combination of careful resource allocation and page fault handling. If a page is not present, but it is needed, the OS will look to the swap file to see if it is there. If it is not, then the OS will raise a fault with the running process. However, if it finds it, it will swap it back in. Paging can be an expensive process. Disks are slow, and memory is essential. This means if a process needs a page that is on disk, we have to pause that process until the disk can have time to load the page into RAM. If you’ve ever used a computer and opened a lot of applications, you’ve no doubt seen the hard disk light turn on. The system grinds to a halt, as the light flickers. What’s happening is you’re seeing a lot of paging activity swapping in and out of memory. We will leave further discussion of paged virtual memory alone for now. Instead, we are going to focus on the virtual page table.<span id="risc-v-virtual-memory"></span> == RISC-V Virtual Memory == Now that we have the basic ideas down, let’s turn our attention to the actual hardware we are using. The RISC-V processor has a clean and powerful method of handling paging. Operating systems like xv6, and HAWX, use this mechanism to separate processes by creating address spaces for each process and for the kernel. The kernel has a 1:1 page table where physical memory is mapped to identical virtual addresses. This makes working with the rest of the system quite easy as the kernel needs to operate close to the metal. The [https://pdos.csail.mit.edu/6.1810/2025/xv6/book-riscv-rev5.pdf xv6 book ] does a very good job explaining this in Chapter 3 - Page tables. I recommend that you take a moment and read through that chapter and the accompanying code before you continue with this section. When you come back, we’re going to hand translate some addresses to get a good feel for how we might go about debugging virtual memory issues. In RISC-V, each page has a variety of flags that can be set for it. These are described in the xv6 reading, as well as in the [https://github.com/riscv/riscv-isa-manual/releases/download/riscv-isa-release-8cf4af7-2026-02-28/riscv-privileged.pdf RISC-V Privileged ISA Manual]. Most should be pretty self explanatory. RISC-V uses four kilobyte pages, and we can map any virtual address onto any physical address. However, a 1:1 mapping table would be impossibly large. The RISC-V has a 64 bit address space. This means that it can address a total of <math display="inline">2^{64} = 18,446,744,073,709,551,616</math> or 16 exabytes of memory. Now, we subdivide this into 4kb pages, of <math display="inline">2^{12} = 4,096</math> byte pages. <math display="inline">2^{64} \div 2^{12} = 2^{52}= 4,503,599,627,370,496</math> possible pages to map. Now, consider that each of these pages have associated with them 10 bits of flag data, and they themselves require 52 bits for physical page numbers. This means that each of these entries must be at least 62 bits long. Rounding to the nearest byte boundary and we find that we need 8 bytes to store these 62 bits. So then the total size of a page table mapping all possible pages for RISC-V would be <math display="inline">8 * 2^{52} = 36,028,797,018,963,968</math>or 32 petabytes per page table. Now, consider that we need a page table for each process, and we need one for the kernel as well. Clearly, we can’t spare that kind of memory! In fact, we probably cannot buy that much memory at all. Very few (if any) real systems will ever approach exhausting the full address space, and so we need a way to sort of “store what we need.” The way this is done in RISC-V is to subdivide the page table into 3 chunks. The L2, L1, and L0 segments of the table. Each of these segments points to an index within a ''page directory''. Each page directory occupies a page of data, and each ''page table entry'' is 8 bytes long. This means we can store 512 page table entries in each page directory, giving us the ability to reference <math display="inline">512^3 = 2^{27}= 134,217,728</math> pages. This is enough to address 512GB in each page table. That should be plenty for our processes! If you observe the address format , you will find that the upper 25 bits of the addresses are labeled “EXT” for “Extended”. Should the need arise for more address space, external hardware could be added to provide additional paging support. This will likely be unnecessary for quite a while though. So now, what we want to do is have a look at some page tables and translate some addresses. For those of you that have been itching to do some binary, you’re in for a treat! As you can tell by the address format and the page table entry format, the fields do not neatly align with [[w:Nibble|nibble]] boundaries. The process for the translation is actually pretty easy, though it can be error prone: # Separate the address into L2, L1, L0, and offset. # Examine the SATP (Supervisor Address Translation and Protection) register to determine if paging is turned on and to find the address of the L2 page directory. # Extract the page table entry from the L2 page directory. # Separate the L2 page table entry into the L1 Physical Page Number and Flags. # Extract the page table entry from the L1 page directory. # Separate the L1 page table entry into the L0 Physical Page Number and Flags. # Extract the page table entry from the L0 page table entry. # Separate the L0 page table entry into the Physical Page Number and Flags. # Calculate the physical address as <code>PA = PPN * 4096 + OFFSET</code> Keep in mind that the RISC-V CPU does this with every byte that it retrieves. It can do this billions of times per second. You’re not going to let a little piece of silicon beat you are you? Ok, yes. Yes you are definitely going to do that. The real question you are probably asking yourself is why you would want to do this by hand? Well, the answer is that if something goes wrong with your page tables, your only hope of debugging it is to examine them manually. (I also rather like asking exam questions where you are presented with a bunch of RAM and asked to translate addresses.) So now, let’s go through that process! I recommend that you follow along, and maybe even try running this experiment yourself. That way you understand how to use the commands that are needed to take a look at the page tables. First, let’s make a program that might do something interesting with virtual memory under xv6. I created the file <code>user/vmfork.c</code> with the following code: <pre>#include &quot;kernel/types.h&quot; #include &quot;kernel/fcntl.h&quot; #include &quot;user/user.h&quot; int main() { int pid; // fork the parent offsets itself slightly and lets the child go first. if(fork()) pause(50); //get the process pid and keep stating it over and over again pid = getpid(); for(;;) { printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); pause(100); } }</pre> To compile and run this, we make the necessary <code>Makefile</code> changes and then execute it: <pre>$ vmfork &amp;pid: 0x0000000000004F9C, pid: 4 &amp;pid: 0x0000000000004F9C, pid: 3 &amp;pid: 0x0000000000004F9C, pid: 4 &amp;pid: 0x0000000000004F9C, pid: 3 &amp;pid: 0x0000000000004F9C, pid: 4 ...</pre> This back and forth between pid 4 and 3 will continue with a little time passing between each one. If you look closely, you will note that both processes have the same address, but different values at that address. That’s virtual memory in action! So now, let’s pry under the hood. The first thing to do is run xv6 in single processor mode. So quit out of [[w:qemu|qemu]], make a set of split terminals, and then run this: <pre>$ CPUS=1 make qemu-gdb</pre> This will start xv6 in gdb mode, but more importantly it will create only one CPU core. This is important because stopping and examining processes is tricky when you are competing among several cores! (You won’t have this problem in HAWX since it runs with only one core active.) Now, in the other terminal start up gdb. <pre>$ gdb-multiarch</pre> We need to find a breakpoint. Stopping inside of a user process can be tricky because the debugger cannot maintain source context easily. So we have to be ready to look at underlying instructions in the process. Open up <code>user/vmfork.asm</code> and look for the main function: <pre>int main() { 0: 1101 addi sp,sp,-32 2: ec06 sd ra,24(sp) 4: e822 sd s0,16(sp) 6: 1000 addi s0,sp,32 int pid; // fork the parent offsets itself slightly and lets the child go first. if(fork()) pause(50); 8: 530000ef jal 538 &lt;fork&gt; c: 87aa mv a5,a0 e: c789 beqz a5,18 &lt;main+0x18&gt; 10: 03200513 li a0,50 14: 5bc000ef jal 5d0 &lt;pause&gt; //get the process pid and keep stating it over and over again pid = getpid(); 18: 5a8000ef jal 5c0 &lt;getpid&gt; 1c: 87aa mv a5,a0 1e: fef42623 sw a5,-20(s0) for(;;) { printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); 22: fec42703 lw a4,-20(s0) 26: fec40793 addi a5,s0,-20 2a: 863a mv a2,a4 2c: 85be mv a1,a5 2e: 00001517 auipc a0,0x1 32: f5250513 addi a0,a0,-174 # f80 &lt;malloc+0x144&gt; 36: 421000ef jal c56 &lt;printf&gt; pause(100); 3a: 06400513 li a0,100 3e: 592000ef jal 5d0 &lt;pause&gt; printf(&quot;&amp;pid: %p, pid: %d\n&quot;, &amp;pid, pid); 42: 0001 nop 44: bff9 j 22 &lt;main+0x22&gt;</pre> I am going to arbitrarily select address <code>0x2c</code> because that is on an instruction boundary and it is within the for loop. Let’s set this as a breakpoint in [[w:Gnu%20Debugger|gdb]] and continue: <pre>(gdb) break *0x2c Breakpoint 1 at 0x2c (gdb) c</pre> This will likely result in xv6 fully starting the shell as it probably doesn’t have an instruction at the same address. However, you may need to run multiple continues to get to where you need to be. Now, once we are there, we are going to want to take a look at physical memory. The only problem is, we can’t do that from the debugger. The debugger only accesses virtual memory, and it is currently inside a user process. That’s where the qemu machine monitor comes in. Click on the qemu terminal and press ''Ctrl-a'' followed by ''c''. That is, press Ctrl-a, release, and then press ''c''. You should see the <code>(qemu)</code> prompt. It’s a lot like gdb, only not quite so beginner friendly. The first thing we need to look at to do this translation is this: <pre>(qemu) info registers CPU#0 V = 0 pc 000000000000002c mhartid 0000000000000000 mstatus 0000000a000000a2 hstatus 0000000200000000 vsstatus 0000000a00000000 mip 0000000000000080 mie 0000000000000220 mideleg 0000000000003666 hideleg 0000000000000000 medeleg 000000000000bfff hedeleg 0000000000000000 mtvec 0000000000000000 stvec 0000003ffffff000 vstvec 0000000000000000 mepc 0000000080001834 sepc 00000000000005c6 vsepc 0000000000000000 mcause 0000000000000000 scause 0000000000000008 vscause 0000000000000000 mtval 0000000000000000 stval 0000000000000000 htval 0000000000000000 mtval2 0000000000000000 mscratch 0000000000000000 sscratch 0000000000000000 satp 8000000000087f54 x0/zero 0000000000000000 x1/ra 000000000000001c x2/sp 0000000000004f90 x3/gp 0505050505050505 x4/tp 0505050505050505 x5/t0 0505050505050505 x6/t1 0505050505050505 x7/t2 0505050505050505 x8/s0 0000000000004fb0 x9/s1 0505050505050505 x10/a0 0000000000000004 x11/a1 0000000000004fe0 x12/a2 0000000000000004 x13/a3 0000000000000001 x14/a4 0000000000000004 x15/a5 0000000000004f9c x16/a6 0505050505050505 x17/a7 000000000000000b x18/s2 0505050505050505 x19/s3 0505050505050505 x20/s4 0505050505050505 x21/s5 0505050505050505 x22/s6 0505050505050505 x23/s7 0505050505050505 x24/s8 0505050505050505 x25/s9 0505050505050505 x26/s10 0505050505050505 x27/s11 0505050505050505 x28/t3 0505050505050505 x29/t4 0505050505050505 x30/t5 0505050505050505 x31/t6 0505050505050505 f0/ft0 0000000000000000 f1/ft1 0000000000000000 f2/ft2 0000000000000000 f3/ft3 0000000000000000 f4/ft4 0000000000000000 f5/ft5 0000000000000000 f6/ft6 0000000000000000 f7/ft7 0000000000000000 f8/fs0 0000000000000000 f9/fs1 0000000000000000 f10/fa0 0000000000000000 f11/fa1 0000000000000000 f12/fa2 0000000000000000 f13/fa3 0000000000000000 f14/fa4 0000000000000000 f15/fa5 0000000000000000 f16/fa6 0000000000000000 f17/fa7 0000000000000000 f18/fs2 0000000000000000 f19/fs3 0000000000000000 f20/fs4 0000000000000000 f21/fs5 0000000000000000 f22/fs6 0000000000000000 f23/fs7 0000000000000000 f24/fs8 0000000000000000 f25/fs9 0000000000000000 f26/fs10 0000000000000000 f27/fs11 0000000000000000 f28/ft8 0000000000000000 f29/ft9 0000000000000000 f30/ft10 0000000000000000 f31/ft11 0000000000000000</pre> This rather impressive looking display is showing all the registers in the CPU, including the privileged ones. (gdb will only ever show you the unprivileged user registers.) The one we are interested in is satp, which has a value of <code>8000000000087f54</code>. (Note: All numbers displayed in the monitor are hexadecimal. Also note that you can use the qemu monitor while the CPU is stopped in the debugger. This is probably the best way to do it.) The satp value has an 8 in the leading [[w:nibble|nibble]], which means that it has a 1 in its most significant bit. This is a flag which means that page protection and translation is active. On the lower end of the satp word, we have <code>87f54</code>. This is the physical page number where we will find the L2 page directory. The address of this page is given by multiplying this number by 4,096 (0x1000 hex): <code>0x87f54000</code>. Before we do that though, we need to choose an address to translate! This program is rather small, so we’ll just use the address that we used for our breakpoint. I’ll carry this process out in a preformatted text section: <pre>1.) Convert 0x2c to binary 0010 1100 2.) The lower 12 bits form the offset, we'll need to pad a bit: 0000 0010 1100 0 2 c offset: 0x02c 3.) The L2, L1, and L0 are given by the next 18 bits, which are all 0: 000000000 000000000 000000000 l2=0 l1=0 l0=0</pre> Ok, so this address is not the most interesting one to work with! But it will make the next parts easy. L2, L1, and L0 are index values into the page directory. Let’s have a look at the page directory by examining physical memory in the qemu monitor: <pre>(qemu) xp/512gx 0x87f54000 0000000087f54000: 0x0000000021fd0401 0x0000000000000000 0000000087f54010: 0x0000000000000000 0x0000000000000000 0000000087f54020: 0x0000000000000000 0x0000000000000000 0000000087f54030: 0x0000000000000000 0x0000000000000000 0000000087f54040: 0x0000000000000000 0x0000000000000000 0000000087f54050: 0x0000000000000000 0x0000000000000000 0000000087f54060: 0x0000000000000000 0x0000000000000000 0000000087f54070: 0x0000000000000000 0x0000000000000000 0000000087f54080: 0x0000000000000000 0x0000000000000000 0000000087f54090: 0x0000000000000000 0x0000000000000000 ... 0000000087f547c0: 0x0000000000000000 0x0000000000000000 0000000087f547d0: 0x0000000000000000 0x0000000000000000 0000000087f547e0: 0x0000000000000000 0x0000000000000000 0000000087f547f0: 0x0000000000000000 0x0000000021fd4c01 0000000087f54800: 0x0000000000000000 0x0000000000000000 0000000087f54810: 0x0000000000000000 0x0000000000000000 ... 0000000087f54f80: 0x0000000000000000 0x0000000000000000 0000000087f54f90: 0x0000000000000000 0x0000000000000000 0000000087f54fa0: 0x0000000000000000 0x0000000000000000 0000000087f54fb0: 0x0000000000000000 0x0000000000000000 0000000087f54fc0: 0x0000000000000000 0x0000000000000000 0000000087f54fd0: 0x0000000000000000 0x0000000000000000 0000000087f54fe0: 0x0000000000000000 0x0000000000000000 0000000087f54ff0: 0x0000000000000000 0x0000000000000000</pre> The regions I skipped were all filled with zeroes. Also, here's what that <code>xp/512gx</code> command means: * xp - Examine Physical Memory * / - begin the formatter (The general format here is <code>xp/fmt physical_address</code> with no spaces) * 512 - This is the number that comes after 511. It is one more than 511 and one less than 513. This is how many units I want the monitor to dump to the screen. * g - Tells the monitor to display "huGe words", which are 64 bit numbers. This is the size of the page table entries. * x - We want the numbers in hexadecimal for ease of use Putting it together, this command will display 512 64 bit numbers, 16 bytes to a row, with row address headings at the beginning. This is an entire 4096 byte page, and therefore the size of the page directory. Now, we need to find the 0th entry, which is at address <code>0x87540000</code> This is the L2 page table entry. It reads <code>0x21fd0401</code>. We need to convert it to binary and examine its flags and PPN. <pre>1.) Convert to binary: 0x21fd0401 0010 0001 1111 1101 0000 0100 0000 0001 2.) Subdivide the number 0010 0001 1111 1101 0000 01 | 00 0000 0001 PPN | FLAGS 3. Convert PPN to hex (Group from right to left in 4 digit increments for greater clarity) 00 1000 0111 1111 0100 0001 0 8 7 f 4 1 4. Thus L2_PPN=0x87f41 and the address of the L1 page directory is 0x87f41000</pre> As a quick sanity check, on this simulated system all physical RAM addresses start with <code>0x8</code> as the first non-zero digit. If you get something else out there, something has gone wrong! Also, note that only the valid flag is set. This is normal for page tables as programs should not be reading or writing them! Now we continue the process with the L1 and L0 side of things (being a little less greedy with our data dumps since we know we only need index 0.): <pre>(qemu) xp/8gx 0x87f41000 0000000087f41000: 0x0000000021fc9001 0x0000000000000000 0000000087f41010: 0x0000000000000000 0x0000000000000000 0000000087f41020: 0x0000000000000000 0x0000000000000000 0000000087f41030: 0x0000000000000000 0x0000000000000000 The L1 index is 0, so we look at the first entry in the table: L1_PTE=0x21fc9001 0010 0001 1111 1100 1001 00 | 00 0000 0001 PPN FLAGS L1_PPN=00 1000 0111 1111 0010 0100 8 7 f 2 4 L0 page directory is at 0x87f24000 (qemu) xp/8gx 0x87f24000 0000000087f24000: 0x0000000021fd145b 0x0000000021fc941b 0000000087f24010: 0x0000000021fc9817 0x0000000021fc9c07 0000000087f24020: 0x0000000021fca0d7 0x0000000000000000 0000000087f24030: 0x0000000000000000 0x0000000000000000 L0 index is 0, so we take the first entry. L0_PTE= 0x21fd145b 0010 0001 1111 1101 0001 01 | 00 0101 1011 PPN FLAGS L0_PPN= 00 1000 0111 1111 0100 0101 8 7 f 4 5 PA = 0x87f45000 + 0x2c = 0x87f4502c</pre> Thus we have shown that the virtual address <code>0x2c</code> translates to physical address <code>0x87f4502c</code>. Also, it’s worth noting the flags that are set: * <code>V</code> - Valid * <code>R</code> - Readable * <code>X</code> - Executable * <code>U</code> - User * <code>G</code> - Global * <code>D</code> - Dirty Valid means the page is present. We can read and execute the code on this page, but not write to it. (That’s what you want for program text.) It is user accessible meaning that the user program can use this memory. I’ll leave off global for now, as well as dirty. (You can look them up if you are curious.) Now, as a final puzzle for you, try translating this address: <pre>0x3ffffff000 0011 1111 1111 1111 1111 1111 1111 0000 0000 0000 L2 L1 L0 0 1111 1111 | 1 1111 1111 | 1 1111 1111 | 0000 0000 0000 0 f f | 1 f f | 1 f f | 0 0 0</pre> This one has different index values. The L2 index is <code>0xff</code>. This is an index into the page directory. Every entry is 8 bytes long, so the total offset into the page directory is <code>0xff * 0x8 = 0x7f8</code>. That’s the only other non-zero entry in our L2 page table! Do you see where it is in my RAM dump above? Can you find it in your own? We’ll find out when you do this week’s written assignment!<span id="conclusion"></span> == Conclusion == Virtual memory allows us to create a custom address space for every program. We can arbitrarily map RAM to any address we like, and we can control access while providing process separation. The details are fairly complex, but the concept is quite simple. Programming it can be a challenge, but if you’ve come this far you should be fine. Good luck!<span id="activities-assignments"></span> == Activities / Assignments == # Continue the address translation of <code>0x3ffffff000</code>. Show your work along with all relevant memory dumps. Answer the following: ## What is the physical address? ## What flags are set here? ## This is the TRAMPOLINE page, used to switch back to the kernel. Why is this mapped here and why are these particular flags set? ## Is there anything surprising about this setup? Why or why not? # [[Operating Systems (Hands-On)/Virtual Memory/HAWX Virtual Memory/|HAWX Virtual Memory]] # [[Operating Systems (Hands-On)/Virtual Memory/Review Quiz/|Review Quiz]] [[Category:Virtual memory]] l3hedo75qg9z87vnte0no6umq20edys 0 328204 2801362 2801180 2026-03-29T21:25:07Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801362 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on the historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC'''. The final deportation and the subsequent destruction of Solomon’s Temple followed in '''586 BC'''. In October '''539 BC''', Babylon fell to the Persian armies, and the captive Judeans were subsequently permitted to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and exactly seventy years since the first recorded deportation in 606 BC. The Second Temple was completed by '''516 BC''', fulfilling the seventy-year "Exile" period calculated from the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400'''. There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 9lrex6jl581x7n2rxfxf2co52s91a3l 2801367 2801362 2026-03-29T22:10:53Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801367 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on the historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC'''. The final deportation and the subsequent destruction of Solomon’s Temple followed in '''586 BC'''. In October '''539 BC''', Babylon fell to the Persian armies, and the captive Judeans were subsequently permitted to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and exactly seventy years since the first recorded deportation in 606 BC. The Second Temple was completed by '''516 BC''', fulfilling the seventy-year "Exile" period calculated from the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400'''. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 6553yt2qmm2e70sh6edbia5asgxqsqk 2801369 2801367 2026-03-29T22:13:30Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801369 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on the historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC'''. The final deportation and the subsequent destruction of Solomon’s Temple followed in '''586 BC'''. In October '''539 BC''', Babylon fell to the Persian armies, and the captive Judeans were subsequently permitted to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and exactly seventy years since the first recorded deportation in 606 BC. The Second Temple was completed by '''516 BC''', fulfilling the seventy-year "Exile" period calculated from the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400'''. === The 4000 year event === [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} c8d2l0gu8e5vikskwjhr93mazyw1d18 2801379 2801369 2026-03-29T23:42:55Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801379 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on the historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400'''. === The 4000 year event === [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} e0i77ob13cudqxkim7b4r3h2x1uktq1 2801380 2801379 2026-03-29T23:44:49Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801380 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400'''. === The 4000 year event === [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} aa09op56m549s70ejp145w6b9rirpu2 2801381 2801380 2026-03-29T23:50:01Z CanonicalMormon 2646631 /* Mathematical Structure of the Later Samaritan Chronology */ 2801381 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> Following this rejection, the Samaritans established their own sanctuary on '''Mount Gerizim'''. Archaeological evidence from the [Mount Gerizim Temple](https://en.wikipedia.org/wiki/Mount_Gerizim_Temple) site indicates that the original temple and sacred precinct were constructed in the mid-5th century BC (c. 450 BC). This initial phase featured a relatively small precinct—roughly 96 by 98 meters—which served as the community's religious hub for approximately 250 years. In the early 2nd century BC, under the reign of '''Antiochus III''', the site underwent a massive reconstruction and expansion. The renovated precinct was significantly larger, utilizing white ashlar stone and featuring a grand entrance staircase and a fortified priestly city (the city of Gerizim) that housed a substantial population. This era appears to have been a significant time of Samaritan prosperity, which might allow the dating of the final Samaritan Pentateuch chronological tradition. If High Priest Amram's chronology was intentionally featuring a significant milestone in the Gerizim temple's construction, by having it occur on creation year 4000 (c. 450 BC), then the final form of the chronology must have been developed around that time. It is unlikely that the chronology was developed earlier and year 4000 just happened to correspond with the building and dedication of the Gerizem Temple. Also, the Samaritan era of prosperity ended in 111 BC when the Hasmonean ruler John Hyrcanus I destroyed both the temple and the adjacent Samaritan city during a military campaign. This destruction was so thorough that the site remained largely abandoned for several centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} iqglh51ppx9c6ucu64zlsbqkgyjc0p4 2801382 2801381 2026-03-29T23:57:37Z CanonicalMormon 2646631 /* The 4000 year event */ 2801382 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} hojdqmukackid8l5u0j68kxsr04yb8y 2801383 2801382 2026-03-29T23:58:15Z CanonicalMormon 2646631 /* The 4000 year event */ 2801383 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 3kkta4nef7w9edt2uimqnmojniiovsf 2801384 2801383 2026-03-30T00:02:47Z CanonicalMormon 2646631 /* The 4000 year event */ 2801384 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries, so the final form of the Samaritan chronology likely took form some time between 450 BC and 111 BC. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} admrs06pjvdva0zj9930gegwm1ysl64 2801385 2801384 2026-03-30T00:03:56Z CanonicalMormon 2646631 /* The 4000 year event */ 2801385 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} okzwcolyw46g1azl1q94dlzb12f736z 2801386 2801385 2026-03-30T00:06:29Z CanonicalMormon 2646631 /* Age at Fatherhood and Miraculous Births */ 2801386 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} gkb2bq6xuz9w52zisos0iuczy8x0qiy 2801396 2801386 2026-03-30T03:07:43Z CanonicalMormon 2646631 /* Summary of Main Arguments */ 2801396 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Comparative Chronology Tables == == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} m3ff4t0dep1mqbr0qvyod9eclt7aps8 2801397 2801396 2026-03-30T03:09:05Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2801397 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} qsvm0a69cb7ra6xymzhlffxf7t6vu1z 2801398 2801397 2026-03-30T03:14:26Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801398 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 6e63rrzged35dissxkjdbzvqi8f8tk2 2801399 2801398 2026-03-30T03:15:29Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801399 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 5e8hybpptw966q0qiki2ct8zlgjclf2 2801401 2801399 2026-03-30T03:20:34Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801401 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} fzllprdz735a9cec3j964qro4edl772 2801402 2801401 2026-03-30T03:22:21Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801402 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} bkmpu8jigk411fhci0rruwohzsm6jdi 2801404 2801402 2026-03-30T03:23:14Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801404 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 21kvyzpt2xhaz6az116ujlraw85ec6f 2801405 2801404 2026-03-30T03:26:49Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801405 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 7uvknt55ikvktf4hyyaiwr9p24bwz2u 2801406 2801405 2026-03-30T03:28:47Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801406 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | colspan="1" style="background-color:#f9f9f9;" | 847 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | colspan="1" style="background-color:#f9f9f9;" | 720 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | colspan="1" style="background-color:#f9f9f9;" | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} 2astb3uabxassjjwhr0pif2bewjl1ez 2801407 2801406 2026-03-30T03:30:33Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801407 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | colspan="1" style="background-color:#f9f9f9;" | 847 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | colspan="1" style="background-color:#f9f9f9;" | 720 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#f9f9f9;" | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | colspan="1" style="background-color:#f9f9f9;" | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="2" style="background-color:#f9f9f9;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} mwpizwbas8w9klgacntqsono6gx4esv 2801409 2801407 2026-03-30T03:33:15Z CanonicalMormon 2646631 /* Preliminary Considerations */ 2801409 wikitext text/x-wiki {{Original research}} This Wikiversity page evaluates and extends mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' written by author Paul D. While the original findings were compelling, the presentation occasionally understated the strength of the underlying data. This page aims to clarify the data's presentation and identify where the mathematical evidence is more robust than originally suggested. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Ages and event years are based on significant numbers (like 7, 40, 49, and 600) rather than biology and history. * '''Alignment with Sacred Cycles:''' The chronologies appear designed to make significant events, like the Exodus and the dedication of Solomon’s Temple, fall on specific "years of the world" (''Anno Mundi''), aligning human history with a perceived divine calendar. * '''The Universal Flood as a Later Insertion:''' The Universality of Noah's Flood was a later insertion into a patriarchal foundation story that originally didn’t have it. Evidence for this includes the way certain lifespans and timelines seem disrupted or recalibrated in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX), to accommodate the Universal scope of the Noah narrative. * '''Chronological Overlaps:''' In the original text (prior to recalibration to accommodate a Universal Flood), the math resulted in overlaps where several patriarchs (such as Methuselah) lived beyond the Flood. To address the article's claims, we must first understand the mathematical foundations of the various witnesses to biblical chronology. The three main patriarchal traditions—the Masoretic Text, Samaritan Pentateuch, and Septuagint—along with the Book of Jubilees, are shown by the Dead Sea Scrolls to have been in use before the Christian era. While surviving fragments do not confirm if every specific begettal and death age matches the versions received today, the writings of early chronographers like Demetrius, Pseudo-Philo, and Josephus indicate that these early traditions were substantially similar. The mathematical logic underlying Jubilees and the Samaritan Pentateuch is detailed in the next section. == Preliminary Considerations == * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | colspan="1" style="background-color:#f9f9f9;" | 847 | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 | colspan="3" style="background-color:#f9f9f9;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | colspan="1" style="background-color:#f9f9f9;" | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | colspan="3" style="background-color:#f9f9f9;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#f9f9f9;" | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | colspan="1" style="background-color:#f9f9f9;" | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="3" style="background-color:#f9f9f9;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == It All Started With Grain == [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209 (compared to year 1207 in existing recensions), which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. === The Mathematical relationship between 40 and 49 === As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> === The Samaritan Pentateuch Connection === Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above diagrams reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch (SP), the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each. His life is juxtaposed against the life of Moses, who lived 120 years, or three generations.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates approximately 2,500 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee, as described in the previous section, are both marked with callouts in this diagram. This includes a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in the year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === The 4000 year event === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). This alignment suggests that the 4000-year milestone may have been architecturally significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After this rejection, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone—aligning the Temple's construction with the year 4000 (c. 450 BC)—then the final form likely developed during this period. It is improbable that such a significant alignment occurred by chance. However, this golden age ended abruptly in 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronology likely reached its definitive form sometime between 450 BC and 111 BC. == The Masoretic Chronology == === Age at Fatherhood and Miraculous Births === Notably, '''twenty patriarchs were over fifty''' at the birth of their sons. This exceeds typical human demographics and creates a tension within the internal logic of Hebrew scripture: if fathering a child at 100 was presented as a unique miracle for Abraham, a long lineage of fathers over fifty—some even nearing that age—effectively '''normalizes''' what should be an exceptional event. Having a long line of ancestors who begat sons at such advanced ages potentially dilutes the miraculous nature of Isaac's birth. It is possible that an earlier tradition—perhaps reflected in the younger begettal ages found in the '''[[wikipedia:Masoretic Text|Masoretic Text]]'''—originally emphasized Abraham’s unique status. However, that message may have been obscured by later attempts to fit the patriarchs into a rigid mathematical structure, such as the "Jubilee of Jubilees" discussed above. == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. === Comparative Chronology Tables === The following tables reconstruct the chronologies of Demetrius and Africanus to illustrate the close corroboration that exists between these early chronographers and High Priest Amram (represented in the '''Samaritan (SP)''' column). Other textual traditions are included for comparison to demonstrate that these reconstructed values are not outliers, but represent typical, probable values within their respective traditions. Note that during the pre-flood era, Priest Amram records ages for Adam through Enoch that are exactly 100 years less than the reconstructed values used by Demetrius and Africanus, accounting for a 700-year difference. Methuselah and Lamech are recorded as 67 and 53 years by Priest Amram, whereas the reconstructed values of 187 and 188 are proposed for the Long Chronology. This results in a total 955-year difference (700 + 120 + 135) in the year of Noah's birth. Subsequently, a minor discrepancy appears in Shem's birth year: Amram and Demetrius utilize Noah's 502nd year, whereas Africanus utilizes the 500th year. This reconstruction completely accounts for the two-year difference between Demetrius and Africanus regarding the year of the Flood (2164 vs. 2162), and it explains the previously noted 953-year discrepancy (955 - 2) between the chronological dates provided by Priest Amram and Africanus. Given this high level of mathematical correlation, the dismissal of these witnesses as "corrupt" or "unreliable" overlooks the evident systemic precision of these early traditions. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} There is an important point here that must be emphasized: the Samaritan chronology concludes exactly in the year 2800—the year the Divine Presence completed the Conquest of Canaan at the head of the armies of Israel. As explained in the previous section, the number 2800 is a highly significant symbolic figure, comprising exactly 70 generations of 40 years each. The previous section also detailed how the Samaritan chronology was likely derived from the ''Book of Jubilees'' framework by systematically increasing the ages of multiple patriarchs. Had the Jubilee or Samaritan chronologies been based strictly on historical or biological data, it is highly improbable that such precise mathematical patterns would exist. As illustrated in the above tables, the year values used in the reconstructed Demetrius and Africanus versions of the Long Chronology are almost identical to the Samaritan chronology for the patriarchs between Shem and the entrance into the Promised Land. Furthermore, the values between Adam and Enoch differ by exactly 100 years. It is unlikely that the long chronologies used by Demetrius and Africanus were originally based on independent historical data that just happened to reduce to exactly 2800 when these mathematical adjustments were applied. It is far more probable that the Samaritan chronology was originally engineered to feature these significant symbolic dates, and the various "Long Chronologies" were simply the result of further systematic adjustments to that primary framework. == A Comparison of Textual Traditions == The following table lists death ages for patriarchs from Adam to Noah. Those later in the chronology and with exceptionally long lives present the possibility of outliving the Flood. Death ages marked in red in the table below and birth ages marked in red in the table above represent potentially problematic figures. It is essential in each textual tradition for either the begettal or death ages to not contain potentially problematic figures. A comparison of the tables reveals how different traditions reconciled patriarchal lifespans with the requirements of a universal Flood. Each source employs a distinct mathematical strategy to handle this chronological "deadline": * '''Book of Jubilees:''' Avoids the conflict entirely by omitting death ages, thus sidestepping the issue of post-Flood survival. * '''Samaritan Pentateuch (SP):''' Maintains lower begettal ages but systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood. * '''Masoretic Text (MT):''' Utilizes lower begettal ages for the earlier patriarchs but shifts to a longer begettal timeframe specifically for the three "problematic" figures to ensure they predecease the Deluge. * '''Septuagint (LXX):''' Generally employs a longer chronology across all generations. The "Whoops" moment identified by Paul D may be a matter of manuscript tradition rather than an original editorial failure. While some witnesses and chronographers like Eusebius record Methuselah's begettal age at 167—creating the "red cell" tension—major witnesses such as '''Codex Alexandrinus''' and later uncial manuscripts preserve the age of 187. This 187-year figure aligns with the Masoretic tradition and resolves the conflict, placing Methuselah’s death six years prior to the Flood. This divergence suggests that the "error" may not have been present in the original translation, but emerged as a variant in later transmission. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="3" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="3" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="3" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="3" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="3" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="2" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#e8e8e8;" | 500 .. 502 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} == The Death of Lamech == The "short chronology" death ages found in the Samaritan Pentateuch (SP) appear to be secondary developments from the "long chronology" traditions. This is supported by the SP's systematic reduction of the lifespans of Jared, Methuselah, and Lamech—a feature absent in the Book of Jubilees, which omits death ages entirely. Conversely, the Masoretic Text (MT) and other "long chronology" witnesses systematically increase the begettal ages for these same three patriarchs, suggesting these higher values are secondary to the older traditions preserved in the "short chronologies." By weighing these textual considerations, we can narrow most patriarchal birth and death ages to a single consensus original value. Shem and Lamech remain the only notable exceptions. The following tables illustrate these reconstructed original values, highlighting the general stability of the source text while isolating the unique instability surrounding Shem’s birth (linked to Gen 11:10) and the divergent traditions for Lamech’s death. * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate a Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at birth of son) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Jubilees ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | Theophilus ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | Africanus ! style="background-color:#fff3e0;" | Demetrius |- ! style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#f0f0f0;" | 130 | colspan="6" style="background-color:#f0f0f0;" | <s>230</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#f0f0f0;" | 105 | colspan="6" style="background-color:#f0f0f0;" | <s>205</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#f0f0f0;" | 90 | colspan="6" style="background-color:#f0f0f0;" | <s>190</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#f0f0f0;" | 70 | colspan="6" style="background-color:#f0f0f0;" | <s>170</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="7" style="background-color:#f0f0f0;" | <s>162</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#f0f0f0;" | 65 | colspan="6" style="background-color:#f0f0f0;" | <s>165</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | <s>187</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>167</s> | colspan="3" style="background-color:#f0f0f0;" | <s>187</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | <s>182</s> | colspan="3" style="background-color:#f0f0f0;" | <s>188</s> | <s>182</s> | colspan="2" style="background-color:#f0f0f0;" | <s>188</s> |- ! style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="8" style="background-color:#f0f0f0;" | 500 | 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | TOTAL YEARS | colspan="2" | 1,207 | 1,556 | colspan="3" | 2,142 | 2,156 | 2,162 | 2,164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Traditions (Age at death) |- ! rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | — ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Septuagint (LXX) ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | Eusebius ! style="background-color:#fff3e0;" | Josephus ! style="background-color:#fff3e0;" | — ! style="background-color:#fff3e0;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | — | <s>847</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | — | <s>720</s> | colspan="7" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | — | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 | — | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | — | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |} There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> == Lectio Difficilior Potior == The principle of Lectio Difficilior Potior (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating the competing death ages for Lamech. In the tables above, Noah is born in Lamech’s 53rd year. If we accept the LXX (Septuagint) figure of 753 as Lamech's death age, a suspiciously neat pattern emerges: Shem is born in Noah’s 500th year, the Flood occurs in Noah’s 600th year, and Lamech dies in Noah’s 700th year. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a 'compressed chronology' (500–600–700) is a hallmark of editorial smoothing—likely a value adjusted during the introduction of the universal flood narrative to create a more 'perfect' structure. Applying Lectio Difficilior, a more 'difficult' and potentially more primitive reading suggests an original 168-year gap—placing Shem’s birth in Noah’s 502nd year and Lamech’s death in Noah’s 670th year. We can therefore reasonably set aside the 753-year figure as being 'too structured to be true,' shifting our focus to the less symmetrical—and likely more authentic—values of 777 or 723. However, the number 777 is itself conspicuously schematic, likely serving as a literary foil to the vengeful Lamech of Cain’s line (Gen 4:24). This raises the question: is 723 the only 'non-schematic' alternative left? Before concluding that any of these individual ages represent a raw, unedited tradition, consider the broader numerical architecture revealed in the table below. When extending the analysis from Adam to Moses, the total lifespan durations in the Masoretic tradition sum to 12,600, while the Eusebius (Armenian) tradition totals 13,200. Both figures are 'perfect' multiples of 600, suggesting that even the seemingly random or 'difficult' individual ages may be precisely calibrated components of an overarching sexagesimal system. (Note: For the patriarchs from Isaac to Moses, where specific variants are not preserved in the Armenian Eusebius tradition, standard consensus lifespans have been utilized to demonstrate how these values remain compatible with the overarching sexagesimal system.) * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} The analysis of these variants reveals a startling conclusion: whether we look at the 'neat' 753, the 'symbolic' 777, or the 'difficult' 723, every candidate for Lamech’s death age eventually resolves into an artificial mathematical structure. This suggests that Lectio Difficilior—the search for the most 'primitive' number—is ultimately a search for the foundational gears of a larger machine. This tradition of engineered longevity mirrors the Sumerian King List, where the ten antediluvian kings are assigned reigns that are all exact multiples of 600. By aligning the total lifespans from Adam to Moses to these same sexagesimal milestones (12,600 and 13,200), the biblical authors were participating in a well-established Near Eastern tradition of sacred mathematics. Ultimately, the ages of the patriarchs in Genesis are not historical records, but a complex mathematical structure designed by ancient authors. Consequently, in the subsequent sections, we will shift our focus: we are no longer looking for 'primitive' numbers, but rather digging into the underlying mathematics that drive this entire chronological engine. We will leave the issue of patriarchal death ages for a while and focus on the birth figures, but before doing so, we should make one more observation to drive home the artificial nature of these number. Paul D made the following observation in the original article: <blockquote> </blockquote> * <span style="color:#b71c1c; font-weight:bold;">Red Cells</span> Indicate figures that could result in patriarchs surviving beyond the stated date of the Flood. * <span style="color:#333333; font-weight:bold;">Blank Cells</span> Indicate where primary sources did not provide specific lifespan or death data. * <span style="color:#333333; font-weight:bold;"><s>Strikethrough</s></span> Indicates values considered secondary developments to accommodate the Universal Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Chronological Tradition Source Values (Age at death) |- ! rowspan="2" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Samaritan (SP) ! style="background-color:#e3f2fd;" | Masoretic (MT) ! style="background-color:#fff3e0;" | Eusebius (Armenian) ! style="background-color:#fff3e0;" | Septuagint (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | <s>847</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | <s>720</s> | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | <s>653</s> | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | <s>753</s> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 436 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 438 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 538 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | - | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 433 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 536 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 464 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 567 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 239 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 342 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 230 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 148 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 198 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 145 | colspan="3" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 136 | style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 137 | colspan="2" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="4" style="background-color:#f0f0f0; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN DURATION SUM | 12,089 | 12,600 | 13,200 | 13,551 |} mx2vgewgifrwjz0sqs87oq91qrorwwk 2 328305 2801321 2800540 2026-03-29T14:51:24Z Tet-Math2 3008895 Accurate 2801321 wikitext text/x-wiki <math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.\ They\ exist\ only\ in\ these</math><math>types\ of\ tetrahedrons.\ Six\ of\ these\ new\ equations\ are\ shown\ side\ by\ side\ with\ the\ Pythagorean\ Theorem</math><math>counterparts\ they\ work\ with.\ Please\ note\ you\ must\ match\ these\ six\ new\ equations\ with\ the\ correct</math><math>Pythagorean\ equations\ as\ shown\ in\ bracketed\ terms.\ Then\ you\ have\ all\ three\ diagonal\ terms\ and\ two\ sets</math><math>of\ the\ same\ rectangular\ terms\ on\ any\ one\ line!\ \ I\ will\ put\ a\ color\ coded\ image\ at\ the\ bottom\ to\ make\ this\ clear.</math> === <math>Wikiversity\ does\ not\ provide\ color\ coding.\ So\ brackets\ are\ used\ instead.</math> === === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>4)\ \ \ c^2-a^2=\bigl(z^2-x^2\bigr)\qquad\ b^2=\bigl(z^2+x^2\bigr)</math></big> === === <big><math>5)\ \ \ b^2-c^2=\bigl(x^2-y^2\bigr)\qquad\ a^2=\bigl(x^2+y^2\bigr)</math></big> === === <big><math>6)\ \ \ a^2-b^2=\bigl(y^2-z^2\bigr)\qquad\ c^2=\bigl(y^2+z^2\bigr)</math></big> === . === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=y^2\qquad\ \ \ \ \ \ a^2-x^2=y^2</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=a^2-x^2</math></big> === === <big><math>\ \ \ \ \ \ 2x^2=a^2+b^2-c^2</math></big> === === <big><math>\ \ \ \ \ \ x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == '''<math>And</math>''' == === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=z^2\qquad\ \ \ \ \ c^2-y^2=z^2</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=c^2-y^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2y^2=a^2+c^2-b^2</math></big> === === <big><math>\ \ \ \ \ \ \ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2}{2}\ }</math></big> === == <math>And</math> == === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2=x^2-z^2\qquad\ \ \ \ b^2=x^2+z^2</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2+z^2=b^2-z^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2z^2=b^2+c^2-a^2</math></big> === === <big><math>\ \ \ \ \ \ \ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === == <math>Again\ these\ 3\ equations\ are\ only\ handy\ side\ effect\ solutions\ or\ Perks.</math> == === <math>x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2\ }{2}}\qquad\ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2\ }{2}}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2\ }{2}}</math> === . === <big><math>These\ \ are\ \ the\ \ other\ \ 5\ \ hidden\ \ equalities\ \ and\ \ by\ \ comparison\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === . <math>Start\ by\ using\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.\ This</math><math>equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem\ section\ at\ the\ bottom.\ Then</math><math>use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === kriy6q6s89t39uh75f7bgkg2bun00bj 2801325 2801321 2026-03-29T16:33:14Z Tet-Math2 3008895 Added tripple Pythagorean Theorem equations 2801325 wikitext text/x-wiki <math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.\ They\ exist\ only\ in\ these</math><math>types\ of\ tetrahedrons.\ Six\ of\ these\ new\ equations\ are\ shown\ side\ by\ side\ with\ the\ Pythagorean\ Theorem</math><math>counterparts\ they\ work\ with.\ Please\ note\ you\ must\ match\ these\ six\ new\ equations\ with\ the\ correct</math><math>Pythagorean\ equations\ as\ shown\ in\ bracketed\ terms.\ Then\ you\ have\ all\ three\ diagonal\ terms\ and\ two\ sets</math><math>of\ the\ same\ rectangular\ terms\ on\ any\ one\ line!\ \ I\ will\ put\ a\ color\ coded\ image\ at\ the\ bottom\ to\ make\ this\ clear.</math> === <math>Wikiversity\ does\ not\ provide\ color\ coding.\ So\ brackets\ are\ used\ instead.</math> === === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>4)\ \ \ c^2-a^2=\bigl(z^2-x^2\bigr)\qquad\ b^2=\bigl(z^2+x^2\bigr)</math></big> === === <big><math>5)\ \ \ b^2-c^2=\bigl(x^2-y^2\bigr)\qquad\ a^2=\bigl(x^2+y^2\bigr)</math></big> === === <big><math>6)\ \ \ a^2-b^2=\bigl(y^2-z^2\bigr)\qquad\ c^2=\bigl(y^2+z^2\bigr)</math></big> === . === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=y^2\qquad\ \ \ \ \ \ a^2-x^2=y^2</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=a^2-x^2</math></big> === === <big><math>\ \ \ \ \ \ 2x^2=a^2+b^2-c^2</math></big> === === <big><math>\ \ \ \ \ \ x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == '''<math>And</math>''' == === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=z^2\qquad\ \ \ \ \ c^2-y^2=z^2</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=c^2-y^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2y^2=a^2+c^2-b^2</math></big> === === <big><math>\ \ \ \ \ \ \ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2}{2}\ }</math></big> === == <math>And</math> == === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2=x^2-z^2\qquad\ \ \ \ b^2=x^2+z^2</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2+z^2=b^2-z^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2z^2=b^2+c^2-a^2</math></big> === === <big><math>\ \ \ \ \ \ \ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === == <math>Again\ these\ 3\ equations\ are\ only\ handy\ side\ effect\ solutions\ or\ Perks.</math> == === <math>x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2\ }{2}}\qquad\ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2\ }{2}}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2\ }{2}}</math> === . === <big><math>These\ \ are\ \ the\ \ other\ \ 5\ \ hidden\ \ equalities\ \ and\ \ by\ \ comparison\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === . <math>Start\ by\ using\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.\ This</math><math>equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem\ section\ at\ the\ bottom.\ Then</math><math>use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>These\ \ are\ \ not\ \ new.\ \ They\ \ are\ \ the\ \ Pythagorean\ \ Theorem\ \ used\ \ 3\ \ times. </math></big> === === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{\ _{_.} a^2+c^2-b^2}{2}\ }</math></big> === . === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === g7kjrcejveidpda8zmpj3t43d4w9czj 2801326 2801325 2026-03-29T16:50:02Z Tet-Math2 3008895 Clarification 2801326 wikitext text/x-wiki <math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.\ They\ exist\ only\ in\ these</math><math>types\ of\ tetrahedrons.\ Six\ of\ these\ new\ equations\ are\ shown\ side\ by\ side\ with\ the\ Pythagorean\ Theorem</math><math>counterparts\ they\ work\ with.\ Please\ note\ you\ must\ match\ these\ six\ new\ equations\ with\ the\ correct</math><math>Pythagorean\ equations\ as\ shown\ in\ bracketed\ terms.\ Then\ you\ have\ all\ three\ diagonal\ terms\ and\ two\ sets</math><math>of\ the\ same\ rectangular\ terms\ on\ any\ one\ line!\ \ I\ will\ put\ a\ color\ coded\ image\ at\ the\ bottom\ to\ make\ this\ clear.</math> === <math>Wikiversity\ does\ not\ provide\ color\ coding.\ So\ brackets\ are\ used\ instead.</math> === === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>4)\ \ \ c^2-a^2=\bigl(z^2-x^2\bigr)\qquad\ b^2=\bigl(z^2+x^2\bigr)</math></big> === === <big><math>5)\ \ \ b^2-c^2=\bigl(x^2-y^2\bigr)\qquad\ a^2=\bigl(x^2+y^2\bigr)</math></big> === === <big><math>6)\ \ \ a^2-b^2=\bigl(y^2-z^2\bigr)\qquad\ c^2=\bigl(y^2+z^2\bigr)</math></big> === . === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=y^2\qquad\ \ \ \ \ \ a^2-x^2=y^2</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=a^2-x^2</math></big> === === <big><math>\ \ \ \ \ \ 2x^2=a^2+b^2-c^2</math></big> === === <big><math>\ \ \ \ \ \ x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == '''<math>And</math>''' == === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=z^2\qquad\ \ \ \ \ c^2-y^2=z^2</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=c^2-y^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2y^2=a^2+c^2-b^2</math></big> === === <big><math>\ \ \ \ \ \ \ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2}{2}\ }</math></big> === == <math>And</math> == === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2=x^2-z^2\qquad\ \ \ \ b^2=x^2+z^2</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2+z^2=b^2-z^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2z^2=b^2+c^2-a^2</math></big> === === <big><math>\ \ \ \ \ \ \ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === == <math>Again\ these\ 3\ equations\ are\ only\ handy\ side\ effect\ solutions\ or\ Perks.</math> == === <math>x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2\ }{2}}\qquad\ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2\ }{2}}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2\ }{2}}</math> === . === <big><math>These\ \ are\ \ the\ \ other\ \ 5\ \ hidden\ \ equalities\ \ and\ \ by\ \ comparison\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === . <math>Start\ by\ using\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.\ This</math><math>equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem\ section\ at\ the\ bottom.\ Then</math><math>use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>These\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{\ _{_.} a^2+c^2-b^2}{2}\ }</math></big> === . === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === stckqwwbtwgvdzkch3i6apu92kei7fj 2801328 2801326 2026-03-29T17:16:02Z Tet-Math2 3008895 2801328 wikitext text/x-wiki <math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.\ They\ exist\ only\ in\ these</math><math>types\ of\ tetrahedrons.\ Six\ of\ these\ new\ equations\ are\ shown\ side\ by\ side\ with\ the\ Pythagorean\ Theorem</math><math>counterparts\ they\ work\ with.\ Please\ note\ you\ must\ match\ these\ six\ new\ equations\ with\ the\ correct</math><math>Pythagorean\ equations\ as\ shown\ in\ bracketed\ terms.\ Then\ you\ have\ all\ three\ diagonal\ terms\ and\ two\ sets</math><math>of\ the\ same\ rectangular\ terms\ on\ any\ one\ line!\ \ I\ will\ put\ a\ color\ coded\ image\ at\ the\ bottom\ to\ make\ this\ clear.</math> === <math>Wikiversity\ does\ not\ provide\ color\ coding.\ So\ brackets\ are\ used\ instead.</math> === === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>4)\ \ \ c^2-a^2=\bigl(z^2-x^2\bigr)\qquad\ b^2=\bigl(z^2+x^2\bigr)</math></big> === === <big><math>5)\ \ \ b^2-c^2=\bigl(x^2-y^2\bigr)\qquad\ a^2=\bigl(x^2+y^2\bigr)</math></big> === === <big><math>6)\ \ \ a^2-b^2=\bigl(y^2-z^2\bigr)\qquad\ c^2=\bigl(y^2+z^2\bigr)</math></big> === . === <big><math>\ \ \ \ \ \ New\ Equations\qquad\quad\ \ \ Pythagorean\ Equations</math></big> === === <big><math>1)\ \ \ c^2-b^2=\bigl(y^2-x^2\bigr)\qquad\ a^2=\bigl(y^2+x^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=y^2\qquad\ \ \ \ \ \ a^2-x^2=y^2</math></big> === === <big><math>\ \ \ \ \ \ c^2-b^2+x^2=a^2-x^2</math></big> === === <big><math>\ \ \ \ \ \ 2x^2=a^2+b^2-c^2</math></big> === === <big><math>\ \ \ \ \ \ x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == '''<math>And</math>''' == === <big><math>2)\ \ \ b^2-a^2=\bigl(z^2-y^2\bigr)\qquad\ c^2=\bigl(z^2+y^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=z^2\qquad\ \ \ \ \ c^2-y^2=z^2</math></big> === === <big><math>\ \ \ \ \ \ \ b^2-a^2+y^2=c^2-y^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2y^2=a^2+c^2-b^2</math></big> === === <big><math>\ \ \ \ \ \ \ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2}{2}\ }</math></big> === == <math>And</math> == === <big><math>3)\ \ \ a^2-c^2=\bigl(x^2-z^2\bigr)\qquad\ b^2=\bigl(x^2+z^2\bigr)</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2=x^2-z^2\qquad\ \ \ \ b^2=x^2+z^2</math></big> === === <big><math>\ \ \ \ \ \ \ a^2-c^2+z^2=b^2-z^2</math></big> === === <big><math>\ \ \ \ \ \ \ 2z^2=b^2+c^2-a^2</math></big> === === <big><math>\ \ \ \ \ \ \ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === == <math>Again\ these\ 3\ equations\ are\ only\ handy\ side\ effect\ solutions\ or\ Perks.</math> == === <math>x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2\ }{2}}\qquad\ y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2\ }{2}}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2\ }{2}}</math> === . === <big><math>These\ \ are\ \ the\ \ other\ \ 5\ \ hidden\ \ equalities\ \ and\ \ by\ \ comparison\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === . <math>Start\ by\ using\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.\ This</math><math>equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem\ section\ at\ the\ bottom.\ Then</math><math>use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>These\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === . === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === . === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === 5aqbj9n88049u5gxmcicptdst0je9xc 2801393 2801328 2026-03-30T02:46:00Z Tet-Math2 3008895 Total Re-Configure 2801393 wikitext text/x-wiki <math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.\ They\ exist\ only\ in\ these</math><math>types\ of\ tetrahedrons.\ Six\ of\ these\ new\ equations\ are\ shown\ side\ by\ side\ with\ the\ Pythagorean\ Theorem</math><math>counterparts\ they\ work\ with.\ Please\ note\ you\ must\ match\ these\ six\ new\ equations\ with\ the\ correct</math><math>Pythagorean\ equations\ as\ shown\ in\ bracketed\ terms.\ Then\ you\ have\ all\ three\ diagonal\ terms\ and\ two\ sets</math><math>of\ the\ same\ rectangular\ terms\ on\ any\ one\ line!\ \ I\ will\ put\ a\ color\ coded\ image\ at\ the\ bottom\ to\ make\ this\ clear.</math> === <big><math>These\ \ are\ \ the\ \ other\ \ 5\ \ hidden\ \ equalities\ \ and\ \ by\ \ comparison\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === . <math>Start\ by\ using\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.\ This</math><math>equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem\ section\ at\ the\ bottom.\ Then</math><math>use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>These\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === . === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === . === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === o88wk2yo790q6or7aoiv64afirkej86 2801395 2801393 2026-03-30T02:59:40Z Tet-Math2 3008895 2801395 wikitext text/x-wiki <big><math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <big><math>Here\ \ are\ \ 5\ \ hidden\ \ equalities\ \ and\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ 5\ equations\ in\ total.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math> == === <big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === . === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === . === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.\ \ It\ \ is\ \ acurate.</math> <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <math>Given\ \ a=14.4\ \ b=10\ \ c=12</math> === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>These\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === . === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === . === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === psnm3p23lztq5zdabl1ve0qi3sun9rz 2801400 2801395 2026-03-30T03:20:20Z Tet-Math2 3008895 Tidy Monster Again 2801400 wikitext text/x-wiki <big><math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <big><math>Here\ \ are\ \ 5\ \ hidden\ \ equalities\ \ and\ \ they\ \ are\ \ magical:</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big><big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> == === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === === <big><math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> === <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ in\ a\ new\ way\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === ecgsdd8y08g39tkjkx1idcny96flpmn 2801403 2801400 2026-03-30T03:22:29Z Tet-Math2 3008895 2801403 wikitext text/x-wiki <big><math>There\ are\ eleven\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <big><math>Here\ \ are\ \ 5\ \ hidden\ \ equalities\ \ and\ \ they\ \ are\ \ the\ \ magical\ \ ones.</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> == <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big><big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> == === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === === <big><math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math></big> === <big><math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <big><math>All\ \ of \ \ these \ \ are \ \ equal</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <big><math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean </math><math>Theorem\ in\ a\ new\ way\ 3\ times\ each\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math></big> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === rvsg9b5en6xez4rf0sufg1er62j7usb 2801422 2801403 2026-03-30T05:00:32Z Tet-Math2 3008895 Clarification 2801422 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <big><math>Here\ \ are\ \ 5\ \ hidden\ \ equalities\ \ and\ \ they\ \ are\ \ the\ \ magical\ \ ones.</math><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K=x^2+y^2+z^2</math></big> == == <big><math>K=a^2+z^2</math></big> == == <big><math>K=b^2+y^2</math></big> == == <big><math>K=c^2+x^2</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big><big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === === <big><math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == === <math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math> === === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === 1gm7t8ktl9jjus0xkg8bnzo1smuai5x 2801426 2801422 2026-03-30T05:13:43Z Tet-Math2 3008895 2801426 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ \ are\ \ the\ \ 5\ \ hidden\ \ equalities\ \ and\ \ they\ \ are\ \ the\ \ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K_2=x^2+y^2+z^2</math></big> == == <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> == == <big><math>K_4=b^2+y^2\qquad\ Black</math></big> == == <big><math>K_5=c^2+x^2\qquad\ Red</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === === <big><math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === gxlilha11cytlhi6382e335ej05e81e 2801427 2801426 2026-03-30T05:15:14Z Tet-Math2 3008895 2801427 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equations\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K_2=x^2+y^2+z^2</math></big> == == <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> == == <big><math>K_4=b^2+y^2\qquad\ Black</math></big> == == <big><math>K_5=c^2+x^2\qquad\ Red</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ connect\ to\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><big><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math></big> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <big><math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math></big> === === <big><math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . == <big><math>PROOF\ -\ Try\ \ these\ \ test\ \ numbers\ \ for\ \ yourself</math></big> == === <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === lh23pydiwkv28ddat2yk45spmezqxv5 2801436 2801427 2026-03-30T05:41:50Z Tet-Math2 3008895 2801436 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K_2=x^2+y^2+z^2</math></big> == == <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> == == <big><math>K_4=b^2+y^2\qquad\ Black</math></big> == == <big><math>K_5=c^2+x^2\qquad\ Red</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ \ is\ \ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === 4pyq1a2sq1ghiri2db26aw61db6kt16 2801438 2801436 2026-03-30T05:46:06Z Tet-Math2 3008895 2801438 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === == <big><math>K_2=x^2+y^2+z^2</math></big> == == <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> == == <big><math>K_4=b^2+y^2\qquad\ Black</math></big> == == <big><math>K_5=c^2+x^2\qquad\ Red</math></big> == . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === s1et5ldcbx4x4x5acgi9jjtmq4dlbxl 2801440 2801438 2026-03-30T05:48:11Z Tet-Math2 3008895 The layout keeps scrambling up on me !!! 2801440 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K_2=x^2+y^2+z^2</math></big> === === <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> === === <big><math>K_4=b^2+y^2\qquad\ Black</math></big> === === <big><math>K_5=c^2+x^2\qquad\ Red</math></big> === . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === ib3z31a2sx174d0ren3tt4vgqc5jjv3 2801441 2801440 2026-03-30T05:48:58Z Tet-Math2 3008895 2801441 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K_2=x^2+y^2+z^2</math></big> === === <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> === === <big><math>K_4=b^2+y^2\qquad\ Black</math></big> === === <big><math>K_5=c^2+x^2\qquad\ Red</math></big> === . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === . === === . === === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === f0838m0uigjy60th51wjc1k3bi1u1nj 2801442 2801441 2026-03-30T05:49:19Z Tet-Math2 3008895 2801442 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K_2=x^2+y^2+z^2</math></big> === === <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> === === <big><math>K_4=b^2+y^2\qquad\ Black</math></big> === === <big><math>K_5=c^2+x^2\qquad\ Red</math></big> === . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === . === === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math><math>ALL\ Triangles\qquad\qquad\ ONLY\ Trirectangular\ Tetrahedrons</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === 1y2skvnnmquqx86ukaauiqp82qh94of 2801444 2801442 2026-03-30T05:54:49Z Tet-Math2 3008895 Oh shit - not again 2801444 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K_2=x^2+y^2+z^2</math></big> === === <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> === === <big><math>K_4=b^2+y^2\qquad\ Black</math></big> === === <big><math>K_5=c^2+x^2\qquad\ Red</math></big> === . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === . === === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === . <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === tinoq0tyncf44fccyol82pfyy94ttje 2801445 2801444 2026-03-30T05:58:59Z Tet-Math2 3008895 2801445 wikitext text/x-wiki <big><math>There\ are\ 5\ hidden\ mathematical\ equalities\ in\ TriRectangular\ Tetrahedrons.</math><math>They\ exist\ only\ in\ these\ types\ of\ tetrahedrons.</math></big> === <math>Here\ are\ the\ 5\ hidden\ equalities\ and\ they\ are\ the\ magical.</math><big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === === <math>Use\ the\ equation\ \ K=\frac{\ a^2+b^2+c^2}{2}\ \ because\ all\ 3\ diagonal\ elements\ a,b,c\ are\ always\ obvious.</math><math>This\ equation\ is\ a\ modified\ version\ of\ Heron's\ semi\ perimeter.\ See\ Heron's\ Theorem</math><math>section\ at\ the\ bottom.\ Then\ use\ a,b,c\ \ to\ calculate\ x,y,z\ \ as\ shown\ below.</math> === .[[File:ABC=ZYX.png|thumb|708x708px|Cali Cat]] === <big><math>K\ equals\ all\ 5\ equations.</math></big> === === <big><math>K_1=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K_2=x^2+y^2+z^2</math></big> === === <big><math>K_3=a^2+z^2\qquad\ Blue</math></big> === === <big><math>K_4=b^2+y^2\qquad\ Black</math></big> === === <big><math>K_5=c^2+x^2\qquad\ Red</math></big> === . <math>And\ they\ interlock\ in\ such\ a\ way\ they\ greatly</math><math>simplify\ TriRectangular\ problems.\ Here\ the</math><math>mating\ pairs\ are\ color\ coded\ for\ easy</math><math>understanding.\ The\ other\ way\ to\ remember</math><math>which\ diagonal\ line\ mates\ with\ it's\ rectangular</math><math>line\ is\ to\ realize\ the\ mating\ pairs\ are\ the\ only</math><math>lines\ that\ never\ touch\ each\ other.</math> <math>Thus\ \ A\ Mates\ only\ with\ Z\ \ \ \ BLUE</math><math>\ \ \ \ \ \ \ \ \ \ \ B\ Mates\ only\ with\ Y\ \ \ \ BLACK</math><math>\ \ \ \ \ \ \ \ \ \ \ C\ Mates\ only\ with\ X\ \ \ \ RED</math> === . === === <math>Given\ \ a,b,c\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ x,y,z</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math>. === <big><math>Given\ \ x,y,z\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ a,b,c</math></big> === <math>Then\ \ a=\sqrt{_{_.}K-z^2\ }\qquad\ b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math> === === <math>Or\ \ \ \ \ \ a=\sqrt{\ x^2+y^2\ }\qquad\ b=\sqrt{_{_.}x^2+z^2\ }\qquad\ c=\sqrt{\ y^2-z^2\ }</math> === == . == === <math>This\ \ amazing\ \ equation\ \ was\ \ found\ \ on\ \ the\ \ internet.\ \ It's\ \ not\ \ my\ \ own\ \ work.</math><big><math>It\ is\ acurate.</math></big> === === <big><math>h=\frac{xyz}{\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ }\ \ \ It\ came\ in\ this\ form\ \ \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === . == <big><math>Semi\ Perimeters</math></big> == <big><math>Heron's\ Theorem\qquad\ \ \ \ TriRectangular\ Theorem\ \ ???</math></big> === <big><math>S=\frac{\ a+b+c}{2}\qquad\ \ \ K=\frac{\ a^2+b^2+c^2}{2}</math></big> === == . == <u><big><math>Proof\ -\ Try\ these\ test\ numbers\ for\ yourself</math></big></u> <big><math>Given\ \ a=14.4\ \ b=10\ \ c=12</math></big> === <math>K_1=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === <big><math>K_2=x^2+y^2+z^2\ \ =225.68</math><math>K_3=a^2+z^2\qquad\ \ \ =225.68\qquad a=14.4\qquad z=4.2801869118065393196504607419769</math><math>K_4=b^2+y^2\qquad\ \ \ =225.68\qquad b=10\qquad \ \ \ \ y=11.210709165793214885544254958172</math><math>K_5=c^2+x^2\qquad\ \ \ =225.68\qquad c=12\qquad \ \ \ \ x=9.0376988221560027818758211287983</math></big> === <math>All\ of \ these \ are \ equal\ therefore \ K=K_1,K_2,K_3,K_4,K_5</math> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === == . == === <math>The\ equations\ below\ are\ not\ new.\ They\ simply\ use\ the\ Pythagorean\ Theorem\ in\ a\ new\ way.</math><math>The\ Pythagorean\ Theorem\ is\ used\ 3\ times\ in\ each\ equation\ to\ directly\ extract\ x,y,z\ from \ a,b,c.</math><math>These\ can\ be\ derived\ more\ easily\ by\ using\ the\ K=K_1,K_2,K_3,K_4,K_5\ system\ above.</math> === == <math>Derive\ x</math> == === <big><math>c^2=y^2+z^2</math></big> === === <big><math>\qquad\ y^2=(a^2-x^2)</math></big> === === <big><math>\qquad\ z^2=(b^2-x^2)</math></big> === === <big><math>c^2=(a^2-x^2)+(b^2-x^2)</math></big> === === <big><math>c^2=a^2+b^2-2x^2</math></big> === === <big><math>2x^2=a^2+b^2-c^2</math></big> === === <big><math>x^2=\frac{\ a^2+b^2-c^2}{2}\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>b^2=x^2+z^2</math></big> === === <big><math>\qquad\ x^2=(a^2-y^2)</math></big> === === <big><math>\qquad\ z^2=(c^2-y^2)</math></big> === === <big><math>b^2=(a^2-y^2)+(c^2-y^2)</math></big> === === <big><math>b^2=a^2+c^2-2y^2</math></big> === === <big><math>2y^2=a^2+c^2-b^2</math></big> === === <big><math>y^2=\frac{\ a^2+c^2-b^2}{2}\qquad\ y=\sqrt{\ \frac{_{_.} a^2+c^2-b^2}{2}\ }</math></big> === == <math>Derive\ y</math> == === <big><math>a^2=x^2+y^2</math></big> === === <big><math>\qquad\ x^2=(b^2-z^2)</math></big> === === <big><math>\qquad\ y^2=(c^2-z^2)</math></big> === === <big><math>a^2=(b^2-z^2)+(c^2-z^2)</math></big> === === <big><math>a^2=b^2+c^2-2z^2</math></big> === === <big><math>2z^2=b^2+c^2-a^2</math></big> === === <big><math>z^2=\frac{\ b^2+c^2-a^2}{2}\qquad\ z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2}{2}\ }</math></big> === c0492sy7x5gxftb93wazdo3l70o0vby 0 328352 2801461 2801221 2026-03-30T06:54:48Z Qianqian Z 3057576 /* Impact on Digital Media Platforms */ 2801461 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional social media that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at Nankai University in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at Microsoft. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data algorithms. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on artificial intelligence and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} nnc97k3ea95mriadnyw6cb4ahbwe9ac 2801466 2801461 2026-03-30T07:08:38Z Qianqian Z 3057576 /* Introduction */ 2801466 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at Nankai University in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at Microsoft. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data algorithms. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on artificial intelligence and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} matfkwhd6ja0750l7ikkdwyejsmvpue 2801467 2801466 2026-03-30T07:11:19Z Qianqian Z 3057576 /* Background of Zhang Yiming */ 2801467 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at Nankai University in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at Microsoft. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on artificial intelligence and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} qwrnwh1vmsy27gii1n4w89jhleofefq 2801470 2801467 2026-03-30T07:13:26Z Qianqian Z 3057576 /* Background of Zhang Yiming */ 2801470 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at Microsoft. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on artificial intelligence and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} ry09gqpwww7q0bny5iocy2f8mpzvjl7 2801471 2801470 2026-03-30T07:14:20Z Qianqian Z 3057576 /* Background of Zhang Yiming */ 2801471 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on artificial intelligence and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} 6mpbhtkw8onbaiork0xjs7qtks2wzkb 2801473 2801471 2026-03-30T07:15:46Z Qianqian Z 3057576 /* Founding of ByteDance and Douyin */ 2801473 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of TikTok. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} r16347csrkjkv4ttu3nxxdckf3wguoh 2801475 2801473 2026-03-30T07:16:27Z Qianqian Z 3057576 /* Founding of ByteDance and Douyin */ 2801475 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees.Douyin became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} hoqxdvqrgfiq9i6ot6ajo9c5osgd92o 2801477 2801475 2026-03-30T07:18:24Z Qianqian Z 3057576 /* Founding of ByteDance and Douyin */ 2801477 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees. [[wikipedia:Douyin|Douyin]], the Chinese version of TikTok, became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that algorithm-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a recommendation system that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} nbh9d9y6f3z2y81lydf1e2svh5dunyn 2801479 2801477 2026-03-30T07:21:27Z Qianqian Z 3057576 /* Algorithm-Driven Content Recommendation */ 2801479 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees. [[wikipedia:Douyin|Douyin]], the Chinese version of TikTok, became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that [[wikipedia:Algorithm|algorithm]]-driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a [[wikipedia:Recommender system|recommendation system]] that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} qmed7iyzxbn5kud1xc8o25bsswkvyii 2801481 2801479 2026-03-30T07:23:14Z Qianqian Z 3057576 /* Algorithm-Driven Content Recommendation */ 2801481 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees. [[wikipedia:Douyin|Douyin]], the Chinese version of TikTok, became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that Algorithm driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a [[wikipedia:Recommender system|recommendation system]] that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the World Internet Conference in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} tvw5iow314ntvkgpf6l4h14f3lh4sqm 2801486 2801481 2026-03-30T07:26:38Z Qianqian Z 3057576 /* Impact on Digital Media Platforms */ 2801486 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees. [[wikipedia:Douyin|Douyin]], the Chinese version of TikTok, became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that Algorithm driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a [[wikipedia:Recommender system|recommendation system]] that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the [[wikipedia:World Internet Conference|World Internet Conference]] in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as Instagram, YouTube, and Facebook have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} jxpepthec6hue8fgztlvgmk3byeoo05 2801488 2801486 2026-03-30T07:28:09Z Qianqian Z 3057576 /* Impact on Digital Media Platforms */ 2801488 wikitext text/x-wiki Zhang Yiming and Algorithm-Driven Content Platforms == Introduction == [[File:The_CEO_Magazine_Cover.jpg|thumb|250px|Zhang Yiming on the cover of The CEO Magazine]] Zhang Yiming is the founder of [[wikipedia:ByteDance|ByteDance]], the company behind TikTok. His work has had a major influence on the development of algorithm-driven content platforms. Unlike traditional [[Social media|social media]] that focuses on friend networks, ByteDance platforms rely heavily on artificial intelligence and recommendation algorithms to deliver personalized content to users. == Background of Zhang Yiming == Zhang Yiming was born in Longyan, Fujian Province, a region known for Hakka culture. He studied software engineering at [[wikipedia:Nankai University|Nankai University]] in Tianjin. Before founding ByteDance, Zhang worked at several internet start-ups, including Kuxun and Fanfou, and briefly worked at [[wikipedia:Microsoft|Microsoft]]. He also founded the real-estate search platform 99fang.com before starting ByteDance in 2012 in Beijing, China. Before founding the company, he worked as a software engineer and developed an interest in information distribution and data [[wikipedia:Algorithm|algorithm]]. His vision was to create platforms that could automatically recommend content based on user interests. In 2012, entrepreneur Zhang Yiming proposed building a platform that would recommend information based on users’ interests. During an early meeting with investors, he explained his idea using sketches drawn on a piece of paper.This idea later became the core concept behind ByteDance and its flagship product, Toutiao (Today’s Headlines). According to Forbes, Zhang Yiming’s net worth was estimated at $35.6 billion in 2021, making the 38-year-old entrepreneur one of the richest individuals in China. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Founding of ByteDance and Douyin == [[File:Tiktok hyper.png|thumb|250px|TikTok logo]] ByteDance was established as a technology company focused on [[wikipedia:Artificial intelligence|artificial intelligence]] and machine learning. It started in a very small office in Beijing, where teams worked in different rooms of a residential apartment. As the company grew rapidly, it moved to larger office buildings and expanded its workforce. One of its earliest successful products was “Toutiao”, a news platform that used algorithms to recommend articles to users. Within a few years, the company had grown from a small startup into a large technology firm with thousands of employees. [[wikipedia:Douyin|Douyin]], the Chinese version of TikTok, became a huge hit in China in 2017. This model later became the foundation for the development of [[wikipedia:TikTok|TikTok]]. With the rise of mobile internet, a new model emerged: information could be recommended to users automatically based on their interests. This marked the transition from “people searching for information” to “information finding people,” which is the foundation of algorithm-based recommendation systems. <ref>{{cite web |title=Who is Zhang Yiming and how he grew ByteDance and TikTok into global platforms |url=https://www.scmp.com/tech/big-tech/article/3134264/who-zhang-yiming-and-how-he-grew-bytedance-and-tiktok-global |website=South China Morning Post |date=2021 |access-date=2026-03-15 }}</ref> == Algorithm-Driven Content Recommendation == Zhang Yiming strongly believed that Algorithm driven recommendation systems would shape the future of digital media. Despite doubts from some colleagues and investors, he insisted on developing this technology. In the early stage of ByteDance, Zhang even wrote internal materials explaining recommendation systems so that the engineering team could build the first version of the algorithm. This early focus on algorithm development later became the company’s main competitive advantage. This idea became the core technology behind ByteDance products. TikTok’s “For You” feed is generated by a [[wikipedia:Recommender system|recommendation system]] that delivers videos likely to interest each individual user. Recommendations are based on several factors, including user interactions such as likes, shares, comments, and follows, as well as video information like captions, sounds, and hashtags. <ref>{{cite web |title=Understanding the Impact of TikTok's Recommendation Algorithm on User Engagement |url=https://www.researchgate.net/publication/382423048_Understanding_the_Impact_of_TikTok's_Recommendation_Algorithm_on_User_Engagement |website=ResearchGate |year=2024 |access-date=2026-03-15 }}</ref> TikTok also states that device and account settings, such as language preference and country setting, are considered, although they carry less weight than behavioral signals. TikTok uses an algorithm-driven recommendation system that analyzes user behavior, such as watch time, likes, comments, and shares. The platform then recommends videos that match the user’s preferences. This approach allows users to discover content quickly and keeps them engaged with personalized video feeds. According to TikTok, stronger indicators of interest, such as whether a user watches a video from beginning to end, receive greater weight in the recommendation process than weaker signals. <ref name="tiktokalgo">{{cite web |title=How TikTok recommends videos for you |url=https://newsroom.tiktok.com/how-tiktok-recommends-videos-for-you |website=TikTok Newsroom |access-date=2026-03-15 }}</ref> A Wall Street Journal investigation found that TikTok only needs one important piece of information to figure out what you want: the amount of time you linger over a piece of content. Every second you hesitate or rewatch, the app is tracking you. <ref>{{cite web |title=Inside TikTok’s Algorithm: A WSJ Investigation |url=https://www.wsj.com/tech/tiktok-algorithm-video-investigation-11626877477 |website=The Wall Street Journal |access-date=2026-03-15 }}</ref> == Impact on Digital Media Platforms == [[File:ByteDance logo English.svg|thumb|300px|ByteDance company logo]] At the [[wikipedia:World Internet Conference|World Internet Conference]] in Wuzhen, Zhang Yiming stated that China’s internet users represent only about one-fifth of the global online population. He argued that Chinese technology companies must expand internationally in order to compete globally. ByteDance later implemented this strategy through TikTok, the international version of Douyin, which uses algorithm-based content recommendation. Despite early skepticism, ByteDance gradually attracted investment from major venture capital firms. Over time, the company completed several funding rounds and its valuation grew rapidly. By the late 2010s, ByteDance had become one of the most valuable technology companies in the world. The success of TikTok has influenced many other social media companies. Platforms such as [[wikipedia:Instagram|Instagram]], [[wikipedia:YouTube|YouTube]], and [[wikipedia:Facebook|Facebook]] have introduced similar recommendation systems to compete with TikTok’s algorithm-driven model. As a result, personalized content feeds have become a dominant feature of modern digital media platforms. ByteDance expanded internationally with products such as TikTok. The platform quickly gained popularity among young users around the world and became one of the most downloaded apps globally. The company established offices in major cities across Asia, Europe, and North America, transforming from a small startup into a global technology company. The platform reached hundreds of millions of users worldwide and became a major competitor to other social media platforms. This shift also changed how users consume information, as content is increasingly selected by algorithms rather than personal networks or manual search. <ref>{{cite web |title=张一鸣的算法暗战 |url=https://www.huxiu.com/article/4810826.html |website=Huxiu |date=2025-11-28 |language=zh |access-date=2026-03-15 }}</ref> == References == {{reflist}} {{BookCat}} rtgmwr6co8ldnefzm66kvx813ua13zt 2 328624 2801370 2801219 2026-03-29T23:25:29Z ~2026-18400-87 3061118 2801370 wikitext text/x-wiki <div style="display: flex; flex-wrap: wrap; gap: 15px; justify-content: center; background: #f9f9f9; padding: 20px; border: 1px dashed #ccc;"> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image One</div> </div> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image Two</div> </div> </div> ffqqlxp8dpbccvjlvwmd1tw309w5hci 2801371 2801370 2026-03-29T23:28:00Z ~2026-18400-87 3061118 2801371 wikitext text/x-wiki <div style="display: flex; flex-wrap: wrap; gap: 15px; justify-content: center; background: #f9f9f9; padding: 20px; border: 1px dashed #ccc;"> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image One</div> </div> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image Two</div> </div> </div> <p> The legendary <span style="border-bottom: 2px dotted #03a9f4; color: #01579b; cursor: help; font-weight: bold;" title="The users who are registered and edit pages."> Editor </span> has entered the wiki. </p> bfobsubkysl7rodwm7yo6y3tssb1nbq 2801375 2801371 2026-03-29T23:31:53Z ~2026-18400-87 3061118 2801375 wikitext text/x-wiki <div style="display: flex; flex-wrap: wrap; gap: 15px; justify-content: center; background: #f9f9f9; padding: 20px; border: 1px dashed #ccc;"> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image One</div> </div> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image Two</div> </div> </div> <p> The legendary <span style="border-bottom: 2px dotted #03a9f4; color: #01579b; cursor: help; font-weight: bold;" title="The users who are registered and edit pages."> Editor </span> has entered the wiki. </p> <div style="background: #0fffae; padding: 10px; border-radius: 5px;"> <h3>Brainrot Unwarning</h3> <p>This page doesn't contain high-level Rizz.</p> </div> {| class="wikitable" style="width: 100%;" ! Header 1 !! Header 2 !! Header 3 |- | Row 1, Col 1 || Row 1, Col 2 || Row 1, Col 3 |- | Row 2, Col 1 || Row 2, Col 2 || Row 2, Col 3 |} <div style="float: right; width: 300px; border: 1px solid #aaa; background: #f9f9f9; margin: 0 0 1em 1em; padding: 5px;"> <div style="background: #228b22; color: white; text-align: center; font-weight: bold; padding: 8px;"> Subject Title </div> <div style="text-align: center; padding: 10px; background: white;"> [[File:Placeholder.png|250px]] </div> <table style="width: 100%; font-size: 0.9em; border-collapse: collapse;"> <tr style="border-bottom: 1px solid #ddd;"> <th style="text-align: left; padding: 5px;">Category</th> <td style="padding: 5px;">Brainrot</td> </tr> </table> </div> <div style="border: 1px solid #c00; border-left: 10px solid #c00; background: #fee; padding: 10px; margin: 10px 0;"> <b>Warning:</b> This section contains restricted info. </div> <div style="border: 1px solid #0fffae; border-left: 10px solid #0fffae; background: #f0fffb; padding: 10px; margin: 10px 0;"> <b>Verified:</b> Fact-checked by the Archive. </div> <div style="display: inline-block; background-color: #03a9f4; color: white; padding: 10px 20px; border-radius: 5px; font-weight: bold; border-bottom: 3px solid #01579b;"> 🚀 Explore the Wiki </div> 0ghn4l8il8loqqhuv5e8pmyfh5mn2q4 2801376 2801375 2026-03-29T23:32:11Z ~2026-18400-87 3061118 2801376 wikitext text/x-wiki <div style="display: flex; flex-wrap: wrap; gap: 15px; justify-content: center; background: #f9f9f9; padding: 20px; border: 1px dashed #ccc;"> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image One</div> </div> <div style="background: white; border: 1px solid #ddd; padding: 10px; text-align: center; width: 160px;"> [[File:Placeholder.png|150px]] <div style="margin-top: 5px; font-weight: bold; font-size: 0.9em;">Image Two</div> </div> </div> <p> The legendary <span style="border-bottom: 2px dotted #03a9f4; color: #01579b; cursor: help; font-weight: bold;" title="The users who are registered and edit pages."> Editor </span> has entered the wiki. </p> <div style="background: #0fffae; padding: 10px; border-radius: 5px;"> <h3>Brainrot Unwarning</h3> <p>This page doesn't contain high-level Rizz.</p> </div> {| class="wikitable" style="width: 100%;" ! Header 1 !! Header 2 !! 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And after being renamed to MirahezeGPDR, Then their user page will be deleted. |} pqe88x23xgzg8n91x9rymx1n3t7jik2 0 328625 2801360 2800829 2026-03-29T20:57:53Z 5heep.doggydog 3055767 /* Bear Management Areas */ minor edits to reduce commas 2801360 wikitext text/x-wiki [[File:Radio tagged bear standing in water up to its belly.jpg|thumb|Brown bear equipped with a transmitter collar]] [https://en.wikipedia.org/wiki/Wildlife_radio_telemetry Radio telemetry] is a form of location tracking commonly used in bear [https://en.wikipedia.org/wiki/Wildlife_conservation conservation] efforts since its initial usage in this field in the 1960s. Radio telemetry uses one or multiple types of signals including [https://en.wikipedia.org/wiki/Very_high_frequency VHF], [https://en.wikipedia.org/wiki/Global_Positioning_System GPS], and [[wikipedia:Satellite_navigation|satellite]], emitted from a transmitter attached to the animal that allows researchers to track its location, which is the primary function of telemetry. Data obtained for bear conservation efforts through radio telemetry includes population density, travel patterns, survival and reproduction rates, and causes of mortality. The data produced from these studies has helped researchers identify and address common hazards that bears face in order to improve the safety of bears and the people that spend time in bear-populated wilderness areas. == History == Radio telemetry was first used in bear conservation by [https://en.wikipedia.org/wiki/Frank_and_John_Craighead Frank and John Craighead] in the 1960s in [https://en.wikipedia.org/wiki/Yellowstone_National_Park Yellowstone National Park].<ref name=":0">{{Cite web|url=https://www.nps.gov/subjects/bears/evolvingtechnologies.htm|title=Evolving Research Technologies - Bears (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-15}}</ref> This study was a crucial step in understanding the way that grizzlies lived and travelled. It also helped establish the groundwork for many of the approaches that are used in bear conservation currently.<ref>{{Cite news|url=https://www.newyorker.com/tech/annals-of-technology/remembering-the-craigheads-pioneers-of-wildlife-biology|title=Remembering the Craigheads, Pioneers of Wildlife Biology|last=Smith|first=Jordan Fisher|date=2016-10-11|work=The New Yorker|access-date=2026-03-15|language=en-US|issn=0028-792X}}</ref> Notably, this effort developed the first radio collars for large mammals, a transmitter that has since become standard for many large species alongside bears.<ref name=":0" /> == Technology == The technology used in radio telemetry for bears has progressed significantly since collars were initially used in the 1960s. === Signal Types === The collars that the Craighead brothers used in the 1960s used VHF signals. These signals are capable of covering distances upwards of 20km, but are dependent on the terrain, becoming less effective when there are obstacles between the transmitter and receiver, such as forests or mountains, which can interfere with the signal transmission. Over time, other signals such as GPS and several other satellite tracking systems have been incorporated into the transmitters used for wildlife tracking. These signals are generally more convenient than VHF because they enable researchers to collect data automatically and do not require researchers to manually track the subject. <ref name=":1">{{Cite web|url=https://wildlifeactinnovations.com/wp-content/uploads/2021/05/A-Guide-to-Wildlife-Tracking-Technologies.pdf|title=A Guide to Wildlife Tracking Technologies|date=May 2021|publisher=AWE Telemetry Systems (PTY) LTD|access-date=2026-03-17}}</ref> === Transmitters === Since the 1960s, transmitters have evolved from collars to include other pieces such as tags and microchips, in the case of smaller animals.<ref name=":3">{{Cite journal|last=Manville|first=Albert M.|last2=Levitt|first2=B. Blake|last3=Lai|first3=Henry C.|date=2024|title=Health and environmental effects to wildlife from radio telemetry and tracking devices-state of the science and best management practices|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10954089/|journal=Frontiers in Veterinary Science|volume=11|pages=1283709|doi=10.3389/fvets.2024.1283709|issn=2297-1769|pmc=10954089|pmid=38511190}}</ref><ref name=":2">{{Cite journal|last=Ross|first=Tyler R.|last2=Thiemann|first2=Gregory W.|last3=Kirschhoffer|first3=B. J.|last4=Kirschhoffer|first4=Jon|last5=York|first5=Geoff|last6=Derocher|first6=Andrew E.|last7=Johnson|first7=Amy C.|last8=Lunn|first8=Nicholas J.|last9=McGeachy|first9=David|date=2024|title=Telemetry without collars: performance of fur- and ear-mounted satellite tags for evaluating the movement and behaviour of polar bears|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11249465/|journal=Animal Biotelemetry|volume=12|issue=1|pages=18|doi=10.1186/s40317-024-00373-2|issn=2050-3385|pmc=11249465|pmid=39022453}}</ref> For bears, collars are still common because of their reliability, especially for long-term studies. Because there have historically been some safety issues surrounding the risk of bears outgrowing collars, among other issues, radio telemetry collars have evolved to include features that address some of these problems such as break-away functions and expandable designs.<ref name=":3" /> Regarding tags, both fur and ear tags have been used in bear studies. In a Polar Bear study, researchers found that ear tags were better at remaining attached to the bear until the time of removal, while fur tags were more likely to fall off before the receiver itself was worn out. Aside from the difference in longevity, both fur and ear tags were found to be similarly effective at tracking location.<ref name=":2" /> == Data Usage == Location tracking is one of the primary forms of data gained from radio telemetry devices. With bears, location tracking has been used to monitor various aspects of their lives. === Environment Usage === Bears travel significant distances through their habitats to locate food, water, and shelter. Using radio telemetry, researchers have been able to identify how bears use their environment in greater detail. Using observation, researchers have identified that bears typically concentrate around areas with high volumes of food. However, food is not the only factor that leads to bears using an area, based on the location data, available cover is also a factor. Aside from some of the factors that persuade bears to an area, there are also factors that dissuade them. In studies from the [https://en.wikipedia.org/wiki/Kenai_Peninsula Kenai Peninsula] and Yellowstone National Park, they were able to detect some patterns in the spaces that bears avoided the most. Based on the tracking data, bears typically avoid areas of around 500m surrounding human infrastructure like roads and houses. Researchers have also noticed that female bears with cubs are more likely to use areas with fewer resources because they tend to experience less traffic from other bears that could pose a threat to the family.<ref>{{Cite journal|last=Suring|first=Lowell H.|last2=Farley|first2=Sean D.|last3=Hilderbrand|first3=Grant V.|last4=Goldstein|first4=Michael I.|last5=Howlin|first5=Shay|last6=Erickson|first6=Wallace P.|date=Dec., 2006|title=Patterns of Landscape Use by Female Brown Bears on the Kenai Peninsula, Alaska|url=https://www-jstor-org.ohlone.idm.oclc.org/stable/4128090?sid=primo|journal=The Journal of Wildlife Management|volume=Vol. 70|pages=pp. 1580-1587|via=JSTOR}}</ref> === Bear Information === Radio telemetry also allows researchers to gather information on the bears themselves, rather than just how they use their environment. Mortality and reproduction rates can be determined by checking tracking data in conjunction with visual observation. Collars that cease movement or transmission for an extended period of time may be checked by researchers to see if the associated bear is still alive and there has been an issue with the collar, or if the bear has passed away. Similar observations are used to estimate reproduction rates. Once bears have emerged from their dens in the spring, researchers will track the location of female bears to manually view if they have cubs and record the number of offspring if any are present.<ref>{{Cite web|url=https://www.nps.gov/articles/aps-v6-i2-c15.htm|title=Grizzly Bear Population Ecology in Denali (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-24}}</ref> Population density can be estimated using a combination of tracking technologies and remote cameras to gain an idea of the number of individual bears that frequent specific areas. Common health hazards to bears can also be identified using radio telemetry data.<ref name=":5">{{Cite journal|last=Matthews|first=Sean M.|last2=Golightly|first2=Richard T.|last3=Higley|first3=J. Mark|date=2008|title=Mark-Resight Density Estimation for American Black Bears in Hoopa, California|url=https://www-jstor-org.ohlone.idm.oclc.org/stable/20204096?sid=primo|journal=Ursus|volume=Vol. 19|pages=pp. 13-21|via=JSTOR}}</ref><ref name=":6">{{Cite web|url=https://www.usgs.gov/news/featured-story/return-yellowstone-grizzly-bear|title=Return of the Yellowstone Grizzly Bear {{!}} U.S. Geological Survey|date=2017-12-13|website=www.usgs.gov|language=en|access-date=2026-03-21}}</ref> == Effects on Conservation Efforts == The use of radio telemetry has aided researchers in developing protections for bear safety through solutions like bear management areas, individual bear management, and population monitoring. === Bear Management Areas === In Yellowstone, [https://en.wikipedia.org/wiki/Glacier_National_Park_(U.S.) Glacier], and other National Parks with notable bear populations, bear management areas have been used. These are typically areas that experience seasonal closures, speed guidelines for roads, or human travel restrictions to decrease the amount of human-bear interactions during periods of frequent bear traffic. These restrictions have been a significant step in improving the safety of both bears and visitors in these parks. The delineation of these areas may be influenced by the location data gathered from radio telemetry devices, as well as human observation.<ref>{{Cite web|url=https://www.nps.gov/yell/learn/management/bear.htm|title=Bear Management - Yellowstone National Park (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-21}}</ref><ref name=":4">{{Cite web|url=https://yosemite.org/saving-species-gps/|title=Saving Species Through GPS Tracking — Yosemite Conservancy|date=2022-06-01|language=en-US|access-date=2026-03-21}}</ref><ref>{{Cite web|url=https://www.nps.gov/glac/learn/news/section-of-highline-trail-remains-closed-after-bear-encounter.htm|title=Section of Highline Trail remains closed after bear encounter - Glacier National Park (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-27}}</ref> === Individual Bear Management === In National Parks, radio telemetry also allows park staff to track specific bears, particularly those that have a record of being overly habituated to people. These bears are more likely to approach and engage with human spaces, which can be a hazard to both people and the bear itself, as bears that become overly desensitized and bold around humans are at an increased risk of being euthanized.<ref>{{Cite web|url=https://idfg.idaho.gov/article/food-conditioned-habituated-grizzly-bear-euthanized-island-park|title=Food-conditioned, habituated grizzly bear euthanized in Island Park {{!}} Idaho Fish and Game|date=2025-06-26|website=idfg.idaho.gov|language=en|access-date=2026-03-23}}</ref> By being able to track a bear's location, staff can be alerted when a bear is approaching a populated area like a campground and remove the bear from the area. This can help reduce the likelihood of that bear becoming increasingly habituated, causing harm to people or property, and for bears that are not yet accustomed to people, prevent habituation entirely.<ref name=":4" /> === Population Monitoring === Because radio telemetry allows researchers to more accurately track population details such as mortality and reproduction rates, population density, and common causes of death, conservationists have been able to closely monitor vulnerable populations. This has helped people more successfully identify and address common risks to these bears, as well as take appropriate action when critical issues arise.<ref name=":5" /><ref name=":6" /><ref>{{Cite journal|last=Connor|first=Thomas|last2=Dheer|first2=Arjun|last3=Dorcy‐Ponce|first3=Janelle|last4=Steinbeiser|first4=Cathleen|last5=Landers|first5=Russ|last6=Klip|first6=Mario|last7=Furnas|first7=Brett|date=2025-07|title=Estimating wildlife populations and their dynamics using multiple data sources and a hierarchical integrated model: The case of California's black bears|url=https://besjournals.onlinelibrary.wiley.com/doi/10.1002/2688-8319.70076|journal=Ecological Solutions and Evidence|language=en|volume=6|issue=3|doi=10.1002/2688-8319.70076|issn=2688-8319}}</ref> == References == {{BookCat}} 06zc2730zb4aneudmcjr06h506nnr48 2801361 2801360 2026-03-29T20:59:15Z 5heep.doggydog 3055767 /* Population Monitoring */ word fix 2801361 wikitext text/x-wiki [[File:Radio tagged bear standing in water up to its belly.jpg|thumb|Brown bear equipped with a transmitter collar]] [https://en.wikipedia.org/wiki/Wildlife_radio_telemetry Radio telemetry] is a form of location tracking commonly used in bear [https://en.wikipedia.org/wiki/Wildlife_conservation conservation] efforts since its initial usage in this field in the 1960s. Radio telemetry uses one or multiple types of signals including [https://en.wikipedia.org/wiki/Very_high_frequency VHF], [https://en.wikipedia.org/wiki/Global_Positioning_System GPS], and [[wikipedia:Satellite_navigation|satellite]], emitted from a transmitter attached to the animal that allows researchers to track its location, which is the primary function of telemetry. Data obtained for bear conservation efforts through radio telemetry includes population density, travel patterns, survival and reproduction rates, and causes of mortality. The data produced from these studies has helped researchers identify and address common hazards that bears face in order to improve the safety of bears and the people that spend time in bear-populated wilderness areas. == History == Radio telemetry was first used in bear conservation by [https://en.wikipedia.org/wiki/Frank_and_John_Craighead Frank and John Craighead] in the 1960s in [https://en.wikipedia.org/wiki/Yellowstone_National_Park Yellowstone National Park].<ref name=":0">{{Cite web|url=https://www.nps.gov/subjects/bears/evolvingtechnologies.htm|title=Evolving Research Technologies - Bears (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-15}}</ref> This study was a crucial step in understanding the way that grizzlies lived and travelled. It also helped establish the groundwork for many of the approaches that are used in bear conservation currently.<ref>{{Cite news|url=https://www.newyorker.com/tech/annals-of-technology/remembering-the-craigheads-pioneers-of-wildlife-biology|title=Remembering the Craigheads, Pioneers of Wildlife Biology|last=Smith|first=Jordan Fisher|date=2016-10-11|work=The New Yorker|access-date=2026-03-15|language=en-US|issn=0028-792X}}</ref> Notably, this effort developed the first radio collars for large mammals, a transmitter that has since become standard for many large species alongside bears.<ref name=":0" /> == Technology == The technology used in radio telemetry for bears has progressed significantly since collars were initially used in the 1960s. === Signal Types === The collars that the Craighead brothers used in the 1960s used VHF signals. These signals are capable of covering distances upwards of 20km, but are dependent on the terrain, becoming less effective when there are obstacles between the transmitter and receiver, such as forests or mountains, which can interfere with the signal transmission. Over time, other signals such as GPS and several other satellite tracking systems have been incorporated into the transmitters used for wildlife tracking. These signals are generally more convenient than VHF because they enable researchers to collect data automatically and do not require researchers to manually track the subject. <ref name=":1">{{Cite web|url=https://wildlifeactinnovations.com/wp-content/uploads/2021/05/A-Guide-to-Wildlife-Tracking-Technologies.pdf|title=A Guide to Wildlife Tracking Technologies|date=May 2021|publisher=AWE Telemetry Systems (PTY) LTD|access-date=2026-03-17}}</ref> === Transmitters === Since the 1960s, transmitters have evolved from collars to include other pieces such as tags and microchips, in the case of smaller animals.<ref name=":3">{{Cite journal|last=Manville|first=Albert M.|last2=Levitt|first2=B. Blake|last3=Lai|first3=Henry C.|date=2024|title=Health and environmental effects to wildlife from radio telemetry and tracking devices-state of the science and best management practices|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10954089/|journal=Frontiers in Veterinary Science|volume=11|pages=1283709|doi=10.3389/fvets.2024.1283709|issn=2297-1769|pmc=10954089|pmid=38511190}}</ref><ref name=":2">{{Cite journal|last=Ross|first=Tyler R.|last2=Thiemann|first2=Gregory W.|last3=Kirschhoffer|first3=B. J.|last4=Kirschhoffer|first4=Jon|last5=York|first5=Geoff|last6=Derocher|first6=Andrew E.|last7=Johnson|first7=Amy C.|last8=Lunn|first8=Nicholas J.|last9=McGeachy|first9=David|date=2024|title=Telemetry without collars: performance of fur- and ear-mounted satellite tags for evaluating the movement and behaviour of polar bears|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11249465/|journal=Animal Biotelemetry|volume=12|issue=1|pages=18|doi=10.1186/s40317-024-00373-2|issn=2050-3385|pmc=11249465|pmid=39022453}}</ref> For bears, collars are still common because of their reliability, especially for long-term studies. Because there have historically been some safety issues surrounding the risk of bears outgrowing collars, among other issues, radio telemetry collars have evolved to include features that address some of these problems such as break-away functions and expandable designs.<ref name=":3" /> Regarding tags, both fur and ear tags have been used in bear studies. In a Polar Bear study, researchers found that ear tags were better at remaining attached to the bear until the time of removal, while fur tags were more likely to fall off before the receiver itself was worn out. Aside from the difference in longevity, both fur and ear tags were found to be similarly effective at tracking location.<ref name=":2" /> == Data Usage == Location tracking is one of the primary forms of data gained from radio telemetry devices. With bears, location tracking has been used to monitor various aspects of their lives. === Environment Usage === Bears travel significant distances through their habitats to locate food, water, and shelter. Using radio telemetry, researchers have been able to identify how bears use their environment in greater detail. Using observation, researchers have identified that bears typically concentrate around areas with high volumes of food. However, food is not the only factor that leads to bears using an area, based on the location data, available cover is also a factor. Aside from some of the factors that persuade bears to an area, there are also factors that dissuade them. In studies from the [https://en.wikipedia.org/wiki/Kenai_Peninsula Kenai Peninsula] and Yellowstone National Park, they were able to detect some patterns in the spaces that bears avoided the most. Based on the tracking data, bears typically avoid areas of around 500m surrounding human infrastructure like roads and houses. Researchers have also noticed that female bears with cubs are more likely to use areas with fewer resources because they tend to experience less traffic from other bears that could pose a threat to the family.<ref>{{Cite journal|last=Suring|first=Lowell H.|last2=Farley|first2=Sean D.|last3=Hilderbrand|first3=Grant V.|last4=Goldstein|first4=Michael I.|last5=Howlin|first5=Shay|last6=Erickson|first6=Wallace P.|date=Dec., 2006|title=Patterns of Landscape Use by Female Brown Bears on the Kenai Peninsula, Alaska|url=https://www-jstor-org.ohlone.idm.oclc.org/stable/4128090?sid=primo|journal=The Journal of Wildlife Management|volume=Vol. 70|pages=pp. 1580-1587|via=JSTOR}}</ref> === Bear Information === Radio telemetry also allows researchers to gather information on the bears themselves, rather than just how they use their environment. Mortality and reproduction rates can be determined by checking tracking data in conjunction with visual observation. Collars that cease movement or transmission for an extended period of time may be checked by researchers to see if the associated bear is still alive and there has been an issue with the collar, or if the bear has passed away. Similar observations are used to estimate reproduction rates. Once bears have emerged from their dens in the spring, researchers will track the location of female bears to manually view if they have cubs and record the number of offspring if any are present.<ref>{{Cite web|url=https://www.nps.gov/articles/aps-v6-i2-c15.htm|title=Grizzly Bear Population Ecology in Denali (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-24}}</ref> Population density can be estimated using a combination of tracking technologies and remote cameras to gain an idea of the number of individual bears that frequent specific areas. Common health hazards to bears can also be identified using radio telemetry data.<ref name=":5">{{Cite journal|last=Matthews|first=Sean M.|last2=Golightly|first2=Richard T.|last3=Higley|first3=J. Mark|date=2008|title=Mark-Resight Density Estimation for American Black Bears in Hoopa, California|url=https://www-jstor-org.ohlone.idm.oclc.org/stable/20204096?sid=primo|journal=Ursus|volume=Vol. 19|pages=pp. 13-21|via=JSTOR}}</ref><ref name=":6">{{Cite web|url=https://www.usgs.gov/news/featured-story/return-yellowstone-grizzly-bear|title=Return of the Yellowstone Grizzly Bear {{!}} U.S. Geological Survey|date=2017-12-13|website=www.usgs.gov|language=en|access-date=2026-03-21}}</ref> == Effects on Conservation Efforts == The use of radio telemetry has aided researchers in developing protections for bear safety through solutions like bear management areas, individual bear management, and population monitoring. === Bear Management Areas === In Yellowstone, [https://en.wikipedia.org/wiki/Glacier_National_Park_(U.S.) Glacier], and other National Parks with notable bear populations, bear management areas have been used. These are typically areas that experience seasonal closures, speed guidelines for roads, or human travel restrictions to decrease the amount of human-bear interactions during periods of frequent bear traffic. These restrictions have been a significant step in improving the safety of both bears and visitors in these parks. The delineation of these areas may be influenced by the location data gathered from radio telemetry devices, as well as human observation.<ref>{{Cite web|url=https://www.nps.gov/yell/learn/management/bear.htm|title=Bear Management - Yellowstone National Park (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-21}}</ref><ref name=":4">{{Cite web|url=https://yosemite.org/saving-species-gps/|title=Saving Species Through GPS Tracking — Yosemite Conservancy|date=2022-06-01|language=en-US|access-date=2026-03-21}}</ref><ref>{{Cite web|url=https://www.nps.gov/glac/learn/news/section-of-highline-trail-remains-closed-after-bear-encounter.htm|title=Section of Highline Trail remains closed after bear encounter - Glacier National Park (U.S. National Park Service)|website=www.nps.gov|language=en|access-date=2026-03-27}}</ref> === Individual Bear Management === In National Parks, radio telemetry also allows park staff to track specific bears, particularly those that have a record of being overly habituated to people. These bears are more likely to approach and engage with human spaces, which can be a hazard to both people and the bear itself, as bears that become overly desensitized and bold around humans are at an increased risk of being euthanized.<ref>{{Cite web|url=https://idfg.idaho.gov/article/food-conditioned-habituated-grizzly-bear-euthanized-island-park|title=Food-conditioned, habituated grizzly bear euthanized in Island Park {{!}} Idaho Fish and Game|date=2025-06-26|website=idfg.idaho.gov|language=en|access-date=2026-03-23}}</ref> By being able to track a bear's location, staff can be alerted when a bear is approaching a populated area like a campground and remove the bear from the area. This can help reduce the likelihood of that bear becoming increasingly habituated, causing harm to people or property, and for bears that are not yet accustomed to people, prevent habituation entirely.<ref name=":4" /> === Population Monitoring === Because radio telemetry allows researchers to more accurately track population details such as mortality and reproduction rates, population density, and common causes of death, conservationists have been able to closely monitor vulnerable populations. This has helped them more successfully identify and address common risks to these bears, as well as take appropriate action when critical issues arise.<ref name=":5" /><ref name=":6" /><ref>{{Cite journal|last=Connor|first=Thomas|last2=Dheer|first2=Arjun|last3=Dorcy‐Ponce|first3=Janelle|last4=Steinbeiser|first4=Cathleen|last5=Landers|first5=Russ|last6=Klip|first6=Mario|last7=Furnas|first7=Brett|date=2025-07|title=Estimating wildlife populations and their dynamics using multiple data sources and a hierarchical integrated model: The case of California's black bears|url=https://besjournals.onlinelibrary.wiley.com/doi/10.1002/2688-8319.70076|journal=Ecological Solutions and Evidence|language=en|volume=6|issue=3|doi=10.1002/2688-8319.70076|issn=2688-8319}}</ref> == References == {{BookCat}} noanr8imv1xdt5o7rxo0flvv60kuplf 2 328626 2801497 2800893 2026-03-30T07:58:35Z ~2026-18400-87 3061118 2801497 wikitext text/x-wiki <pre style="color: blue; background-color: #f0f0f0; width: 50%; border: 1px solid #ccc; padding: 10px;"> La Vacca Atomo Atomita Anpali Babel Capibara Sushi Ti Ti Ti Sahur Roti Bohay Zip Zip Zip Zip Zip Zip Zip Zip Orange Cat UIA UIA Mucca Campana Diego Armadillo Maranzillo Banananito Bandito Octopusini Zucchinim Ble Ble Blu Blu Tiramisubmarini Hipotano Watermelono Pollo Pasticciotto Pastino Charlie Munin Ananasso Il Titanico Cabalini Perini Rang Ring Reng Bus Ireng </pre> ewki32m4fmc36koriao23wc7vkmjtq8 2801498 2801497 2026-03-30T08:00:09Z ~2026-18400-87 3061118 2801498 wikitext text/x-wiki <pre style="color: blue; background-color: #f0f0f0; width: 50%; border: 1px solid #ccc; padding: 10px;"> La Vacca Atomo Atomita Anpali Babel Capibara Sushi Ti Ti Ti Sahur Roti Bohay Zip Zip Zip Zip Zip Zip Zip Zip Orange Cat UIA UIA Mucca Campana Diego Armadillo Maranzillo Banananito Bandito Octopusini Zucchinim Ble Ble Blu Blu Tiramisubmarini Hipotano Watermelono Pollo Pasticciotto Pastino Charlie Munin Ananasso Il Titanico Cabalini Perini Rang Ring Reng Bus Ireng </pre> <pre style="color: green; background-color: #f0f0f0; width: 50%; border: 1px solid #ccc; padding: 10px;"> Dr Dr dinosaur is LA pajohotpato rang ring bus ireng pipi caramel anomaliiiiiiiii..... trr trr sahur Him him sahur dolphinera dolphina tung tung sahur br br patapim dididisahur vilti Volta </pre> 1iqsmhik1xci0j94r7fcwypy2osjl5l 2 328662 2801335 2800560 2026-03-29T18:26:07Z ThinkingScience 3061446 a few notes for March 28 2801335 wikitext text/x-wiki This is a continuation of [[User:Dekatriofovia/Draftspace/Coursera]]. I'm continuing to make notes on a computer with a screen that allows me to see text more clearly. March 26: We are still at the transcripts and I'm supposed to read from page 4 to page 16...which I doubt I'll be able to do since I have much less time right now. Tomorrow or later in the future I can reflect on how this went if I don't gain any more insights today. Me and my mother made this more organized this time. We decided that during certain days and hours of the week we will 'study together' but that also includes homework. My homework was page 4-16. We'll see how that goes... Today was a day we had scheduled. 2 hours studying, taking turns reading paragraphs. We began at page 16, which is 2 pages after a chapter about Bernard Mandeville and Adam Smith begins, and went until page 25 of the transcript where the chapter about Auguste Comte begins. Now we got a new day scheduled along with a suitable time and then we'll read another 2 hours I guess. '''March 28''': This is the 2nd day we studied for 2 hours at a specific day and a specific hour. With other words before March 26 we managed to make a schedule and now the schedule is complete. 6 hours per week, 3 days a week and 2 hours per day. We read from page 16 and until page 33 where "3.6 Comte’s Religion of Humanity" begins. lhesf6qilijr84cpefg89tfrz23372y 0 328719 2801329 2801211 2026-03-29T17:21:14Z ~2026-19504-42 3062117 /* Digital Marketing Strategies in K-Pop */ Added photos and text. 2801329 wikitext text/x-wiki = Digital Marketing Strategies in K-Pop = [[File:STAYC Metamorphic Showcase 240701.jpg|thumb|319x319px|Idol Group "STAYC"]] == Introduction of K-Pop == K-Pop, or Korean Pop, is a popular genre originating in South Korea in the [[wikipedia:K-pop|1990’s]]. The genre has reached a peak in recent years, particularly in this new digital era, thanks to their creative visual concepts and music creation. South Korean singer, <ref>{{Cite journal|date=2026-03-13|title=Psy|url=https://en.wikipedia.org/w/index.php?title=Psy&oldid=1343242822|journal=Wikipedia|language=en}}</ref>, known for his hit “Gangnam Style” in 2012 impacted the music industry globally reaching ‘‘Billboard’’ Hot 100 at number 2<ref>{{Cite journal|date=2026-03-25|title=Gangnam Style|url=https://en.wikipedia.org/w/index.php?title=Gangnam_Style&oldid=1345305552|journal=Wikipedia|language=en}}</ref> which was the highest charting song by a South Korean artist during that time. Its quirky choreography made it a dance challenge that everyone wanted to participate in, which is why it was so successful. HYBE, JYP Entertainment, and YG Entertainment are among the most well-known entertainment companies for the artists they have produced. A few examples of major groups that have been created are BTS, BLACKPINK, and Stray Kids. One thing unites them all: they utilize a variety of digital marketing techniques to achieve success on a worldwide scale, achieving success in streams and international visibility. == How its Digital Marketing Strategies are Different Compared to Western Artists == [[File:TriJ_Kpop_Collection.jpg|thumb|'''K-Pop Collection.jpg'''|263x263px]] How is it different? Western Artists focus on individual branding while K-Pop artists focus on collectivism <ref>{{Cite web|url=https://admindagency.com/branding-tactics-east-and-west/|title=Branding tactics - differences between East and West|last=admind|date=2023-02-21|website=Admind|language=en-GB|access-date=2026-03-29}}</ref>. Western artists often use their personal social media accounts to engage with their fans in an attempt to establish an image of themselves individually. Creative marketing is essential to K-Pop. Artist-to-fan engagement platforms, trendy song composition, scheduled teasers, and, most importantly, different visual concepts for every comeback (releasing of new music, most times along with a physical album <ref>{{Cite web|url=https://www.cosmo.ph/entertainment/preparing-for-a-k-pop-comeback-a4850-20210522-lfrm|title=New To K-Pop? Here's How You Can Prepare For A Group's Comeback|last=May 22|first=Sam Tan Published|last2=2021|website=COSMO.PH|language=en-ph|access-date=2026-03-29}}</ref>). These techniques (plus many more) are intended to build excitement and anticipation, which makes their digital marketing strategies unique. == Social Media Marketing == Social platforms such as [https://www.tiktok.com/en/ TikTok], [https://www.instagram.com/ Instagram], [http://YouTube/ YouTube], and [https://x.com/ X] (formally known as Twitter) are where artists get the most engagement because of their features such as creating short form content or trending hashtags. They take a different approach, creating hashtags for trends to increase engagement, making mini-dance challenges for their music, and dancing to popular songs with their own style. This allows it to reach audiences where K-Pop isn't normally incorporated in their (the users) algorithm. It gets views and likes that long-form content, like YouTube behind-the-scenes footage or Vlogs, might not garner. It’s a simple and easy way to create engagement, especially for the current generation who uses these social media platforms the most. Additionally, they use foreign apps that have features similar to those of Instagram and TikTok, such as Xiaohongshu, sometimes referred as Red Note, which is the third most popular social media platform in China <ref>{{Cite web|url=https://www.eliteasia.co/top-china-social-media-platforms/|title=10 Top China Social Media Platforms 2026 & How to Market on Them|last=Marketing|first=Elite Asia|date=2026-02-08|website=Elite Asia|language=en-US|access-date=2026-03-29}}</ref>. This helps them reach a larger audience. == Fan Engagement Platforms == Apps such as Bubble and Weverse are made specifically for idol-to-fan engagement. [[File:Bubble Logo.svg|thumb|Bubble Logo|254x254px]] === Bubble === "''Make your day more special with a message like a gift from your favorite idol!''"<ref>{{Cite web|url=https://www.dearu.com/en/pages/business_bubble.php|title=DEARU|website=www.dearu.com|language=ko|access-date=2026-03-29}}</ref> A monthly paid subscription where fans receive private messages from their favorite artists. It goes by a ticket purchase basis, so 1 ticket equals one artist even if they’re part of a group. The majority of the artists featured on the app are from JYP Entertainment and SM Entertainment. It is a dedicated platform made by [[wikipedia:Dear U|DearU]] for artist-to-fan or fan-to-artist messaging. In addition, it comes with additional features like exclusive text stickers and anniversaries that show how long you've been a subscriber. [[File:Weverse logo.svg|thumb|Weverse app logo|211x211px]] === Weverse === An app created by HYBE with a primary focus on various communities and artists. The majority of its functions include artist updates, live streaming, a first peek at the content of physical albums, and announcements of live events, which are accessible to all users. Exclusive content on the other hand is only available with a paid membership. Access to exclusive content, artist-to-fan messaging, merch and concert ticket presales, and their quick translation feature (that is handy for non-Korean speakers) are all included. == Streaming Platforms and Algorithmic Promotion == The two most popular music streaming services worldwide are Spotify and Apple Music <ref>{{Cite web|url=https://www.businessofapps.com/data/music-streaming-market/|title=Music App Revenue and Usage Statistics (2026)|website=Business of Apps|language=en-US|access-date=2026-03-29}}</ref>. Both of them have the ability of making personalized playlists with titles that could attract someone's curiosity. “[https://open.spotify.com/playlist/37i9dQZF1DX0018ciYu6bM?si=f-ao45hpQiyD49pInJdmNw KimBops!]” and “[https://music.apple.com/us/playlist/kpopwrld/pl.48229b41bbfc47d7af39dae8e8b5276e KPOPWRLD]” are examples of that. The algorithm functions as a marketing tool in and of itself without the need for direct promotion. Autoplay is one feature that can boost a user's algorithm to play related tracks, like a western Pop song auto-playing into K-Pop. Existing K-Pop listeners are also driven to listen to more K-Pop due to the end-of-the-year wrapped. It compiles their favorite singers and genres and lastly, shows their percentage/ranking as a listener. Additionally, there is the fan-driven streaming culture, which increases visibility and chart rankings. == Global Top K-Pop Artists of the Year on Spotify in 2025 <ref>{{Cite web|url=https://www.hellokpop.com/news/spotify-reveals-global-top-k-pop-artists-and-songs-of-2025/|title=Spotify Reveals Global Top K-Pop Artists and Songs of 2025|last=Abby|date=2025-12-05|website=HELLOKPOP|language=en-US|access-date=2026-03-29}}</ref> == # BTS # Stray Kids # JENNIE # ROSE # HUNTRIX # TWICE # BLACKPINK # Jung Kook # ENHYPEN # KATSEYE == Video Content Strategy == === Youtube === [[File:스트레이키즈 '마주치면 두근' - Stray Kids Airport Arrival (4K).webm|thumb|308x308px|Stray Kids Airport Arrival]] Visual media plays a major role in the K-Pop industry. Music videos, dance routines/practices, behind-the-scenes footage, and concert snippets are examples of media that is just as valuable as short-form content because they foster stronger emotional connections with fans and increase loyalty. Even videos not directly uploaded from the entertainment companies gain traction, a common one in South Korean idol culture is airport arrivals and departures. This is the main opportunity for idols to showcase which brands they are ambassadors for by being fully dressed in said brand. == See also == * [[wikipedia:Korean_Wave#References|Korean Wave]] * [[wikipedia:Fandom_culture_in_South_Korea|Fandom Culture in South Korea]] * [[wikipedia:Hybe|HYBE]] * [[wikipedia:JYP_Entertainment|JYP Entertainment]] * [[wikipedia:YG_Entertainment|YG Entertainment]] * [[wikipedia:SM_Entertainment|SM Entertainment]] == References == <references /> {{BookCat}} [[Category:Marketing]] [[Category:K-Pop]] [[Category:Hallyu]] __NOINDEX__ __NONEWSECTIONLINK__ 4sp3zcqgusoua5rn3ppwy5g4ld9a855 0 328723 2801320 2026-03-29T14:45:33Z Ireicher2 2797952 first edit 2801320 wikitext text/x-wiki enter your content here. Let me know if it works. d5ujs48pm29oq3zfdjv70ykg16nzdth 2801387 2801320 2026-03-30T00:53:04Z Aali19 3062035 Added sources and insights on why enthusiasts like the Mezger Engine in their Porsche 911s. 2801387 wikitext text/x-wiki {{DISPLAYTITLE:The Mezger Engine}} =The Mezger Engine: Porsche’s Racing DNA in Road Cars = === Introduction === [[https://news.dupontregistry.com/blogs/buying-guides/last-productionporsche-mezger-flat-six-engine-997-gt3rs-40?utm_source=]] The Mezger engine is widely considered one of the most important and respected engines in the history of Porsche. Named after engineer Hans Mezger, this engine stands out because it directly connects Porsche’s racing technology to its road cars. Unlike standard production engines, the Mezger engine was built with motorsport in mind, which is why it became a key component in high-performance models like the GT3, Turbo, and GT2. == Background and Origins == The Mezger engine did not start as a typical road car engine. Instead, it evolved from Porsche’s racing programs, particularly endurance racing and Le Mans 24hr racing prototypes. The design traces back to engines used in race cars like the GT1, which influenced its structure and performance capabilities. Because of this, the Mezger engine carried over many racing features, making it far more durable and performance-focused than standard engines. This is one of the main reasons why enthusiasts respect it so much today. == Technical Design and Features == [[https://www.hemmings.com/stories/inside-the-porsche-mezger-engine-in-the-911-turbo-gt3-and-gt2/]] Flat-Six Configuration The Mezger engine uses a flat-six (boxer) layout, meaning the cylinders are horizontally opposed. This design improves balance, reduces vibration, and lowers the center of gravity, which helps with handling. Dry-Sump Lubrication One of the most important features is its true dry-sump oil system. Unlike standard engines, this system prevents oil starvation during high-speed cornering, which is critical for track driving. The engine is known for “durability” and its ability to handle extreme conditions due to this design. Racing-Based Crankcase Another major difference is the crankcase design, which is derived from racing engines. This allows the engine to handle higher stress and higher RPMs compared to normal production engines == What Makes the Mezger Engine Special == [[https://www.magnetomagazine.com/articles/whats-so-special-about-the-mezger-flat-six/]] According to Magneto Magazine, the Mezger engine developed over decades into a high-performance machine that could produce up to 500 horsepower in its final form while maintaining extremely high engine speeds. What really makes it special is not just the power, but the way it delivers it. The engine is known for its sharp throttle response, high-revving nature, and unique sound. It represents a time when engineering was focused more on performance than cost or emissions. This allows the the 911 997 GT3 RS 4.0 to have a top speed of 194 mph and a 0-60 in 3.8 seconds. =Differences from Standard Porsche Engines= The Mezger engine is very different from engines like the M96 and M97, which were used in regular Porsche models.They were not designed to be massproduced. Mezger engines use a true dry-sump system, while others use simpler designs. Mezger engines are derived from racing, while others are built for cost efficiency. Mezger engines are known for higher reliability, especially compared to issues like IMS bearing failures in other engines These differences can explain why Mezger-powered cars are more desirable among collectors. == References == 1: Mezger, H. (1978). Turbocharging engines for racing and passenger cars. SAE Transactions, 87, 2779–2797. http://www.jstor.org/stable/44632583 2: Frankel, A., et al. (2020, June 16). The Mezger engine: Porsche legend created by a genius. Motor Sport Magazine. https://www.motorsportmagazine.com/articles/road-cars/the-mezger-engine-porsche-legend-created-by-a-genius 3: Hinsdale, P. (1972). The fabulous Porsche 917. https://www.amazon.com/Fabulous-Porsche-917-Peter-Hinsdale/dp/0877990522 4: Hughes, E. (2023, March 27). What’s so special about the Porsche Mezger flat-six engine? Magneto Magazine. https://www.magnetomagazine.com/articles/whats-so-special-about-the-mezger-flat-six/ 5: Hemmings. (n.d.). Inside the Porsche “Mezger” engine in the 911 Turbo, GT3, and GT2. Hemmings. https://www.hemmings.com/stories/inside-the-porsche-mezger-engine-in-the-911-turbo-gt3-and-gt2/ {{DEFAULTSORT:}} {{DEFAULTSORT:Mezger Porsche Turbo Engine}} [[Category:Engineering]] [[Category:Engineering and Technology]] [[Category:Automotive Technology]] atoar0bev9wkibej4by3gd1e7ge95dj 0 328725 2801330 2026-03-29T17:55:26Z G.SilvaTaraska 3046674 Generated in Greta_Gemini_Hypertext 4.2 2801330 wikitext text/x-wiki This MediaWiki document synthesizes your research and analysis of hypertext theory from the **DesignWriteStudio** project. It combines your historical research, "In the Wild" platform audits, and contemporary scholarship expansions into a single portfolio structure. ```mediawiki __TOC__ = Hypertext Theory and Application: A DesignWriteStudio Portfolio = This portfolio documents the systematic exploration of hypertext by '''Greta Silva-Taraska'''. [cite_start]It traces the evolution of digital connectivity from 1945 theoretical models to 2026 algorithmic environments, focusing on the pedagogical implications for [[Instructional Design and Technology]] (IDT)[cite: 205, 1176]. == Section 1: Foundational Research and History == <div class="mw-collapsible mw-collapsed"> === The Evolution of Hypertext === The conceptual roots of hypertext are grounded in the transition from linear paper constraints to associative digital networks. * [cite_start]'''The Memex (1945):''' [[Vannevar Bush]] proposed a mechanized desk for storing and linking microfilm, introducing the concept of "associative trails"[cite: 14, 55, 81]. * [cite_start]'''Project Xanadu (1960s):''' [[Ted Nelson]] coined the terms "hypertext" and "hypermedia" to describe non-sequential writing and "zippered lists" that maintain link integrity during document revision[cite: 11, 16, 76, 103]. * [cite_start]'''The NLS (1968):''' [[Douglas Engelbart]] demonstrated the first functional hypertext interface, including the computer mouse, during "The Mother of All Demos"[cite: 18, 59]. * [cite_start]'''The World Wide Web (1989):''' [[Tim Berners-Lee]] integrated hypertext with the Internet at CERN, creating a global, decentralized information space[cite: 19, 60]. === Multi-Scalar Definitions === [cite_start]Hypertext is defined by its ability to overcome linear constraints through interactive hyperlinks[cite: 12, 46]. * [cite_start]'''10-Word Definition:''' Non-linear electronic text connected by interactive hyperlinks for instant navigation[cite: 46]. * '''50-Word Definition:''' Hypertext is a non-linear method of organizing digital information using clickable links. [cite_start]Coined by Ted Nelson, it allows readers to navigate interconnected documents instantly, transforming sequential reading into a dynamic, associative, and interactive experience [cite: 48-50]. </div> == Section 2: Hypertext "In the Wild" Analysis == <div class="mw-collapsible mw-collapsed"> [cite_start]Using a specific framework of '''Six Essential Hypertext Features''', three platforms were audited to observe how hypertext influences user behavior[cite: 150, 310, 583]. {| class="wikitable" style="width: 100%;" ! Feature !! [[Goodreads]] (Social) !! [[Barnes & Noble]] (Retail) !! [[YouTube]] (Streaming) |- | '''Link''' || [cite_start]Bridges personal profiles to global book data [cite: 178-180]. || [cite_start]Structural anchors for commercial subcategories [cite: 230-231]. || [cite_start]Algorithmic bridges between video nodes[cite: 280]. |- | '''Tag''' || [cite_start]User-generated "shelves" (e.g., "to-read")[cite: 182]. || [cite_start]Industry metadata for sub-genres[cite: 233]. || [cite_start]Pill-shaped metadata for global trends[cite: 282]. |- | '''Template''' || [cite_start]Uniform profile layouts for social comparison[cite: 186]. || [cite_start]Rigid product cards focused on purchase friction [cite: 236-237]. || [cite_start]Standardized thumbnail grids for scannability[cite: 286]. |- | '''Transclusion''' || [cite_start]Mirroring user stats into a personal shelf [cite: 196-198]. || [cite_start]Real-time price and stock updates [cite: 244-245]. || [cite_start]Live view counts and engagement data [cite: 295-296]. |- | '''Filter''' || [cite_start]Personal library refinement by year or status [cite: 190-194]. || [cite_start]Catalog narrowing by price or trope [cite: 239-240]. || [cite_start]Instant feed re-sorting by topic pills [cite: 289-291]. |- | '''Objectification''' || [cite_start]The book as a "social unit" to be rated [cite: 201-204]. || [cite_start]The book as a "buyable product" [cite: 247-248]. || [cite_start]The video as a "media unit" [cite: 300-302]. |} </div> == Section 3: Contemporary Scholarship and Expansion == <div class="mw-collapsible mw-collapsed"> Recent research addresses the evolution of linking and the cognitive challenges of digital navigation. * [cite_start]'''Bidirectional Linking:''' Contemporary "tools for thought" like [[Obsidian (software)|Obsidian]] use backlinking to turn one-way signs into multidimensional intersections[cite: 766, 899]. * [cite_start]'''Hypertextual Friction:''' Systems like [[Wikipedia]] and Are.na prioritize authored links over opaque [[Algorithmic bias|algorithms]] to preserve user agency and traceable provenance [cite: 853, 901-902]. * [cite_start]'''Cognitive Disorientation:''' Research into "cognitive overhead" explores how non-linear navigation can lead to a loss of context, requiring new designs for "visible trails" in AI-mediated interfaces [cite: 904-905, 1906]. </div> == References and DOI Citations == <div class="mw-collapsible"> * Anderson, M. W. R., & Millard, D. E. (2023). [cite_start]''Seven Hypertexts''. https://doi.org/10.1145/3603163.3609048 [cite: 900, 935] * Beckermeyer, R. (1970). [cite_start]''Interactive graphic consoles: environment and software''. https://doi.org/10.1145/1478462.1478506 [cite: 984, 1115] * Glantz, R. S. (1970). [cite_start]''SHOEBOX: a personal file handling system for textual data''. https://doi.org/10.1145/1478462.1478541 [cite: 972, 1119] * Halasz, F. G., Moran, T. P., & Trigg, R. H. (1987). [cite_start]''NoteCards in a nutshell''. https://doi.org/10.1145/29933.30859 [cite: 736, 882] * Liu, S., & Almeda, S. G. (2025). [cite_start]''Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web''. https://doi.org/10.1145/3720533.3750065 [cite: 902, 936] * Nelson, T. H. (1965). [cite_start]''Complex information processing: a file structure for the complex, the changing and the indeterminate''. https://doi.org/10.1145/800197.806036 [cite: 71, 144, 741, 1123] * Rahdari, B., & Brusilovsky, P. (2025). ''From Links to Dialogue; [cite_start]Hypertext Challenges and Opportunities in Conversational Navigation''. https://doi.org/10.1145/3720533.3750064 [cite: 904, 937] * Trigg, R. H., & Weiser, M. (1986). [cite_start]''TEXTNET: a network-based approach to text handling''. https://doi.org/10.1145/5401.5402 [cite: 744, 887] </div> [[Category:Hypertext Theory]] [[Category:Instructional Design]] [[Category:Portfolio]] ``` 9nbi9yunsdureozzo5ur2dpdomjojt4 2801331 2801330 2026-03-29T18:00:02Z G.SilvaTaraska 3046674 additions done manually 2801331 wikitext text/x-wiki This MediaWiki document synthesizes your research and analysis of hypertext theory from the **DesignWriteStudio** project. It combines your historical research, "In the Wild" platform audits, and contemporary scholarship expansions into a single portfolio structure. ```mediawiki __TOC__ = Hypertext Theory and Application: A DesignWriteStudio Portfolio = This portfolio documents the systematic exploration of hypertext by '''Greta Silva-Taraska'''. [cite_start]It traces the evolution of digital connectivity from 1945 theoretical models to 2026 algorithmic environments, focusing on the pedagogical implications for [[Instructional Design and Technology]] (IDT)[cite: 205, 1176]. == Section 1: Foundational Research and History == <div class="mw-collapsible mw-collapsed"> === The Evolution of Hypertext === The conceptual roots of hypertext are grounded in the transition from linear paper constraints to associative digital networks. * [cite_start]'''The Memex (1945):''' [[Vannevar Bush]] proposed a mechanized desk for storing and linking microfilm, introducing the concept of "associative trails"[cite: 14, 55, 81]. * [cite_start]'''Project Xanadu (1960s):''' [[Ted Nelson]] coined the terms "hypertext" and "hypermedia" to describe non-sequential writing and "zippered lists" that maintain link integrity during document revision[cite: 11, 16, 76, 103]. * [cite_start]'''The NLS (1968):''' [[Douglas Engelbart]] demonstrated the first functional hypertext interface, including the computer mouse, during "The Mother of All Demos"[cite: 18, 59]. * [cite_start]'''The World Wide Web (1989):''' [[Tim Berners-Lee]] integrated hypertext with the Internet at CERN, creating a global, decentralized information space[cite: 19, 60]. === Multi-Scalar Definitions === [cite_start]Hypertext is defined by its ability to overcome linear constraints through interactive hyperlinks[cite: 12, 46]. * [cite_start]'''10-Word Definition:''' Non-linear electronic text connected by interactive hyperlinks for instant navigation[cite: 46]. * '''50-Word Definition:''' Hypertext is a non-linear method of organizing digital information using clickable links. [cite_start]Coined by Ted Nelson, it allows readers to navigate interconnected documents instantly, transforming sequential reading into a dynamic, associative, and interactive experience [cite: 48-50]. </div> == Section 2: Hypertext "In the Wild" Analysis == <div class="mw-collapsible mw-collapsed"> [cite_start]Using a specific framework of '''Six Essential Hypertext Features''', three platforms were audited to observe how hypertext influences user behavior[cite: 150, 310, 583]. {| class="wikitable" style="width: 100%;" ! Feature !! [[Goodreads]] (Social) !! [[Barnes & Noble]] (Retail) !! [[YouTube]] (Streaming) |- | '''Link''' || [cite_start]Bridges personal profiles to global book data [cite: 178-180]. || [cite_start]Structural anchors for commercial subcategories [cite: 230-231]. || [cite_start]Algorithmic bridges between video nodes[cite: 280]. |- | '''Tag''' || [cite_start]User-generated "shelves" (e.g., "to-read")[cite: 182]. || [cite_start]Industry metadata for sub-genres[cite: 233]. || [cite_start]Pill-shaped metadata for global trends[cite: 282]. |- | '''Template''' || [cite_start]Uniform profile layouts for social comparison[cite: 186]. || [cite_start]Rigid product cards focused on purchase friction [cite: 236-237]. || [cite_start]Standardized thumbnail grids for scannability[cite: 286]. |- | '''Transclusion''' || [cite_start]Mirroring user stats into a personal shelf [cite: 196-198]. || [cite_start]Real-time price and stock updates [cite: 244-245]. || [cite_start]Live view counts and engagement data [cite: 295-296]. |- | '''Filter''' || [cite_start]Personal library refinement by year or status [cite: 190-194]. || [cite_start]Catalog narrowing by price or trope [cite: 239-240]. || [cite_start]Instant feed re-sorting by topic pills [cite: 289-291]. |- | '''Objectification''' || [cite_start]The book as a "social unit" to be rated [cite: 201-204]. || [cite_start]The book as a "buyable product" [cite: 247-248]. || [cite_start]The video as a "media unit" [cite: 300-302]. |} </div> == Section 3: Contemporary Scholarship and Expansion == <div class="mw-collapsible mw-collapsed"> Recent research addresses the evolution of linking and the cognitive challenges of digital navigation. * [cite_start]'''Bidirectional Linking:''' Contemporary "tools for thought" like [[Obsidian (software)|Obsidian]] use backlinking to turn one-way signs into multidimensional intersections[cite: 766, 899]. * [cite_start]'''Hypertextual Friction:''' Systems like [[Wikipedia]] and Are.na prioritize authored links over opaque [[Algorithmic bias|algorithms]] to preserve user agency and traceable provenance [cite: 853, 901-902]. * [cite_start]'''Cognitive Disorientation:''' Research into "cognitive overhead" explores how non-linear navigation can lead to a loss of context, requiring new designs for "visible trails" in AI-mediated interfaces [cite: 904-905, 1906]. == Section 3: Contemporary Scholarship and Expansion == <div class="mw-collapsible mw-collapsed"> === From Navigation to Synthesis: "Tools for Thought" === In contemporary practice, the "navigable network" has transitioned from simple document retrieval to active knowledge synthesis[cite: 844, 897]. * '''Bidirectional Linking:''' Modern "feral" hypertext systems—such as [[Obsidian (software)|Obsidian]], Roam Research, and Logseq—utilize bidirectional links to manifest Nelson’s "associative trails" as visual graph databases. * '''Multidimensional Intersections:''' Unlike the "one-way signs" of early HTTP, these systems allow a user to see every node that points back to a current piece of information, creating a holistic map of personal knowledge [cite: 766-767, 814, 899]. * '''Scholarly Source:''' Anderson, M. W. R., & Millard, D. E. (2023). "Seven Hypertexts". ''Proceedings of the 34th ACM Conference on Hypertext and Social Media''. https://doi.org/10.1145/3603163.3609048 [cite: 803, 841, 907] === Hypertextual Friction vs. Algorithmic Opacity === As the web moves toward "seamless" AI-driven feeds, scholars argue for the reintroduction of "friction" to preserve human agency [cite: 852-853, 901]. * '''Traceable Provenance:''' Platforms like [[Wikipedia]] and Are.na prioritize visible, authored links over automated recommendations, ensuring users can trace the origin of information [cite: 855-856, 902]. * '''Countering the "Filter Bubble":''' Explicit linking acts as a counter to the opaque nature of algorithmic content feeds, requiring the reader to make conscious navigational choices [cite: 853, 901-903]. * '''Scholarly Source:''' Liu, S., & Almeda, S. G. (2025). "Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web". ''ACM Conference on Hypertext and Social Media''. https://doi.org/10.1145/3720533.3750065 [cite: 902, 908] === The Disorientation Challenge in Modern Interfaces === The "lost in hyperspace" phenomenon remains a critical concern in [[Instructional Design]], particularly as conversational AI begins to synthesize non-linear information [cite: 904-905, 1906]. * '''Cognitive Overhead:''' Non-linear navigation requires users to maintain a "mental map" of their location, which can lead to cognitive fatigue if the interface lacks clear "associative trails". * '''Conversational Navigation:''' Modern LLMs (like ChatGPT or Gemini) often displace traditional "blue-link" navigation, potentially causing a loss of critical context if the "visible trails" are removed [cite: 847, 851, 904-905]. * '''Scholarly Source:''' Rahdari, B., & Brusilovsky, P. (2025). "From Links to Dialogue; Hypertext Challenges and Opportunities in Conversational Navigation". https://doi.org/10.1145/3720533.3750064 [cite: 904, 909] === Section 3 References === * Anderson, M. W. R., & Millard, D. E. (2023). "Seven Hypertexts". https://doi.org/10.1145/3603163.3609048 [cite: 907] * Liu, S., & Almeda, S. G. (2025). "Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web". https://doi.org/10.1145/3720533.3750065 [cite: 908] * Rahdari, B., & Brusilovsky, P. (2025). "From Links to Dialogue; Hypertext Challenges and Opportunities in Conversational Navigation". https://doi.org/10.1145/3720533.3750064 [cite: 910] </div> </div> == References and DOI Citations == <div class="mw-collapsible"> * Anderson, M. W. R., & Millard, D. E. (2023). [cite_start]''Seven Hypertexts''. https://doi.org/10.1145/3603163.3609048 [cite: 900, 935] * Beckermeyer, R. (1970). [cite_start]''Interactive graphic consoles: environment and software''. https://doi.org/10.1145/1478462.1478506 [cite: 984, 1115] * Glantz, R. S. (1970). [cite_start]''SHOEBOX: a personal file handling system for textual data''. https://doi.org/10.1145/1478462.1478541 [cite: 972, 1119] * Halasz, F. G., Moran, T. P., & Trigg, R. H. (1987). [cite_start]''NoteCards in a nutshell''. https://doi.org/10.1145/29933.30859 [cite: 736, 882] * Liu, S., & Almeda, S. G. (2025). [cite_start]''Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web''. https://doi.org/10.1145/3720533.3750065 [cite: 902, 936] * Nelson, T. H. (1965). [cite_start]''Complex information processing: a file structure for the complex, the changing and the indeterminate''. https://doi.org/10.1145/800197.806036 [cite: 71, 144, 741, 1123] * Rahdari, B., & Brusilovsky, P. (2025). ''From Links to Dialogue; [cite_start]Hypertext Challenges and Opportunities in Conversational Navigation''. https://doi.org/10.1145/3720533.3750064 [cite: 904, 937] * Trigg, R. H., & Weiser, M. (1986). [cite_start]''TEXTNET: a network-based approach to text handling''. https://doi.org/10.1145/5401.5402 [cite: 744, 887] </div> [[Category:Hypertext Theory]] [[Category:Instructional Design]] [[Category:Portfolio]] ``` 7t6iwc3wgmf1ku6sd3er4rabhlw9hav 2801333 2801331 2026-03-29T18:08:29Z G.SilvaTaraska 3046674 introduction added 2801333 wikitext text/x-wiki == Introduction: Methodology and AI Collaborative Workflow == This section documents the procedural workflow and iterative research conducted through the Gemini LLM interface. The following logs represent a comprehensive archive of the analytical process used to develop this portfolio. While technical synchronization issues were encountered with transcripts 3.1, 3.2, and 3.3, the remaining data was successfully processed to audit the evolution of hypertext systems. === Reflective Analysis of AI Interaction === A critical meta-analysis of these interactions revealed a distinct "Command-and-Control" user profile. The feedback indicates that my operational style is characterized by a high degree of '''passive acceptance''' regarding the model's generated content. While I maintained strict technical "guardrails" through the use of specific DOI citations and MediaWiki syntax requirements, I frequently moved through linear task escalation without significantly challenging the model’s internal logic [cite: 65-66, 171-174, 487-488]. In the context of [[Hypertext]] theory, this mirrors the challenge of '''cognitive agency'''. Just as early pioneers like [[Ted Nelson]] warned against rigid, "frozen" information structures, an effective AI collaborator must avoid becoming a "linear processor" of LLM output [cite: 11-12, 133-134, 851]. Moving forward, my goal is to transition from a "System Architect" to a "Critical Evaluator." I intend to: * **Increase Hypertextual Friction:** Intentionally questioning synthesized responses to ensure deeper "traceable provenance"[cite: 853, 902]. * **Challenge Algorithmic Opacity:** Moving beyond mere production to actively audit the model's interpretive leaps [cite: 852-853]. * **Dig Deeper into the "Node":** Ensuring that the AI serves as an adjunct to my own creativity rather than a substitute for it, fulfilling Nelson’s vision of a "dream file" for the changing and the indeterminate [cite: 90-91, 138]. It is a notable and progressive irony that an Artificial Intelligence model provided the catalyst for this human-centric reflection on the need for greater pushback and intellectual skepticism in digital environments. </div> ''This MediaWiki document synthesizes your research and analysis of hypertext theory from the **DesignWriteStudio** project. It combines your historical research, "In the Wild" platform audits, and contemporary scholarship expansions into a single portfolio structure.'' </div> __TOC__ = Hypertext Theory and Application: A DesignWriteStudio Portfolio = This portfolio documents the systematic exploration of hypertext by '''Greta Silva-Taraska'''. [cite_start]It traces the evolution of digital connectivity from 1945 theoretical models to 2026 algorithmic environments, focusing on the pedagogical implications for [[Instructional Design and Technology]] (IDT)[cite: 205, 1176]. == Section 1: Foundational Research and History == <div class="mw-collapsible mw-collapsed"> === The Evolution of Hypertext === The conceptual roots of hypertext are grounded in the transition from linear paper constraints to associative digital networks. * [cite_start]'''The Memex (1945):''' [[Vannevar Bush]] proposed a mechanized desk for storing and linking microfilm, introducing the concept of "associative trails"[cite: 14, 55, 81]. * [cite_start]'''Project Xanadu (1960s):''' [[Ted Nelson]] coined the terms "hypertext" and "hypermedia" to describe non-sequential writing and "zippered lists" that maintain link integrity during document revision[cite: 11, 16, 76, 103]. * [cite_start]'''The NLS (1968):''' [[Douglas Engelbart]] demonstrated the first functional hypertext interface, including the computer mouse, during "The Mother of All Demos"[cite: 18, 59]. * [cite_start]'''The World Wide Web (1989):''' [[Tim Berners-Lee]] integrated hypertext with the Internet at CERN, creating a global, decentralized information space[cite: 19, 60]. === Multi-Scalar Definitions === [cite_start]Hypertext is defined by its ability to overcome linear constraints through interactive hyperlinks[cite: 12, 46]. * [cite_start]'''10-Word Definition:''' Non-linear electronic text connected by interactive hyperlinks for instant navigation[cite: 46]. * '''50-Word Definition:''' Hypertext is a non-linear method of organizing digital information using clickable links. [cite_start]Coined by Ted Nelson, it allows readers to navigate interconnected documents instantly, transforming sequential reading into a dynamic, associative, and interactive experience [cite: 48-50]. </div> == Section 2: Hypertext "In the Wild" Analysis == <div class="mw-collapsible mw-collapsed"> [cite_start]Using a specific framework of '''Six Essential Hypertext Features''', three platforms were audited to observe how hypertext influences user behavior[cite: 150, 310, 583]. {| class="wikitable" style="width: 100%;" ! Feature !! [[Goodreads]] (Social) !! [[Barnes & Noble]] (Retail) !! [[YouTube]] (Streaming) |- | '''Link''' || [cite_start]Bridges personal profiles to global book data [cite: 178-180]. || [cite_start]Structural anchors for commercial subcategories [cite: 230-231]. || [cite_start]Algorithmic bridges between video nodes[cite: 280]. |- | '''Tag''' || [cite_start]User-generated "shelves" (e.g., "to-read")[cite: 182]. || [cite_start]Industry metadata for sub-genres[cite: 233]. || [cite_start]Pill-shaped metadata for global trends[cite: 282]. |- | '''Template''' || [cite_start]Uniform profile layouts for social comparison[cite: 186]. || [cite_start]Rigid product cards focused on purchase friction [cite: 236-237]. || [cite_start]Standardized thumbnail grids for scannability[cite: 286]. |- | '''Transclusion''' || [cite_start]Mirroring user stats into a personal shelf [cite: 196-198]. || [cite_start]Real-time price and stock updates [cite: 244-245]. || [cite_start]Live view counts and engagement data [cite: 295-296]. |- | '''Filter''' || [cite_start]Personal library refinement by year or status [cite: 190-194]. || [cite_start]Catalog narrowing by price or trope [cite: 239-240]. || [cite_start]Instant feed re-sorting by topic pills [cite: 289-291]. |- | '''Objectification''' || [cite_start]The book as a "social unit" to be rated [cite: 201-204]. || [cite_start]The book as a "buyable product" [cite: 247-248]. || [cite_start]The video as a "media unit" [cite: 300-302]. |} </div> == Section 3: Contemporary Scholarship and Expansion == <div class="mw-collapsible mw-collapsed"> Recent research addresses the evolution of linking and the cognitive challenges of digital navigation. * [cite_start]'''Bidirectional Linking:''' Contemporary "tools for thought" like [[Obsidian (software)|Obsidian]] use backlinking to turn one-way signs into multidimensional intersections[cite: 766, 899]. * [cite_start]'''Hypertextual Friction:''' Systems like [[Wikipedia]] and Are.na prioritize authored links over opaque [[Algorithmic bias|algorithms]] to preserve user agency and traceable provenance [cite: 853, 901-902]. * [cite_start]'''Cognitive Disorientation:''' Research into "cognitive overhead" explores how non-linear navigation can lead to a loss of context, requiring new designs for "visible trails" in AI-mediated interfaces [cite: 904-905, 1906]. == Section 3: Contemporary Scholarship and Expansion == <div class="mw-collapsible mw-collapsed"> === From Navigation to Synthesis: "Tools for Thought" === In contemporary practice, the "navigable network" has transitioned from simple document retrieval to active knowledge synthesis[cite: 844, 897]. * '''Bidirectional Linking:''' Modern "feral" hypertext systems—such as [[Obsidian (software)|Obsidian]], Roam Research, and Logseq—utilize bidirectional links to manifest Nelson’s "associative trails" as visual graph databases. * '''Multidimensional Intersections:''' Unlike the "one-way signs" of early HTTP, these systems allow a user to see every node that points back to a current piece of information, creating a holistic map of personal knowledge [cite: 766-767, 814, 899]. * '''Scholarly Source:''' Anderson, M. W. R., & Millard, D. E. (2023). "Seven Hypertexts". ''Proceedings of the 34th ACM Conference on Hypertext and Social Media''. https://doi.org/10.1145/3603163.3609048 [cite: 803, 841, 907] === Hypertextual Friction vs. Algorithmic Opacity === As the web moves toward "seamless" AI-driven feeds, scholars argue for the reintroduction of "friction" to preserve human agency [cite: 852-853, 901]. * '''Traceable Provenance:''' Platforms like [[Wikipedia]] and Are.na prioritize visible, authored links over automated recommendations, ensuring users can trace the origin of information [cite: 855-856, 902]. * '''Countering the "Filter Bubble":''' Explicit linking acts as a counter to the opaque nature of algorithmic content feeds, requiring the reader to make conscious navigational choices [cite: 853, 901-903]. * '''Scholarly Source:''' Liu, S., & Almeda, S. G. (2025). "Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web". ''ACM Conference on Hypertext and Social Media''. https://doi.org/10.1145/3720533.3750065 [cite: 902, 908] === The Disorientation Challenge in Modern Interfaces === The "lost in hyperspace" phenomenon remains a critical concern in [[Instructional Design]], particularly as conversational AI begins to synthesize non-linear information [cite: 904-905, 1906]. * '''Cognitive Overhead:''' Non-linear navigation requires users to maintain a "mental map" of their location, which can lead to cognitive fatigue if the interface lacks clear "associative trails". * '''Conversational Navigation:''' Modern LLMs (like ChatGPT or Gemini) often displace traditional "blue-link" navigation, potentially causing a loss of critical context if the "visible trails" are removed [cite: 847, 851, 904-905]. * '''Scholarly Source:''' Rahdari, B., & Brusilovsky, P. (2025). "From Links to Dialogue; Hypertext Challenges and Opportunities in Conversational Navigation". https://doi.org/10.1145/3720533.3750064 [cite: 904, 909] === Section 3 References === * Anderson, M. W. R., & Millard, D. E. (2023). "Seven Hypertexts". https://doi.org/10.1145/3603163.3609048 [cite: 907] * Liu, S., & Almeda, S. G. (2025). "Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web". https://doi.org/10.1145/3720533.3750065 [cite: 908] * Rahdari, B., & Brusilovsky, P. (2025). "From Links to Dialogue; Hypertext Challenges and Opportunities in Conversational Navigation". https://doi.org/10.1145/3720533.3750064 [cite: 910] </div> </div> == References and DOI Citations == <div class="mw-collapsible"> * Anderson, M. W. R., & Millard, D. E. (2023). [cite_start]''Seven Hypertexts''. https://doi.org/10.1145/3603163.3609048 [cite: 900, 935] * Beckermeyer, R. (1970). [cite_start]''Interactive graphic consoles: environment and software''. https://doi.org/10.1145/1478462.1478506 [cite: 984, 1115] * Glantz, R. S. (1970). [cite_start]''SHOEBOX: a personal file handling system for textual data''. https://doi.org/10.1145/1478462.1478541 [cite: 972, 1119] * Halasz, F. G., Moran, T. P., & Trigg, R. H. (1987). [cite_start]''NoteCards in a nutshell''. https://doi.org/10.1145/29933.30859 [cite: 736, 882] * Liu, S., & Almeda, S. G. (2025). [cite_start]''Agency Among Agents: Designing with Hypertextual Friction in the Algorithmic Web''. https://doi.org/10.1145/3720533.3750065 [cite: 902, 936] * Nelson, T. H. (1965). [cite_start]''Complex information processing: a file structure for the complex, the changing and the indeterminate''. https://doi.org/10.1145/800197.806036 [cite: 71, 144, 741, 1123] * Rahdari, B., & Brusilovsky, P. (2025). ''From Links to Dialogue; [cite_start]Hypertext Challenges and Opportunities in Conversational Navigation''. https://doi.org/10.1145/3720533.3750064 [cite: 904, 937] * Trigg, R. H., & Weiser, M. (1986). [cite_start]''TEXTNET: a network-based approach to text handling''. https://doi.org/10.1145/5401.5402 [cite: 744, 887] </div> [[Category:Hypertext Theory]] [[Category:Instructional Design]] [[Category:Portfolio]] ``` onrwnl35islc5vudrr0zcw1226jzklz 0 328726 2801338 2026-03-29T18:27:38Z Ireicher2 2797952 Created page with "put your content here." 2801338 wikitext text/x-wiki put your content here. 4pewze3vuo5bvtr2rovz4payl2wtvd3 2801340 2801338 2026-03-29T18:36:31Z Ireicher2 2797952 2801340 wikitext text/x-wiki StudyFetch is an AI program that is transforming the way students can learn and get instant help on their class work and homework. StudyFetch is available on the internet, google play store, and apple app store. Some students find there are a lot of challenges with school work, and StudyFetch gives those students who are struggling an opportunity to learn and succeed in the difficult classes. You can use StudyFetch for any class or classes that you need help with. StudyFetch was created by Esan Durrani and Ryan Trattner in 2023. They were both around 20 years old when they launched StudyFetch.<ref>{{Cite web|url=https://www.forbes.com/profile/studyfetch/|title=StudyFetch|website=Forbes|language=en|access-date=2026-03-29}}</ref> 3e9vwfc7gjd5z6o0bwmc248cv4p513b 2801341 2801340 2026-03-29T18:39:35Z Ireicher2 2797952 2801341 wikitext text/x-wiki StudyFetch is an AI program that is transforming the way students can learn and get instant help on their class work and homework. StudyFetch is available on the internet, google play store, and apple app store. Some students find there are a lot of challenges with school work, and StudyFetch gives those students who are struggling an opportunity to learn and succeed in the difficult classes. You can use StudyFetch for any class or classes that you need help with. StudyFetch was created by Esan Durrani and Ryan Trattner in 2023. They were both around 20 years old when they launched StudyFetch<ref>{{Cite web|url=https://www.forbes.com/profile/studyfetch/|title=StudyFetch|website=Forbes|language=en|access-date=2026-03-29}}</ref>. ==References== 2hqkro0pg2umhdotkux03hxzqftoue6 2801342 2801341 2026-03-29T18:43:44Z Ireicher2 2797952 2801342 wikitext text/x-wiki ==Introduction== StudyFetch is an AI program that is transforming the way students can learn and get instant help on their class work and homework. StudyFetch is available on the internet, google play store, and apple app store. Some students find there are a lot of challenges with school work, and StudyFetch gives those students who are struggling an opportunity to learn and succeed in the difficult classes. You can use StudyFetch for any class or classes that you need help with. StudyFetch was created by Esan Durrani and Ryan Trattner in 2023. They were both around 20 years old when they launched StudyFetch<ref>{{Cite web|url=https://www.forbes.com/profile/studyfetch/|title=StudyFetch|website=Forbes|language=en|access-date=2026-03-29}}</ref>. ==Benefits== In StudyFetch you are able to record lectures, it will generate notes for you, create study materials for you to practice, and create images. Whether you want an image of a molecular compound, box plot, histogram, or something else, StudyFetch can generate it for you. In the classroom, teachers and professors talk a lot, it can be difficult to listen, remember and take note of everything. By using StudyFetch, you can listen and remember as much as you can while paying full attention to the lectures. This reduces the risk of not hearing your teacher or Professor talk about something important, because you were distracted by typing or writing what was said. In the event someone is a slow writer or typer, the notes that StudyFetch took for you eliminates that factor. For students that have jobs, or another class, it could be a struggle to attend office hours or if they can't get to the campus tutoring centers. Having an online tutor like StudyFetch, can help by tutoring students at the students convenience. ==References== hk8qpl8t9innge9bfor0w67cpbr6687 2801421 2801342 2026-03-30T04:59:21Z Matthew0824 3056136 I added more information to my own article 2801421 wikitext text/x-wiki ==== Introduction ==== StudyFetch is an AI program that is transforming the way students can learn and get instant help on their class work and homework. StudyFetch is available on the internet, google play store, and apple app store. Some students find there are a lot of challenges with school work, and StudyFetch gives those students who are struggling an opportunity to learn and succeed in the difficult classes. You can use StudyFetch for any class or classes that you need help with. StudyFetch was created by Esan Durrani and Ryan Trattner in 2023. They were both around 20 years old when they launched StudyFetch. StudyFetch headquarters is in Los Angeles, CA . <ref name=":0">{{Cite web|url=https://finance.yahoo.com/news/studyfetch-launches-free-studyfetch-read-130000540.html?guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAAM_ZWyruZkxwSTtpMFwAFEbzDn4pDEARyMU4gr3FXvV_OKRo_76ScFJ_lFFPHy8kAs0_USTMcNkLHc4VmGZ6rpwp8HFNBjsJlfRFnt7maolIrxw1pmJEfwUkHxs6ASB9axqPt_gqDyxK43aAsBTH4g3gN8-seeJrgxQXA9YG-Tzs&guccounter=2|title=StudyFetch launches Free platform to solve reading crisis}}</ref> == Reading and learning issues and effects within the United States of America == StudyFetch created a free AI reading feature to help adults in need, and to resolve the major problem of more than half of American adults only being able to read within the elementry school difficulty level. Around one-third of children have either not learned how to read or have poor reading skills. <ref name=":0" /> The Literacy crisis alone caused the United States to add over two trillion dollars in debt each year. Due to this debt crisis, medical care prices went up, and job layoffs increased. <ref name=":0" /> StudyFetch uses statistically analyzed learning strategies to summarize topics, in a way that anyone can understand. <ref name=":0" /> ==Benefits== StudyFetch not only benefits Students, but, it also benefits teachers and professors. Teachers and professors can use StudyFetch to evaluate there students coursework and determine level-based material for each students needs.<ref name=":0" /> StudyFetch will take teachers and professors instructions and turn it into terms the student can understand.<ref name=":0" /> This can create the best way each student can learn the class assignments and material.<ref name=":0" /> Ryan Trattner (CEO), wanted to ensure that every student has the opportunity to succeed in there courses, so he created StudyFetch to help students achieve that opportunity.<ref name=":0" /> StudyFetch platform is available for free, with no required subscription charges.<ref name=":0" /> StudyFetch records lectures, makes notes, turns them into flashcards, podcasts, and practice quizzes. [https://medium.com/@abebellini/advancing-your-education-with-ai-facts-about-studyfetch-45b130d1c9f9 <nowiki>[2]</nowiki>] Since StudyFetch has these features, Students can just pay full attention to what the teacher or professor is talking about, while StudyFetch takes key notes and writes down everything that the teacher or professor is saying. [https://medium.com/@abebellini/advancing-your-education-with-ai-facts-about-studyfetch-45b130d1c9f9 <nowiki>[2]</nowiki>] AI technology is able to teach students about topics that they were clueless about within a few hours. [https://techedpodcast.com/whitaker/ <nowiki>[3]</nowiki>] == Cons == AI programs are taking away learning from students. <ref name=":0" /> As technology enhances, grades are decreasing on tests. <ref name=":0" /> Creators thought that more and better technology within education, the better the academic learning outcome. <ref name=":0" /> A neuroscientist named Jared Cooney Horvarth believes that the more and better technology, the education outcomes will worsen. <ref name=":0" /> Using AI programs causes lazyness and makes people feel like they don't need to think, since they can just look up anything within a minute. An MIT survey showed that students finish there homework about 37% faster when using AI tools. As AI gets more powerful, it has the potential to replace jobs, and we may see a future where work is optional or jobs are completely replaced. [https://techedpodcast.com/whitaker/ <nowiki>[3]</nowiki>] Students can complete High School and a college degree without learning anything if you use StudyFetch or any other AI platform for quick answers. [https://techedpodcast.com/whitaker/ <nowiki>[3]</nowiki>] ==References== 1. [https://finance.yahoo.com/news/studyfetch-launches-free-studyfetch-read-130000540.html?guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAAM_ZWyruZkxwSTtpMFwAFEbzDn4pDEARyMU4gr3FXvV_OKRo_76ScFJ_lFFPHy8kAs0_USTMcNkLHc4VmGZ6rpwp8HFNBjsJlfRFnt7maolIrxw1pmJEfwUkHxs6ASB9axqPt_gqDyxK43aAsBTH4g3gN8-seeJrgxQXA9YG-Tzs&guccounter=2 StudyFetch Launches Free 'StudyFetch Read' Platform to Combat America's Literacy Crisis] 2. [https://medium.com/@abebellini/advancing-your-education-with-ai-facts-about-studyfetch-45b130d1c9f9 Advancing Your Education with AI — Facts About StudyFetch | by Abe Bellini | Medium] 3. [https://techedpodcast.com/whitaker/ AI can close the learning gap in Underserved Classrooms. But We Have to Guide, Not Just Give] ckgvik60xhvmlm1o3qlbs3j91tcg75d 6 328727 2801389 2026-03-30T01:31:33Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260330 - 20260328) |Source={{own|Young1lim}} |Date=2026-03-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2801389 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260330 - 20260328) |Source={{own|Young1lim}} |Date=2026-03-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3x0istf3b4266ver9xb3kv02kytan3o 6 328728 2801391 2026-03-30T01:37:41Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260330 - 20260328) |Source={{own|Young1lim}} |Date=2026-03-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2801391 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260330 - 20260328) |Source={{own|Young1lim}} |Date=2026-03-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} dluiyvz6edacgubamwgwcfaq9vyx6oe 0 328729 2801392 2026-03-30T02:05:24Z ThoEkm382 3062144 Created page with "{{DISPLAYTITLE:Steve Jobs leadership}} = What the Turing Machine Influenced in the Modern Day = === Background === The Turing machine is a device created by Alan Turing in 1937-1937 that were specifically made for the computability of numbers. The Turing Machine got its name from Alonzo Church during the review of Turing's paper in 1937. Turing machines are seen as a device that built the foundation for future models of computability devices and computer science.<ref>{..." 2801392 wikitext text/x-wiki {{DISPLAYTITLE:Steve Jobs leadership}} = What the Turing Machine Influenced in the Modern Day = === Background === The Turing machine is a device created by Alan Turing in 1937-1937 that were specifically made for the computability of numbers. The Turing Machine got its name from Alonzo Church during the review of Turing's paper in 1937. Turing machines are seen as a device that built the foundation for future models of computability devices and computer science.<ref>{{Cite web|url=https://plato.stanford.edu/entries/turing-machine/|title=Turing Machine|date=Summer 2025|website=2025-05-21|language=en|access-date=2026-03-19|last=Del Mol|first=Liesbeth}}</ref> == Influences in Modern Computing == [[File:A Turing Machine.png|thumb|308x308px|'''A Turing Machine.png''']] The Turing machine is used as the foundation in modern computers. For example, every new computer built uses a finite approximation of a Turing machine. This means that programs that run on the computers correspond exactly as programs run on Turing machines. All computers of today share the same structure as the Turing tape-head-control paradigm.<ref>{{Cite web|url=https://intuitionlabs.ai/articles/what-is-a-turing-machine|title=Turing Machine Explained: The Model of Modern Computation|website=intuitionlabs.ai|language=en|access-date=2026-03-21|date=2026-03-20|last=Laurent|first=Adrien}}</ref> == Influences on Artificial Intelligence == In the modern artificial intelligence field, the Turing Test is one of the most popular questions that helped share AI to what it is today. The Turing Test was Alan Turing's idea, where a human judge would be in one room, with another human in another room and one room with a machine inside of it. The judge will receive a text-based conversation from each room, and if the judge isn't able to reliably decide which text is made by who, then the machine showed intelligence in a practical sense. The Turing machine is also the foundation used for AI, which helps the AI understand things such as problem-solving.<ref>{{Cite web|url=https://medium.com/@igquinteroch/alan-turings-computing-machinery-and-intelligence-foundations-of-ai-and-its-modern-relevance-797756081e79|title=Alan Turing’s “Computing Machinery and Intelligence”: Foundations of AI and Its Modern Relevance|last=Quintero|first=Ignacio|website=medium.com|language=en|access-date=2026-03-21|date=2025-05-28}}</ref> == Influence in Modern Cryptography == The Turing machine was able to set the foundation for modern cryptography, being able to shape how we understand computation and security. This was first shown during World War II, where one of Turing's devices, not the Turing machine, was able to break the Enigma code, showcasing just how powerful machines could be in cryptography. In today's time, the Turing machine is used as the backbone in how modern cryptographic systems are designed. Although not directly used in today's age, the Turing machine was able to forge cryptography into how it performs in the modern days.<ref>{{Cite web|url=https://www.nist.gov/blogs/taking-measure/alan-turings-everlasting-contributions-computing-ai-and-cryptography|title=Alan Turing’s Everlasting Contributions to Computing, AI and Cryptography|last=Peralta|first=Rene|date=2022-06-23|website=nist.gov|access-date=2026-03-23}}</ref> == References == 1: May 2025, De Mol, Liesbeth, "Turing Machines," ''The Stanford Encyclopedia of Philosophy'' (Summer 2025 Edition, Edward N. Zalta & Uri Nodelman (eds.) ''plato.stanford.edu''. Retrieved 2026-03-19. 2: March 2026, Adrien Laurent. "Turing Machine Explained: The Model of Modern Computation." ''intuitionlabs.ai''. Retrieved 2026-03-21. 3: May 2025, Ignacio Quintero. "Alan Turing's 'Computing Machinery and Intelligence': Foundations of AI and Its Modern Relevance." ''medium.com''. Retrieved 2026-03-21. 4: June 2022, Rene Peralta. "Alan Turing's Everlasting Contributions to Computing, Ai and Cryptography." ''nist.gov''. Retrieved 2026-03-23. {{DEFAULTSORT:"Can't Help Myself" Robot}} {{DEFAULTSORT:Steve Jobs Leadership}} [[Category:Robotics]] [[Category:Social problems]] [[Category:Installation art]] [[Category:Video]] {{BookCat}} <references /> qz9wa33vro1l47j79kcsiezme47dc7o 0 328730 2801418 2026-03-30T04:41:02Z RamonVela5 3062311 Created page with "== Introduction == == Background == == How TikTok Works == == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References ==" 2801418 wikitext text/x-wiki == Introduction == == Background == == How TikTok Works == == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References == 5cdtvyc2ns4aorjuyrxizyjoxyz8m1k 2801428 2801418 2026-03-30T05:16:04Z RamonVela5 3062311 /* Introduction */ 2801428 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == == How TikTok Works == == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References == qwt0de730ajosiecuwoce8ij088n9hk 2801432 2801428 2026-03-30T05:29:44Z RamonVela5 3062311 /* Background */ 2801432 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References == 8ykamww3sn09eg4le7le18yjkufl95u 2801437 2801432 2026-03-30T05:45:53Z RamonVela5 3062311 /* How TikTok Works */ 2801437 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer period of time before even realizing how many hours have passed. == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References == 4r5e4405mzxna5z8nklvt4rdkekomkb 2801439 2801437 2026-03-30T05:46:08Z RamonVela5 3062311 /* How TikTok Works */ 2801439 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == == Examples == == Impact on Young Adults == == Conclusion == == References == bch9immjv0b3pktjz263pljicnnb29d 2801443 2801439 2026-03-30T05:53:05Z RamonVela5 3062311 /* Effects on Attention Span */ 2801443 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == == Impact on Young Adults == == Conclusion == == References == 0n8kx81t2jjrurm1qset3b0hl7hx8kn 2801446 2801443 2026-03-30T05:59:18Z RamonVela5 3062311 /* Examples */ 2801446 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == == Conclusion == == References == p2oeh7vlpq0zn4h04zlhyabt8zyoa0b 2801449 2801446 2026-03-30T06:13:14Z RamonVela5 3062311 /* Impact on Young Adults */ 2801449 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == == References == pnbu82lt3i46ejn3gg2r1d42uh3ct5o 2801453 2801449 2026-03-30T06:19:38Z RamonVela5 3062311 /* Conclusion */ 2801453 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == qf63n8nruwv19i0lp6cr1bboj7noz2i 2801456 2801453 2026-03-30T06:32:21Z RamonVela5 3062311 /* References */ 2801456 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://time.com/3858309/attention-spans-goldfish/ Time – Attention Spans in the Digital Age] 7b5izzx2wcjaaio1ia16xt3kozo20rg 2801457 2801456 2026-03-30T06:39:01Z RamonVela5 3062311 /* Examples */ 2801457 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. This endless scrolling has even created a new term on the internet in the last few years, known as "Doomscrolling". In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://time.com/3858309/attention-spans-goldfish/ Time – Attention Spans in the Digital Age] dmqysb5vx0xgnsnawwalb0vutwsr9fs 2801458 2801457 2026-03-30T06:42:23Z RamonVela5 3062311 /* References */ 2801458 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. This endless scrolling has even created a new term on the internet in the last few years, known as "Doomscrolling". In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://time.com/3858309/attention-spans-goldfish/ Time – Attention Spans in the Digital Age] [https://www.health.harvard.edu/mind-and-mood/doomscrolling-dangers Harvard Health - The Dangers of Doomscrolling] co28v5qgu1dhzri1n9olqh41d9o856b 2801459 2801458 2026-03-30T06:43:06Z RamonVela5 3062311 /* References */ 2801459 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. This endless scrolling has even created a new term on the internet in the last few years, known as "Doomscrolling". In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://www.health.harvard.edu/mind-and-mood/doomscrolling-dangers Harvard Health - The Dangers of Doomscrolling] nvy9bej82w7jpzrgrue0qdvvcznmrq6 2801462 2801459 2026-03-30T06:55:09Z RamonVela5 3062311 /* Effects on Attention Span */ 2801462 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content. Behaviors and habits such as "doomscrolling," where users continually consume large amounts of fast-paced or negative content have been linked to increased stress, reduced focus and difficulty engaging with slower paced tasks; this exposure to overwhelming information has the potential to decrease an individual's ability to maintain attention during everyday activities that are normally performed at a slower pace than those found online such as studying, reading, or completing assignments == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. This endless scrolling has even created a new term on the internet in the last few years, known as "Doomscrolling". In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://www.health.harvard.edu/mind-and-mood/doomscrolling-dangers Harvard Health - The Dangers of Doomscrolling] mvacl7xthn8dnxn8vllbhaed9nue0xa 2801463 2801462 2026-03-30T06:55:31Z RamonVela5 3062311 /* Examples */ 2801463 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content. Behaviors and habits such as "doomscrolling," where users continually consume large amounts of fast-paced or negative content have been linked to increased stress, reduced focus and difficulty engaging with slower paced tasks; this exposure to overwhelming information has the potential to decrease an individual's ability to maintain attention during everyday activities that are normally performed at a slower pace than those found online such as studying, reading, or completing assignments == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take rge time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://www.health.harvard.edu/mind-and-mood/doomscrolling-dangers Harvard Health - The Dangers of Doomscrolling] 0492z2eg34pzmrami88umsuj21e84yi 2801465 2801463 2026-03-30T07:02:30Z RamonVela5 3062311 /* Examples */ 2801465 wikitext text/x-wiki == Introduction == Short Form content has become one of the most widely used forms of digital media today. Tiktok is one of the many platforms where short Form video content is available for user to view and share with others. Most TikTok videos range from a couple of seconds to as much as a minute or two. The reason behind creating these types of digital media was to create an environment that will continue to engage users by providing them with a variety of new, quickly changing information. One major concern has been created around the idea of how short-form digital media can affect the attention spans in young adults.With increasing amounts of time being spent viewing and consuming short-form information, it could become increasingly difficult for young adults to stay focused on reading, watching or completing longer more detailed type of tasks. == Background == The large amount of short-form content today is due in part to the rapid growth of video-sharing apps (like TikTok) since its launch in 2016. While TikTok allows for longer than Instagram Reel or YouTube Short, the focus is on creating content that can be easily consumed while being watched in a small format repeatedly. A key feature of TikTok's success has been an advanced recommendation engine that analyzes how viewers use the app; this allows users to continuously watch endless amounts of videos that have been specifically curated to match viewer interests. As a result, there is now a whole new way that users consume media by viewing multiple, high-energy content pieces at a very fast pace. In addition to TikTok's dominance of the short-form video market, many other social media platforms(such as Instagram Reels and YouTube Shorts) have followed suit and produced a mass market for short-form content. == How TikTok Works == The way TikTok is structured keeps you constantly entertained with video after video. Upon opening, you are first presented with your "For You" page; this is created using algorithms that consider the past activities of users on the app, including likes, time spent viewing videos, and engagement. Videos are generally very brief in length (a few seconds up to just less than one minute) and can be easily consumed. Also, when scrolling down on the screen, new content will appear almost instantly at all times; therfore there is no natural or logical end point to viewing. Thus, users are continually encouraged to view longer and longer periods of time before even realizing how many hours have passed. == Effects on Attention Span == Short-form content formats like TikTok, which feature fast-paced video segments that are consumed quickly by viewers, can also reduce the attention span of the viewer, In addition to having an effect on how long viewers will engage in certain types of content(e.g., reading, viewing longer videos), the rapid delivery of short-form content is likely to cause viewers to develope habits of consuming content rapidly, potentially reducing their tolerance of longer forms of content and contributing to developing a habit of seeking immediate entertainment, thereby increasing the likelihood that consumers will lose the patience with slower paced or less stimulating content. Behaviors and habits such as "doomscrolling," where users continually consume large amounts of fast-paced or negative content have been linked to increased stress, reduced focus and difficulty engaging with slower paced tasks; this exposure to overwhelming information has the potential to decrease an individual's ability to maintain attention during everyday activities that are normally performed at a slower pace than those found online such as studying, reading, or completing assignments == Examples == One way that short-form content affects our focus is by how we use TikTok. Some users will scroll endlessly through short videos(which they sometimes do for hours) because it feels easier to watch video after video than take the time to read something or study. In addition, many people tend to multitask as they are viewing content, which all makes it harder to maintain an attentive state. == Impact on Young Adults == The effect of short-term content on young adults has a strong presence because most young adults obtain their entertainment, communication, and news from digital media. With the potential reduction of attention span, young adults may find it increasingly difficult to be attentive at school; complete homework or other assignment; or complete long tasks which require prolonged concentration. The changed in attention span can also have an effect on day-to-day routines;thus, providing instant gratification through rapid stimulation. Ultimately, over time this pattern will contribute to diminished production; decreased ability to learn; and reduced ability to concentrate in both academic settings and in general. == Conclusion == Young adults are increasingly consuming short-form content that provides instant gratification on social media platforms such as TikTok. The benefits of short-form content could be an indicator for how users are focusing and processing information differently. To create awareness among the public about the potential negative impacts of this technology; we need to recognize what types of habits or patterns of behavior young adults are developing by continuously accessing digital forms of media. == References == [https://www.health.harvard.edu/blog/keeping-your-smartphone-nearby-may-not-be-so-smart-2017080212163 Harvard Health – Smartphone Distraction and Cognitive Cost] [https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you TikTok – How the “For You” Algorithm Works] [https://www.health.harvard.edu/mind-and-mood/doomscrolling-dangers Harvard Health - The Dangers of Doomscrolling] frbh8c0dz72e0pnypq6zg11phugom9w 0 328731 2801419 2026-03-30T04:41:24Z Vania Lat 3057605 added a lot of the content 2801419 wikitext text/x-wiki == Esports - Overview == Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref>{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''Spacewar!'' which was organized by students. Fast forward a few years later (1980), a ''Space Invaders'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres (MOBA, Shooters, Fighting, etc.) with people playing these games around the world. (PDF Citation needed) ==== Rise of the Personal Computer Industry ==== Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to personal computers (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments. When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s internet cafes (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest League of Legends (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the Riot Games team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the 2020 LoL Worlds. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the 2017 LoL Worlds opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, first person shooters (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, Application Programming Interfaces (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock Overwatch team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in DOTA 2, Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. Twitch and YouTube are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * Twitch: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and Bits<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * YouTube: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * Facebook Gaming: Alongside the live chat feature, it has subscriptions, Stars, and utilizes cloud gaming<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing (PDF NEEDED). Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology (PDF NEEDED). ==== The Players & Future Successors ==== As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> As players and teams become more popular, the esports sector grows everyday. 8m1pu49lzmrourwbaf90cisg8n4v7i8 2801420 2801419 2026-03-30T04:50:05Z Vania Lat 3057605 added reference header 2801420 wikitext text/x-wiki == Esports - Overview == Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref>{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''Spacewar!'' which was organized by students. Fast forward a few years later (1980), a ''Space Invaders'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres (MOBA, Shooters, Fighting, etc.) with people playing these games around the world. (PDF Citation needed) ==== Rise of the Personal Computer Industry ==== Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to personal computers (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments. When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s internet cafes (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest League of Legends (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the Riot Games team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the 2020 LoL Worlds. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the 2017 LoL Worlds opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, first person shooters (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, Application Programming Interfaces (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock Overwatch team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in DOTA 2, Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. Twitch and YouTube are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * Twitch: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and Bits<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * YouTube: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * Facebook Gaming: Alongside the live chat feature, it has subscriptions, Stars, and utilizes cloud gaming<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing (PDF NEEDED). Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology (PDF NEEDED). ==== The Players & Future Successors ==== As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> As players and teams become more popular, the esports sector grows everyday. ==References== 4kmjh2flgu7odlj2bdeu8yubzjnslcr 2801433 2801420 2026-03-30T05:30:20Z Vania Lat 3057605 added external links 2801433 wikitext text/x-wiki == Esports - Overview == Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref>{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''[[wikipedia:Spacewar!|Spacewar!]]'' which was organized by students. Fast forward a few years later (1980), a ''[[wikipedia:Space_Invaders|Space Invaders]]'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres ([[wikipedia:Multiplayer_online_battle_arena|MOBA]], Shooters, Fighting, etc.) with people playing these games around the world. (PDF Citation needed) ==== Rise of the Personal Computer Industry ==== Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to [[wikipedia:Personal_computer|personal computers]] (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments.<ref name=":0" /> When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s [[wikipedia:PC_bang|internet cafes]] (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest [[wikipedia:League_of_Legends|League of Legends]] (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the [[wikipedia:Riot_Games|Riot Games]] team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the [[wikipedia:2020_League_of_Legends_World_Championship|2020 LoL Worlds]]. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the [[wikipedia:2017_League_of_Legends_World_Championship|2017 LoL Worlds]] opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, [[wikipedia:First-person_shooter|first person shooters]] (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, [[wikipedia:API|Application Programming Interfaces]] (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock [[wikipedia:Overwatch|Overwatch]] team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in [[wikipedia:Dota_2|Defense of the Ancients 2]] (DOTA 2), Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. [https://www.twitch.tv/ Twitch] and [https://www.youtube.com/ YouTube] are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * [https://www.twitch.tv/ Twitch]: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and Bits<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * [https://www.youtube.com/ YouTube]: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * [https://www.facebook.com/gaming/ Facebook Gaming]: Alongside the live chat feature, it has subscriptions, Stars, and utilizes cloud gaming<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing (PDF NEEDED). Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology (PDF NEEDED). ==== The Players & Future Successors ==== As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> As players and teams become more popular, the esports sector grows everyday. ==References== 067d4c8n08my1qsvjsjx16dg2zmzrap 2801450 2801433 2026-03-30T06:13:16Z Vania Lat 3057605 added images 2801450 wikitext text/x-wiki == Esports - Overview == [[File:LGD Gaming at the 2015 LPL Summer Finals.jpg|alt=The Esports team "LDG Gaming" is shown competing at the 2015 League of Legends Professional Summer Finals. The players are shown to be playing on PCs and wearing headphones.|thumb|312x312px|Esports team "LDG Gaming" competing in the 2015 League of Legends Professional Summer Finals]] Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref>{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''[[wikipedia:Spacewar!|Spacewar!]]'' which was organized by students. Fast forward a few years later (1980), a ''[[wikipedia:Space_Invaders|Space Invaders]]'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres ([[wikipedia:Multiplayer_online_battle_arena|MOBA]], Shooters, Fighting, etc.) with people playing these games around the world. (PDF Citation needed) ==== Rise of the Personal Computer Industry ==== [[File:T1 Base Camp PC bang interior - 2023-07-21.jpg|alt=A huge room filled with rentable PCs that come equipped with keyboards, mousepads, mouses, headphones, and other gear using for gaming. The room reps the "T1" team's brand through logos within the cafe.|thumb|360x360px|Esports "T1" team's internet cafe]] Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to [[wikipedia:Personal_computer|personal computers]] (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments.<ref name=":0" /> When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s [[wikipedia:PC_bang|internet cafes]] (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== [[File:Shanghai Media Tech Studio for Worlds 2020.jpg|alt=The image depicts a huge stage with lights at the top and LED screens surrounding the stage. The PCs for each team is situated on opposite sides with one team on the left and the other team positioned on the right.|thumb|415x415px|The stage for the League of Legends 2020 Worlds Championship.]] These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest [[wikipedia:League_of_Legends|League of Legends]] (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the [[wikipedia:Riot_Games|Riot Games]] team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the [[wikipedia:2020_League_of_Legends_World_Championship|2020 LoL Worlds]]. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the [[wikipedia:2017_League_of_Legends_World_Championship|2017 LoL Worlds]] opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, [[wikipedia:First-person_shooter|first person shooters]] (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, [[wikipedia:API|Application Programming Interfaces]] (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock [[wikipedia:Overwatch|Overwatch]] team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in [[wikipedia:Dota_2|Defense of the Ancients 2]] (DOTA 2), Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. [https://www.twitch.tv/ Twitch] and [https://www.youtube.com/ YouTube] are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * [https://www.twitch.tv/ Twitch]: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and Bits<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * [https://www.youtube.com/ YouTube]: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * [https://www.facebook.com/gaming/ Facebook Gaming]: Alongside the live chat feature, it has subscriptions, Stars, and utilizes cloud gaming<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing (PDF NEEDED). Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology (PDF NEEDED). ==== The Players & Future Successors ==== [[File:OpTic Gaming win Valorant Masters Reykjavík 2022.jpg|alt=The esports team "OpTic Gaming" is shown holding the victory trophy up together with smiles on their faces at the Valorant Masters 2022 Championship in Reyjavík.|thumb|317x317px|Esports team "OpTic Gaming" wins Valorant Masters 2022 in Reyjavík.]] As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> As players and teams become more popular, the esports sector grows everyday. ==References== mc4psvabm4btsaeg5acey69wl4c76mg 2801455 2801450 2026-03-30T06:31:48Z Vania Lat 3057605 finished adding sources & minor grammatical/sentences fixes 2801455 wikitext text/x-wiki == Esports - Overview == [[File:LGD Gaming at the 2015 LPL Summer Finals.jpg|alt=The Esports team "LDG Gaming" is shown competing at the 2015 League of Legends Professional Summer Finals. The players are shown to be playing on PCs and wearing headphones.|thumb|312x312px|Esports team "LDG Gaming" competing in the 2015 League of Legends Professional Summer Finals.]] Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref name=":1">{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''[[wikipedia:Spacewar!|Spacewar!]]'' which was organized by students. Fast forward a few years later (1980), a ''[[wikipedia:Space_Invaders|Space Invaders]]'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres ([[wikipedia:Multiplayer_online_battle_arena|MOBA]], Shooters, Fighting, etc.) with people playing these games around the world.<ref name=":0" /> ==== Rise of the Personal Computer Industry ==== [[File:T1 Base Camp PC bang interior - 2023-07-21.jpg|alt=A huge room filled with rentable PCs that come equipped with keyboards, mousepads, mouses, headphones, and other gear using for gaming. The room reps the "T1" team's brand through logos within the cafe.|thumb|360x360px|Esports "T1" team's internet cafe]] Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to [[wikipedia:Personal_computer|personal computers]] (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments.<ref name=":0" /> When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s [[wikipedia:PC_bang|internet cafes]] (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== [[File:Shanghai Media Tech Studio for Worlds 2020.jpg|alt=The image depicts a huge stage with lights at the top and LED screens surrounding the stage. The PCs for each team is situated on opposite sides with one team on the left and the other team positioned on the right.|thumb|415x415px|The stage for the League of Legends 2020 Worlds Championship.]] These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest [[wikipedia:League_of_Legends|League of Legends]] (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the [[wikipedia:Riot_Games|Riot Games]] team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the [[wikipedia:2020_League_of_Legends_World_Championship|2020 LoL Worlds]]. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the [[wikipedia:2017_League_of_Legends_World_Championship|2017 LoL Worlds]] opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, [[wikipedia:First-person_shooter|first person shooters]] (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, [[wikipedia:API|Application Programming Interfaces]] (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock [[wikipedia:Overwatch|Overwatch]] team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in [[wikipedia:Dota_2|Defense of the Ancients 2]] (DOTA 2), Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. [https://www.twitch.tv/ Twitch] and [https://www.youtube.com/ YouTube] are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * [https://www.twitch.tv/ Twitch]: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and Bits<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * [https://www.youtube.com/ YouTube]: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * [https://www.facebook.com/gaming/ Facebook Gaming]: Alongside the live chat feature, it has subscriptions, Stars, and utilizes cloud gaming<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing.<ref name=":1" /> Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology.<ref name=":1" /> ==== The Players & Future Successors ==== [[File:OpTic Gaming win Valorant Masters Reykjavík 2022.jpg|alt=The esports team "OpTic Gaming" is shown holding the victory trophy up together with smiles on their faces at the Valorant Masters 2022 Championship in Reyjavík.|thumb|317x317px|Esports team "OpTic Gaming" wins Valorant Masters 2022 in Reyjavík.]] As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> As players and teams become more popular, the esports sector grows everyday. ==References== 3noq7pfcsg081u7l6dxy2jd8i272u92 2801460 2801455 2026-03-30T06:48:49Z Vania Lat 3057605 added content (links), formatting 2801460 wikitext text/x-wiki == Esports - Overview == [[File:LGD Gaming at the 2015 LPL Summer Finals.jpg|alt=The Esports team "LDG Gaming" is shown competing at the 2015 League of Legends Professional Summer Finals. The players are shown to be playing on PCs and wearing headphones.|thumb|312x312px|Esports team "LDG Gaming" competing in the 2015 League of Legends Professional Summer Finals.]] Esports refers to an industry where gamers worldwide professionally compete in a team or solo setting across various games. These esports players often compete in tournaments in their respective games while representing a team and/or country. The evolution of the internet and growth of technology allowed gamers to have greater accessibility to resources for practicing, with esports maintaining its momentum from the 1990s.<ref>{{Cite web|url=https://www.amu.apus.edu/area-of-study/health-sciences/resources/the-future-of-esports/|title=The Future of Esports: Key Trends Shaping Competitive Gaming|website=American Military University|language=en|access-date=2026-03-30}}</ref> As esports becomes more popular, the industry itself becomes more lucrative in terms of opportunities for both players and businesses. The industry’s popularity can be attributed to these segments: Publishers (video game distributors and developers), organizations and their teams, players, and the community (fanbase).<ref name=":1">{{Cite web|url=https://www.videogameseurope.eu/vge-esports/esports-a-complete-guide-by-the-video-games-industry/|title=Video Games Europe - The Guide to Esports|website=VIDEOGAMES EUROPE|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Esports Tournaments == === Tournaments - Overview & History === Esports tournaments are defined by the competition of the top players within a respective game competing against each other. These tournaments can be held locally or internationally with each tournament having its own unique bracketing system. The first unofficial esports tournament (1971) involved the game ''[[wikipedia:Spacewar!|Spacewar!]]'' which was organized by students. Fast forward a few years later (1980), a ''[[wikipedia:Space_Invaders|Space Invaders]]'' tournament with 10,000 players started the momentum for esports to take off.<ref>{{Cite web|url=https://aixr.org/insights/the-rise-and-future-of-esports-2/|title=The Rise and Future of eSports|last=Pontes|first=Henrique|date=2023-04-21|website=AIXR|language=en-US|access-date=2026-03-30}}</ref> Now, there are tournaments that cater to many different game genres ([[wikipedia:Multiplayer_online_battle_arena|MOBA]], Shooters, Fighting, etc.) with people playing these games around the world.<ref name=":0" /> ==== Rise of the Personal Computer Industry ==== [[File:T1 Base Camp PC bang interior - 2023-07-21.jpg|alt=A huge room filled with rentable PCs that come equipped with keyboards, mousepads, mouses, headphones, and other gear using for gaming. The room reps the "T1" team's brand through logos within the cafe.|thumb|360x360px|Esports "T1" team's internet cafe]] Asia is the powerhouse behind the momentum for esports as they are heavily invested in the gaming and technological industry. Since the start, countries like South Korea, China, and Japan have been experimenting with technology from consoles to [[wikipedia:Personal_computer|personal computers]] (PCs). With both of these industries working hand-in-hand, virtual reality and artificial intelligence have been widely applied especially in tournaments.<ref name=":0" /> When South Korea was able to combine high-speed internet and PCs on a nationwide scale before everyone else, the idea of PC gaming rapidly took off. People from all over the world came to Korea to practice and watch esports. This led to Korea’s [[wikipedia:PC_bang|internet cafes]] (PC cafes) being a popular hit as it allowed anyone to sit at a cafe, rent a computer and practice/play any game they wanted. Since then PCs have been widely used in major tournaments all around the world.<ref name=":0">{{Cite journal|last=Chap|first=Weing Kham|last2=Rao|first2=Anindita Chatterjee|last3=Pandey|first3=Praveen Kumar|date=2022-12-23|title=A Technological Review on Rise of Esports in World Economy|url=https://dl.acm.org/doi/fullHtml/10.1145/3590837.3590916|journal=Proceedings of the 4th International Conference on Information Management &amp; Machine Intelligence|location=New York, NY, USA|publisher=ACM|pages=1–8|doi=10.1145/3590837.3590916}}</ref> ==== Exploring Artificial Intelligence, Virtual Reality, and Extended Reality ==== [[File:Shanghai Media Tech Studio for Worlds 2020.jpg|alt=The image depicts a huge stage with lights at the top and LED screens surrounding the stage. The PCs for each team is situated on opposite sides with one team on the left and the other team positioned on the right.|thumb|415x415px|The stage for the League of Legends 2020 Worlds Championship.]] These major esports tournaments often add artificial intelligence (AI), virtual reality (VR), extended reality (XR), or other features to build hype for the audience. For instance, the biggest [[wikipedia:League_of_Legends|League of Legends]] (LoL) tournament, ‘Worlds,’ brings the game and players’ performance to life using extended reality. Similar to how major league sports events are hosted in different states, LoL Worlds follows this same format. To bring homage to that respective host country, the [[wikipedia:Riot_Games|Riot Games]] team used LED screens and augmented reality (AR) to create a stage that incorporates elements from that country during live broadcasting and competition in the [[wikipedia:2020_League_of_Legends_World_Championship|2020 LoL Worlds]]. The combination of XR and AR along with advanced production technology supplied by companies like Lux Machina allows VR worlds to be instantly rendered on a screen with specs of “32K resolution and 60 frames per second.”<ref>{{Cite web|url=https://www.sportsvideo.org/2020/10/30/riot-games-brings-revolutionary-xr-tech-to-lol-world-championship-in-shanghai/|title=Riot Games Brings Revolutionary XR Tech to LoL World Championship in Shanghai - Sports Video Group|date=2020-10-30|language=en-US|access-date=2026-03-30}}</ref> Opening tournament performances often incorporate augmented reality LoL characters, particle effects, and other game aspects to enhance and bring life to these performances. For instance, the [[wikipedia:2017_League_of_Legends_World_Championship|2017 LoL Worlds]] opening features the opening song “Legends Never Dies” paired with a LED screen displaying the music video and an AR LoL Elder Dragon to enhance the live singing/dancing performance.<ref>{{Cite web|url=https://www.youtube.com/watch?v=mP3fGkpmVM0&list=RDmP3fGkpmVM0&start_radio=1|title=- YouTube|website=www.youtube.com|access-date=2026-03-30}}</ref> Besides this, AR is used as an overlay tool to display a team/players’ stats and their real name/gamer tag which is helpful for differentiating players especially as a newcomer to the esports scene. Similarly, AR organizes relevant game information such as timers for abilities or objectives, map placement, and battles in an easily visualized format during tournaments. In contrast to the MOBA LoL, [[wikipedia:First-person_shooter|first person shooters]] (FPS) esports competitors can often benefit from AR by using it to create a visual representation of the map to plan out strategies before tournaments or scrims.<ref>{{Cite web|url=https://esportselite.co.uk/the-role-of-ar-in-esports-events/|title=The Role of AR in Esports Events|last=esportselite03|date=2025-10-09|website=Esports Elite|language=en-US|access-date=2026-03-30}}</ref> AI engines have been used in many games to quickly and concisely organize match data, playstyles of each team, and explain game terminology in simpler terms. This not only benefits the audience and casters who are providing live commentary, but allows the players to easily grasp what mistakes they made.<ref>{{Cite web|url=https://cacm.acm.org/news/how-ai-is-driving-the-esports-boom/|title=How AI Is Driving the Esports Boom – Communications of the ACM|date=2022-09-01|language=en-US|access-date=2026-03-30}}</ref> ==== Impact of Application Programming Interfaces towards Success ==== For esports teams especially, [[wikipedia:API|Application Programming Interfaces]] (APIs) are commonly used to prepare for matches. These tools include Shadow.gg, Mobalytics, and Overwolf that help players analyze match results and the best strategies to implement in the future. Using data gathered from previous matches, the San Francisco Shock [[wikipedia:Overwatch|Overwatch]] team were able to achieve victory in major champion leagues for two consecutive years (2019-2020). Similarly, in [[wikipedia:Dota_2|Defense of the Ancients 2]] (DOTA 2), Team Liquid used APIs to gather information about their enemies and playstyles which contributed to them winning The International 2017 (one of the biggest DOTA tournaments).<ref>{{Cite web|url=https://www.statology.org/the-analytics-behind-esports-data-driven-strategies-in-competitive-gaming/|title=The Analytics Behind Esports: Data-Driven Strategies in Competitive Gaming|last=Chugani|first=Vinod|date=2024-09-26|website=Statology|language=en-US|access-date=2026-03-30}}</ref> == The Technological Landscape of Streaming & Content Creation == === Streaming & Content Creation - Overview === Gamers are known for streaming or uploading videos about the games they are playing while providing live commentary. [https://www.twitch.tv/ Twitch] and [https://www.youtube.com/ YouTube] are popular streaming platforms that are used to broadcast live esports matches to audiences worldwide due to their advanced technical capabilities.<ref name=":0" /> Video game publishers often allow streaming platforms rights to broadcast their games as a way to get people interested in playing said game. As a result, the video game publishers garner more popularity and strategies to improve their game while the platform rakes in money. With major tech companies starting to focus on esports, streaming company subsidiaries as a result have a lot of influence when it comes to gaining rights to broadcast games.<ref>{{Cite web|url=https://www.polytechnique-insights.com/en/columns/digital/esports-everything-you-need-to-know-about-this-exploding-digital-market/|title=Esports: everything you need to know about this exploding digital market|last=Simon|first=Pierre-Jean Benghozi, Jean-Paul|date=2025-10-01|website=Polytechnique Insights|language=en-GB|access-date=2026-03-30}}</ref> ==== Streaming Platforms & Their Impact in Esports ==== Popular streaming platforms and their popular features include: * [https://www.twitch.tv/ Twitch]: Has a live chat feature to talk with the audience, channel subscriptions with different tiers, and [https://www.twitch.tv/bits Bits].<ref>{{Cite web|url=https://streamworks.ae/article/esports-streaming|title=Esports Streaming: The Ultimate Guide to Captivating Gaming Audiences|website=streamworks.ae|language=en|access-date=2026-03-30}}</ref> * [https://www.youtube.com/ YouTube]: Besides the live chat feature, it offers channel memberships which unlock “members-only” accessible content from videos, emotes, posts, and more.<ref>{{Cite web|url=https://www.youtube.com/join/|title=Channel memberships|website=Channel memberships|language=en|access-date=2026-03-30}}</ref> * [https://www.facebook.com/gaming/ Facebook Gaming]: Alongside the live chat feature, it has subscriptions, [https://www.facebook.com/fbgaminghome/creators/bootcamp/monetization/earn-stars Stars], and utilizes cloud gaming.<ref>{{Cite web|url=https://restream.io/learn/what-is/facebook-gaming/|title=What Is Facebook Gaming? {{!}} Restream Learn|date=2024-03-28|website=Restream {{!}} Learn|language=en|access-date=2026-03-30}}</ref> Besides these platforms, there are DouYu and Huya being used in China. Anyone is freely allowed to utilize these streaming platforms without cost since these platforms make money through partnerships! As a result, there are influencers who play the same games as popular esports streamers that help grow the popularity despite not competing.<ref name=":1" /> Since these platforms offer the ability to co-stream with other streamers and host watch parties, influencers use this feature as a way to collaborate or commentate on live games using their own experience. More than 50% of the viewership that results from watching esports tournaments can be attributed to co-streaming.<ref>{{Cite web|url=http://outof.games/news/8380-your-guide-to-esports-streaming-platforms/|title=Your Guide to Esports Streaming Platforms|website=Out of Games|language=en|access-date=2026-03-30}}</ref> Towards the start of esports, streaming platforms often enjoyed partnerships from technological brands such as ASUS, Intel, Nvidia, Razer, and Logitech that were exposed to the esports industry. However, the momentum of esports has influenced non-technological brands like Adidas, Cola, KIA, KitKat, and more to get involved. These brands often plug their merchandise during live tournaments by players consuming their products (such as food or drinks) and cameras panning to sponsored technology.<ref name=":1" /> ==== The Players & Future Successors ==== [[File:OpTic Gaming win Valorant Masters Reykjavík 2022.jpg|alt=The esports team "OpTic Gaming" is shown holding the victory trophy up together with smiles on their faces at the Valorant Masters 2022 Championship in Reyjavík.|thumb|317x317px|Esports team "OpTic Gaming" wins Valorant Masters 2022 in Reyjavík.]] As a result of streaming, the audience is able to learn more about a respective team and their players. Streaming allows the audience to discover a player’s individual talent, personality, and a potential role model that would otherwise go unnoticed in a team setting. While some players became famous as a result of their competitive talent, others build their popularity from the content they stream or upload. This can also be applied to upcoming talents that haven’t made a name for themselves to eventually gain enough recognition to be recruited for an esports team. Having an online audience (or fanbase) that supports the team or player(s) contributes to their popularity which in turn makes their name more known to partnerships down the line.<ref>{{Cite web|url=https://www.higglo.io/post/how-streaming-platforms-drive-esports-growth|title=How Streaming Platforms Fuel Esports Growth|last=Higglo|date=2025-10-22|website=Higglo|language=en|access-date=2026-03-30}}</ref> ==References== 5x4o6wxeop8zu99dd3dgigyobj79p8v 0 328732 2801431 2026-03-30T05:23:48Z SwarG07 3062365 Created page with " == Overview == Kanye West (legally known as Ye) is widely known as a musician, but his work in the 2010s and 2020s has pushed the boundaries of digital media. Unlike traditional artists who release a "final" version of an album, West has introduced the concept of the "Living Album"—a digital product that is constantly updated, edited, and changed even after it has been released to the public. == The "Living Album" Concept == The best example of this digital media shi..." 2801431 wikitext text/x-wiki == Overview == Kanye West (legally known as Ye) is widely known as a musician, but his work in the 2010s and 2020s has pushed the boundaries of digital media. Unlike traditional artists who release a "final" version of an album, West has introduced the concept of the "Living Album"—a digital product that is constantly updated, edited, and changed even after it has been released to the public. == The "Living Album" Concept == The best example of this digital media shift was the 2016 release of ''The Life of Pablo''. After the album was put on streaming services, West continued to change lyrics, adjust vocal mixes, and even add entire songs to the tracklist. In a digital media context, this treats music more like **software** than a traditional physical record. Just as a phone app receives "Version 1.1" or "Version 2.0" updates to fix bugs or add features, West used the flexibility of digital streaming to treat his art as an ongoing project. This challenges the idea of what a "finished" piece of art actually is in the digital age. == The Stem Player and Distribution == In 2021, West moved further into digital technology by releasing the **Stem Player**. This was a small, puck-shaped hardware device that allowed users to: * **Isolate Tracks:** Users could turn off the drums, vocals, or bass with physical sliders. * **Remix in Real-Time:** It gave the consumer the power to change how the media sounded, which is a form of interactive digital art. * **Bypass Platforms:** By only releasing the album ''Donda 2'' on this device, he attempted to create a "closed loop" digital ecosystem, moving away from big tech companies like Apple and Spotify. == Digital Rollouts as Performance Art == West also uses digital livestreams (on platforms like Apple Music and Amazon Prime) to turn the "uploading" of an album into a global event. These livestreams often show him finishing the songs in real-time. For a student of digital media, this is interesting because it shows how the **process** of creating digital content can become the content itself == References == * Hess, M. (2018). ''Icons of Hip Hop''. Greenwood Press. ** This book provides a historical look at Kanye West’s career. I used it to understand how his early work set the stage for his later experiments with digital technology and media. * Kano Computing. (2021). ''Stem Player: A New Way to Experience Music''. ** This is the technical guide for the Stem Player hardware. It helped me explain the specific digital features of the device, like how it isolates vocals and drums in real-time. * Digital Media Weekly. (2016). ''How Streaming Changed the Concept of a 'Finished' Album''. ** This article analyzes the shift from physical CDs to streaming. It was the main source for my section on the "Living Album" and how digital files can be updated after they are released. hpsz68hhzina03q6etzuk7o89nmpdxx 2 328733 2801448 2026-03-30T06:12:19Z Senad Dizdarević 3062367 I added the About Me page, which includes my bio, research interests, current projects, publications, and Wikiversity goals. 2801448 wikitext text/x-wiki [https://en.wikipedia.org/wiki/Personal_development] [https://en.wikipedia.org/wiki/Self-realization] 9nxumoeyb6qqlds2ixhp807nx48dpit 2801454 2801448 2026-03-30T06:21:10Z Senad Dizdarević 3062367 I added the About Me page, which includes my bio, research interests, current projects, publications, and Wikiversity goals. 2801454 wikitext text/x-wiki == About Me == I am a Slovenian journalist, author of 12 books for personal development, including two book series, '''''It’s Finally PROVEN! God Does NOT Exist The FIRST Valid EVIDENCE in History''', and '''Letters to Palkies Messages to My Friends on Another Planet''''', and independent digital creator working globally across multiple disciplines. My work focuses on identity restructuring, experiential empiricism, rational philosophy, and the development of educational frameworks that support personal and societal transformation. I am the creator of the '''AIPA Method (Awakening Into Pure Awareness)''', a structured approach to awareness‑driven identity reconstruction. == Research Interests == * Awareness-driven identity restructuring * Experiential empiricism * Philosophy of consciousness * Faith deconstruction and worldview transitions * Rational, non-theistic philosophy * Psychological resilience and existential transformation * Educational design and structured learning systems * Digital publishing and knowledge architecture == Current Projects == === AIPA Method (Awakening Into Pure Awareness) === A comprehensive framework for understanding and guiding identity transitions through structured awareness practices. The project includes: * theoretical foundations * empirical hypotheses * applied modules * case-based analysis * educational materials * visual models and diagrams === Experiential Empiricism === A methodological proposal that reframes empirical investigation as structured pattern analysis within experience, aiming to dissolve classical philosophical problems such as the hard problem of consciousness, the explanatory gap, and the mind–body divide. === Faith Deconstruction Application === An applied extension of the AIPA Method focused on belief revision, worldview restructuring, and the psychological processes involved in transitioning out of inherited or dogmatic belief systems. == '''Authority Identifiers''' == * '''ORCID:''' 0009-0008-9369-2734 * '''ISNI:''' 0000 0005 3005 8622, https://isni.org/isni/0000000530058622 * '''VIAF:''' 97154440103035341417 (Personal), Permalink: http://viaf.org/viaf/97154440103035341417 * '''WORLDCAT:''' https://search.worldcat.org/search?q=Senad+Dizdarevi%C4%87&limit=10&offset=1 == Wikidata: == '''Author: https://www.wikidata.org/wiki/Q138599746''' '''Book Series: Letters to Palkies:''' <code>https://www.wikidata.org/wiki/Q138600860</code> '''Book Series: God does not exist:''' <code>https://www.wikidata.org/wiki/Q138600298</code> '''AIPA Method (CreativeWork) -''' <code>https://www.wikidata.org/wiki/Q138601057</code> == Publications == A selection of my work is available on: * '''Zenodo:''' '''https://doi.org/10.5281/zenodo.19155200''' * '''PhilPapers:''' '''https://philpapers.org/archive/DIZAMA.pdf''' * '''OSF:''' '''https://osf.io/qgxmw/wiki?wiki=qp4tw''' * '''HAL:''' '''https://hal.science/user/index''' * '''Personal''' '''websites: https://god-doesntexist.com/ and https://www.letterstopalkies.com/''' '''Recent publication:''' '''Dizdarević vs Dawkins Atheism Comparison: Revolutionary Evidence That Changes Everything About God’s Existence''' Zenodo: https://zenodo.org/records/18666530 == Wikiversity Goals == On Wikiversity, I aim to: * develop open educational resources * publish structured modules of the AIPA Method * contribute to discussions on consciousness, identity, and experiential research * create accessible learning pathways for global audiences * support the Wikimedia mission of open knowledge == Keywords: == Faith deconstruction, religious trauma, Religious Trauma Syndrome (RTS), leaving religion, abandoning belief, Pure Awareness, identity reconstruction, cognitive-phenomenological model, autoethnography, mind-stopping, partial personalities, awakening, self-realization, consciousness, AIPA method, personal development, mindfulness comparison, addiction recovery, stress management, burnout, complex PTSD, digital overload, social media addiction, relationship development, leadership psychology, adolescent development, loneliness, prisoner rehabilitation. == Contact == For professional communication, please refer to my external profiles and official website. [https://en.wikipedia.org/wiki/Personal_development] [https://en.wikipedia.org/wiki/Self-realization] b3gv6lmx826rp1f3nv98wtphvggmmll 3 328734 2801482 2026-03-30T07:24:10Z Juandev 2651 Welcome template 2801482 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Fundingpipscouponcode!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Juandev|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. 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See you around Wikiversity! --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:24, 30 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} du7ru4vqqcb0322vj51aqw6tsxf3d0y 3 328735 2801492 2026-03-30T07:31:01Z Juandev 2651 Welcome template 2801492 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Senad Dizdarević!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Juandev|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. 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See you around Wikiversity! --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:33, 30 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 71ykocvhnejuqpp5bxc2pfena2z73cv 2 328737 2801495 2026-03-30T07:51:51Z ~2026-18400-87 3061118 Created page with "<syntaxhighlight lang="wikitext"> <div style="float: right; width: 300px; border: 1px solid #aaa; background: #f9f9f9; margin: 0 0 1em 1em; padding: 5px;"> <div style="background: #228b22; color: white; text-align: center; font-weight: bold; padding: 8px;"> Subject Title </div> <div style="text-align: center; padding: 10px; background: white;"> [[File:Placeholder.png|250px]] </div> <table style="width: 100%; font-size: 0.9em; border-collapse:..." 2801495 wikitext text/x-wiki <syntaxhighlight lang="wikitext"> <div style="float: right; width: 300px; border: 1px solid #aaa; background: #f9f9f9; margin: 0 0 1em 1em; padding: 5px;"> <div style="background: #228b22; color: white; text-align: center; font-weight: bold; padding: 8px;"> Subject Title </div> <div style="text-align: center; padding: 10px; background: white;"> [[File:Placeholder.png|250px]] </div> <table style="width: 100%; font-size: 0.9em; border-collapse: collapse;"> <tr style="border-bottom: 1px solid #ddd;"> <th style="text-align: left; padding: 5px;">Category</th> <td style="padding: 5px;">Hil</td> </tr> </table> </div> </syntaxhighlight> To: <div style="float: right; width: 300px; border: 1px solid #aaa; background: #f9f9f9; margin: 0 0 1em 1em; padding: 5px;"> <div style="background: #228b22; color: white; text-align: center; font-weight: bold; padding: 8px;"> Subject Title </div> <div style="text-align: center; padding: 10px; background: white;"> [[File:Placeholder.png|250px]] </div> <table style="width: 100%; font-size: 0.9em; border-collapse: collapse;"> <tr style="border-bottom: 1px solid #ddd;"> <th style="text-align: left; padding: 5px;">Category</th> <td style="padding: 5px;">Hil</td> </tr> </table> </div> hq3sq9c2ut3o1fas9vg586i5hgumyyx 0 328739 2801549 2026-03-30T10:14:39Z Strivenword 3045506 Created page with "{{:DesignWriteStudio/SiteElements/Navbox}} = Sovereign States of South America: Population and Area = This page presents a structured overview of sovereign states in South America using data retrieved from Wikidata via SPARQL queries.<ref>[https://query.wikidata.org/ Wikidata Query Service]</ref> == Data Table == {| class="wikitable sortable" |- ! Country !! Population !! Area (km²) !! Density (per km²) |- | Brazil || 213421037 || 8515767 || {{#expr: 213421037 / 85..." 2801549 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} = Sovereign States of South America: Population and Area = This page presents a structured overview of sovereign states in South America using data retrieved from Wikidata via SPARQL queries.<ref>[https://query.wikidata.org/ Wikidata Query Service]</ref> == Data Table == {| class="wikitable sortable" |- ! Country !! Population !! Area (km²) !! Density (per km²) |- | Brazil || 213421037 || 8515767 || {{#expr: 213421037 / 8515767 round 1}} |- | Colombia || 51874024 || 1141748 || {{#expr: 51874024 / 1141748 round 1}} |- | Argentina || 45195777 || 2780400 || {{#expr: 45195777 / 2780400 round 1}} |- | Peru || 33050325 || 1285216 || {{#expr: 33050325 / 1285216 round 1}} |- | Venezuela || 28435940 || 916445 || {{#expr: 28435940 / 916445 round 1}} |- | Chile || 19603733 || 756102 || {{#expr: 19603733 / 756102 round 1}} |- | Ecuador || 17643060 || 283561 || {{#expr: 17643060 / 283561 round 1}} |- | Bolivia || 11731281 || 1098581 || {{#expr: 11731281 / 1098581 round 1}} |- | Paraguay || 7132530 || 406752 || {{#expr: 7132530 / 406752 round 1}} |- | Uruguay || 3485151 || 176215 || {{#expr: 3485151 / 176215 round 1}} |- | Guyana || 804567 || 214969 || {{#expr: 804567 / 214969 round 1}} |- | Suriname || 612985 || 163821 || {{#expr: 612985 / 163821 round 1}} |} == Top 5 Countries by Population == # Brazil # Colombia # Argentina # Peru # Venezuela == Top 5 Countries by Area == # Brazil # Argentina # Peru # Colombia # Bolivia == Comparative Analysis == The two rankings reveal both overlap and divergence in the spatial and demographic structure of South America. Brazil appears at the top of both lists, indicating that it is simultaneously the largest territorial state and the most populous. This suggests a central structural role in the continent’s geography and human distribution.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP1082%20%3Fpopulation%20.%20%3Fcountry%20wdt%3AP2046%20%3Farea%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America with population and area]</ref> Argentina and Peru appear in both lists, though their rankings differ. Argentina ranks higher in area than in population, indicating relatively lower population density. Peru shows a similar pattern, though less extreme. Colombia ranks second in population but fourth in area, indicating a comparatively higher population density and more concentrated settlement patterns. Venezuela appears in the population top five but not in the area top five, reinforcing the pattern of higher density relative to its size. Bolivia appears in the area top five but not in population, indicating a lower density and a more spatially extensive but less populated territory.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Farea%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP2046%20%3Farea%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America by area]</ref> Suriname, the smallest sovereign state in South America by both population and area in this dataset, provides a useful contrast. Its population density is low relative to more densely settled countries such as Colombia and Ecuador, despite its much smaller total area. This indicates that its limited population is distributed across its territory in a relatively sparse manner. Compared to larger countries like Brazil or Argentina, which combine large territories with substantial populations, Suriname’s case illustrates that smaller size does not necessarily correspond to higher density.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP1082%20%3Fpopulation%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America by population]</ref> Taken together, these comparisons illustrate that territorial size and population are related but not determinative. Population density serves as the mediating factor that clarifies how human settlement is distributed across space. == Data Quality == The data is sourced from Wikidata, a collaboratively edited knowledge graph. Several considerations apply: * Population values may represent estimates from different years and are not guaranteed to be temporally aligned. * Area values are generally more stable but may vary depending on measurement standards. * The classification "sovereign state" depends on Wikidata’s ontology and may exclude disputed or partially recognized entities. * As with any open dataset, entries may be incomplete or subject to revision. Because the data is retrieved through queries, it reflects a specific snapshot of Wikidata at the time of access. == References == {{Reflist}} {{:DesignWriteStudio/SiteElements/Footer}} 3962d04k0hysjvq7n6zoo9tezxvjry4 2801550 2801549 2026-03-30T10:16:03Z Strivenword 3045506 Deleted the Data Quality section, which seems to be inappropriate to the assignment. 2801550 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} = Sovereign States of South America: Population and Area = This page presents a structured overview of sovereign states in South America using data retrieved from Wikidata via SPARQL queries.<ref>[https://query.wikidata.org/ Wikidata Query Service]</ref> == Data Table == {| class="wikitable sortable" |- ! Country !! Population !! Area (km²) !! Density (per km²) |- | Brazil || 213421037 || 8515767 || {{#expr: 213421037 / 8515767 round 1}} |- | Colombia || 51874024 || 1141748 || {{#expr: 51874024 / 1141748 round 1}} |- | Argentina || 45195777 || 2780400 || {{#expr: 45195777 / 2780400 round 1}} |- | Peru || 33050325 || 1285216 || {{#expr: 33050325 / 1285216 round 1}} |- | Venezuela || 28435940 || 916445 || {{#expr: 28435940 / 916445 round 1}} |- | Chile || 19603733 || 756102 || {{#expr: 19603733 / 756102 round 1}} |- | Ecuador || 17643060 || 283561 || {{#expr: 17643060 / 283561 round 1}} |- | Bolivia || 11731281 || 1098581 || {{#expr: 11731281 / 1098581 round 1}} |- | Paraguay || 7132530 || 406752 || {{#expr: 7132530 / 406752 round 1}} |- | Uruguay || 3485151 || 176215 || {{#expr: 3485151 / 176215 round 1}} |- | Guyana || 804567 || 214969 || {{#expr: 804567 / 214969 round 1}} |- | Suriname || 612985 || 163821 || {{#expr: 612985 / 163821 round 1}} |} == Top 5 Countries by Population == # Brazil # Colombia # Argentina # Peru # Venezuela == Top 5 Countries by Area == # Brazil # Argentina # Peru # Colombia # Bolivia == Comparative Analysis == The two rankings reveal both overlap and divergence in the spatial and demographic structure of South America. Brazil appears at the top of both lists, indicating that it is simultaneously the largest territorial state and the most populous. This suggests a central structural role in the continent’s geography and human distribution.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP1082%20%3Fpopulation%20.%20%3Fcountry%20wdt%3AP2046%20%3Farea%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America with population and area]</ref> Argentina and Peru appear in both lists, though their rankings differ. Argentina ranks higher in area than in population, indicating relatively lower population density. Peru shows a similar pattern, though less extreme. Colombia ranks second in population but fourth in area, indicating a comparatively higher population density and more concentrated settlement patterns. Venezuela appears in the population top five but not in the area top five, reinforcing the pattern of higher density relative to its size. Bolivia appears in the area top five but not in population, indicating a lower density and a more spatially extensive but less populated territory.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Farea%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP2046%20%3Farea%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America by area]</ref> Suriname, the smallest sovereign state in South America by both population and area in this dataset, provides a useful contrast. Its population density is low relative to more densely settled countries such as Colombia and Ecuador, despite its much smaller total area. This indicates that its limited population is distributed across its territory in a relatively sparse manner. Compared to larger countries like Brazil or Argentina, which combine large territories with substantial populations, Suriname’s case illustrates that smaller size does not necessarily correspond to higher density.<ref>[https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20WHERE%20%7B%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%20.%20%3Fcountry%20wdt%3AP30%20wd%3AQ18%20.%20%3Fcountry%20wdt%3AP1082%20%3Fpopulation%20.%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%20%7D SPARQL query: sovereign states in South America by population]</ref> Taken together, these comparisons illustrate that territorial size and population are related but not determinative. Population density serves as the mediating factor that clarifies how human settlement is distributed across space. == References == {{Reflist}} {{:DesignWriteStudio/SiteElements/Footer}} f0x2acr0tu3at1gek4l0gtrf4ow1nqe