Image:KleinBottle-01.png - Simple English Wikipedia

# Image:KleinBottle-01.png

### From Wikipedia, a free encyclopedia written in simple English for easy reading.

- Image
- File history
- Links

KleinBottle-01.png‎ (65KB, MIME type: `image/png`)

This is a file from the Wikimedia Commons, a repository of free content hosted by the Wikimedia Foundation. The description on its description page there and licensing information is visible below.

## [edit] Summary

Standard immersion of a Klein bottle into R3. Made with Mathematica.

## [edit] See also

Image:Klein bottle.svg

## [edit] Licensing

I, the author of this work, hereby release it into the public domain. This applies worldwide.

In case this is not legally possible:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Afrikaans | Alemannisch | Aragonés | العربية | Български | Català | Česky | Cymraeg | Dansk | Deutsch | Ελληνικά | English | Español | Esperanto | فارسی | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית | Latina | Lietuvių | Magyar | Bahasa Melayu | Nederlands | Norsk (bokmål) | Norsk (nynorsk) | 日本語 | Polski | Português | Ripoarish | Română | Русский | Slovenčina | Slovenščina | Српски | Svenska | ไทย | Türkçe | Українська | Tiếng Việt | Walon | 简体中文 | 繁體中文 | 粵語 | +/-

## [edit] Parameterization

This immersion of the Klein bottle into R3 is given by the following parameterization. Here the parameters u and v run from 0 to 2π and r is a fixed positive constant.

For :

For :

## [edit] Mathematica source

    KleinBottle[r_:1] =
     Function[{u, v},
       UnitStep[Sin[u]]
       {
           6 Cos[u](1 + Sin[u]) + 4r(1 - Cos[u]/2) Cos[u]Cos[v],
           16 Sin[u] + 4r(1 - Cos[u]/2) Sin[u]Cos[v],
           4r(1 - Cos[u]/2) Sin[v]
       }
       + (1 - UnitStep[Sin[u]])
       {
           6 Cos[u](1 + Sin[u]) - 4r(1 - Cos[u]/2) Cos[v],
           16 Sin[u],
           4r(1 - Cos[u]/2) Sin[v]
       }
     ]

     ParametricPlot3D[Evaluate[KleinBottle[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
       PlotPoints -> {50, 19}, Boxed -> False, Axes -> False,
       ViewPoint -> {0.454, -2.439, -2.301}]

## Links

The following pages link to this file:

- Klein bottle

Retrieved from "http://simple.wikipedia.org../../../k/l/e/Image%7EKleinBottle-01.png_3d68.html"

##### Views

- File
- Talk
- Newest version

##### Search