# File:KleinBottle-Figure8-01.png

KleinBottle-Figure8-01.png(560 × 400 pixels, file size: 149 KB, MIME type: image/png)

## Summary

Description Figure-eight immersion of a Klein bottle into R3. Made with Mathematica.
English: The "figure 8" immersion of the Klein bottle.
Italiano: L'immersione a "figura 8" della bottiglia di Klein.
Русский: Реализация бутылки Клейна в виде восьмерки
Date 05/08/06
Source Own drawing, Mathematica 5.1
Permission
( Reusing this file)

The original image was released into the public domain by Fropuff:

 This work has been released into the public domain by its author, Fropuff. This applies worldwide.In some countries this may not be legally possible; if so: Fropuff grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

The derived, redrawn, edited image was released into the public domain by Inductiveload:

 This work has been released into the public domain by its author, Inductiveload. This applies worldwide.In some countries this may not be legally possible; if so: Inductiveload grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
 File:KleinBottle-Figure8-01.svg is a vector version of this file. It should be used in place of this raster image when superior. File:KleinBottle-Figure8-01.png File:KleinBottle-Figure8-01.svg For more information about vector graphics, read about Commons transition to SVG. There is also information about MediaWiki's support of SVG images.

## 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 some fixed positive constant.

$x = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u$
$y = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u$
$z = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v$

## Mathematica source

Klein8[r_:2] =
Function[{u, v},
{
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Cos[u],
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Sin[u],
Sin[u/2]Sin[v] + Cos[u/2]Sin[2v]
}
]

ParametricPlot3D[Evaluate[Klein8[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
PlotPoints -> 60, Boxed -> False, Axes -> False, ImageSize -> 800]


This image was then antialised with Chris Hill's code, made transparent around the surface and had stray pixels removed in an image editor.

The following pages on Schools Wikipedia link to this image (list may be incomplete):