Summary

DescriptionKleinBottle-Figure8-01.png	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
Author	Fropuff, Inductiveload
Permission (Reusing this file)	The original image was released into the public domain by Fropuff: Public domainPublic domainfalsefalse 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: Public domainPublic domainfalsefalse 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.
Other versions	For a cut-away version see Image:KleinBottle-Figure8-02.png.

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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.

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

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

${\displaystyle z=\sin {\frac {u}{2}}\sin v+\cos {\frac {u}{2}}\sin 2v}$

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.