Image:KleinBottle-Figure8-01.png
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KleinBottle-Figure8-01.png (560 × 400 pixel, file size: 149 KB, MIME type: image/png)
| This is a file from the Wikimedia Commons. The description on its description page there is shown below. |
| 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
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| Source |
Own drawing, Mathematica 5.1 |
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| Date |
05/08/06 |
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| Permission |
The original image was released into the public domain by Fropuff:
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| Other versions | For a cut-away version see Image:KleinBottle-Figure8-02.png. |
[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 some fixed positive constant.
[edit] 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.
