File:KleinBottle-Figure8-01.png
KleinBottle-Figure8-01.png (560 × 400 pixels, file size: 149 KB, MIME type: image/png)
Captions
Summary[edit]
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 |
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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:
The derived, redrawn, edited image was released into the public domain by Inductiveload:
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Other versions | For a cut-away version see Image:KleinBottle-Figure8-02.png. |
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Parameterization[edit]
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.
Mathematica source[edit]
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.
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 06:09, 16 February 2007 | 560 × 400 (149 KB) | Inductiveload (talk | contribs) | ||
22:55, 4 March 2005 | 300 × 226 (75 KB) | Dbenbenn (talk | contribs) | losslessly compressed with pngcrush, 18% smaller | ||
17:45, 3 March 2005 | 300 × 226 (91 KB) | Fropuff~commonswiki (talk | contribs) | Figure-eight immersion of a Klein bottle. {{PD}} |
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