Image:KleinBottle-Figure8-01.png

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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
Source

Own drawing, Mathematica 5.1
Date

05/08/06
Author

Fropuff, Inductiveload
Permission
(Reusing this image)

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

Public domain This image 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.

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

Public domain This image 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.

العربية | Български | Deutsch | Ελληνικά | English | Español | Français | Italiano | 日本語 | 한국어 | Polski | Русский | Српски / Srpski | Svenska | தமிழ் | Українська | ‪中文(繁體)‬ | ‪中文(简体)‬  +/-
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.

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

[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.

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeDimensionsUserComment
current06:09, 16 February 2007560×400 (149 KB)Inductiveload
22:55, 4 March 2005300×226 (75 KB)Dbenbenn (losslessly compressed with pngcrush, 18% smaller)
17:45, 3 March 2005300×226 (91 KB)Fropuff (Figure-eight immersion of a Klein bottle. {{PD}})
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