Image:KleinBottle-Figure8-01.png

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

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.

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

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.

Other versions

For a cut-away version see Image:KleinBottle-Figure8-02.png.

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.

File history

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

	Date/Time	Dimensions	User	Comment
current	06:09, 16 February 2007	560×400 (149 KB)	Inductiveload
	22:55, 4 March 2005	300×226 (75 KB)	Dbenbenn	(losslessly compressed with pngcrush, 18% smaller)
	17:45, 3 March 2005	300×226 (91 KB)	Fropuff	(Figure-eight immersion of a Klein bottle. {{PD}})

File links

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

Klein bottle

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