Keywords: KleinBottle-Figure8-01.png Figure-eight immersion of a Klein bottle into R<sup>3</sup> Made with Mathematica The figure 8 immersion of the Klein bottle L'immersione a figura 8 della bottiglia di Klein Реализация бутылки Клейна в виде восьме ки Own drawing Mathematica 5 1 05/08/06 Fropuff Inductiveload The original image was released into the public domain by Fropuff Fropuff The derived redrawn edited image was released into the public domain by Inductiveload Inductiveload For a cut-away version see Image KleinBottle-Figure8-02 png KleinBottle-Figure8-01 svg Parameterization This immersion of the Klein bottle into R<sup>3</sup> is given by the following parameterization Here the parameters u and v run from 0 to 2 and r is some fixed positive constant <math>x \left r + \cos\frac u 2 \sin v - \sin\frac u 2 \sin 2v\right \cos u</math> <math>y \left r + \cos\frac u 2 \sin v - \sin\frac u 2 \sin 2v\right \sin u</math> <math>z \sin\frac u 2 \sin v + \cos\frac u 2 \sin 2v</math> Mathematica source Klein8r_ 2 Function u v r + Cosu/2Sinv - Sinu/2Sin2v Cosu r + Cosu/2Sinv - Sinu/2Sin2v Sinu Sinu/2Sinv + Cosu/2Sin2v ParametricPlot3DEvaluateKlein8u v u 0 2Pi v 0 2Pi PlotPoints -> 60 Boxed -> False Axes -> False ImageSize -> 800 This image was then antialised with http //members wri com/jeffb/visualization/aa2 shtml Chris Hill's code made transparent around the surface and had stray pixels removed in an image editor Klein bottle Images with Mathematica source code |