Imej:KleinBottle-Figure8-01.png
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Figure-eight immersion of a Klein bottle into R3. Made with Mathematica. For a cut-away version see Image:KleinBottle-Figure8-02.png.
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[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 -> 40, Boxed -> False, Axes -> False]
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