File:KleinBottle-01.png

Parameterization can be better found at en.wikipedia.org/wiki/File:KleinBottle-01.png as we currently lack the ability to write in mathematical script. This comes from their webpage on this subject.

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 2&pi; and r is a fixed positive constant.

Mathematica source
KleinBottle[r_:1] = Function[{u, v}, UnitStep[Sin[u]] {       6 Cos[u](1 + Sin[u]) + 4r(1 - Cos[u]/2) Cos[u]Cos[v], 16 Sin[u] + 4r(1 - Cos[u]/2) Sin[u]Cos[v], 4r(1 - Cos[u]/2) Sin[v] }   + (1 - UnitStep[Sin[u]]) {       6 Cos[u](1 + Sin[u]) - 4r(1 - Cos[u]/2) Cos[v], 16 Sin[u], 4r(1 - Cos[u]/2) Sin[v] } ]  ParametricPlot3D[Evaluate[KleinBottle[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi}, PlotPoints -> {50, 19}, Boxed -> False, Axes -> False, ViewPoint -> {0.454, -2.439, -2.301}]

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