Klein Bottle

Differential Equations, Mechanics, and Computation
klein bottle
Rotate Me

Here is a parametric formula for the Klein bottle.

aa=3;s=1;
ParametricPlot3D[
  s*{(aa+Cos[v/2]*Sin[u]-Sin[v/2]*Sin[2*u])*Cos[v],
     (aa+Cos[v/2]*Sin[u]-Sin[v/2]*Sin[2*u])*Sin[v],
      Sin[v/2]*Sin[u]+Cos[v/2]*Sin[2*u]},
      {u,0,2*Pi},{v,0,6}]

The aa constant controls the overall radius of the torus shape. If aa is large, the object will be like a thin tube. The s is the scaling factor for the whole thing. u is the sweep of the figure-8 profile. 0 to 2 π. v is the sweep of the torus tube. 0 to 2 π.

twisted-figure-8.nb.zip

klein_bottle.gcf.

Klein Bottle.

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