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Contents

Summary

Standard immersion of a Klein bottle into R3. Made with Mathematica.

See also

Image:Klein bottle.svg

Licensing

Public domain I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide.

In case this is not legally possible:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.


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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 a fixed positive constant.

For 0 \leq u < \pi:

x = 6 \cos u(1 + \sin u) + 4r\left(1 - \frac{\cos u}{2}\right) \cos u \cos v
y = 16 \sin u + 4r\left(1 - \frac{\cos u}{2}\right) \sin u \cos v
z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v

For \pi \leq u < 2\pi:

x = 6 \cos u(1 + \sin u) - 4r\left(1 - \frac{\cos u}{2}\right) \cos v
y = 16 \sin u\,
z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v

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}]

The following pages on the English Wikipedia link to this file (pages on other projects are not listed):
  • Klein bottle
  • Partition of unity
  • Talk:Klein bottle
  • Euler characteristic
  • Hsiang-Lawson's conjecture
  • Meagre set
  • Fibre bundle
  • Zero set
  • Knot polynomial
  • Sigma-compact space
  • Spinor bundle
  • Affine representation
  • Submersion (mathematics)
  • Unicoherent
  • Continuum (mathematics)
  • F-sigma set
  • Toroid
  • Weak operator topology
  • Spectral space
  • Banach-Alaoglu theorem
  • Plus construction
  • Nerve of an open covering
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  • Von Neumann conjecture
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  • Lyusternik–Schnirelmann category
  • Lebesgue's number lemma
  • Bundle (mathematics)
  • Cocycle class
  • Upper topology
  • HOMFLY polynomial
  • Critical value
  • Whitney immersion theorem
  • Reidemeister move
  • Link (knot theory)
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  • Nagata–Smirnov metrization theorem
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  • Portal:Mathematics/Featured picture/2006 11