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This image was selected as picture of the month on the Mathematics Portal for November 2006 
czech:Kleinova láhev je těleso,ve kterém nelze přejít přes okraj. Technicky vzato má jen jednu stranu. V knize Hravá matematika od Radka Chajdy jsem našel otázku: lze do Kleinovy láhve něco nalít? Ano lze do ní něco nalít a ještě není potřeba víčko.
Lukáš HOZDA 1.11.2009

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See also
Image:Klein bottle.svg
Licensing
Public domainPublic domainfalsefalse 

I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse

Parameterization
This immersion of the Klein bottle into R^{3} 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 :
For :
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}]
File usage
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