File:KleinBottle-01.png

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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

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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 :

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 history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current23:39, 12 December 2006Thumbnail for version as of 23:39, 12 December 2006240 × 300 (64 KB)Mahahahaneapneappngcrushed
10:21, 15 September 2006Thumbnail for version as of 10:21, 15 September 2006240 × 300 (77 KB)Dark knightAdded transparency, unfortunately dimension rose
02:23, 4 March 2005Thumbnail for version as of 02:23, 4 March 2005240 × 300 (57 KB)Dbenbennlosslessly compressed with pngcrush, 20% smaller
17:44, 3 March 2005Thumbnail for version as of 17:44, 3 March 2005240 × 300 (71 KB)Fropuff~commonswikiStandard immersion of a Klein bottle. {{PD}}

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