# File:KleinBottle-01.png

KleinBottle-01.png(240 × 300 pixels, file size: 64 KB, MIME type: image/png)

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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## 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 ${\displaystyle 0\leq u<\pi }$:

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

For ${\displaystyle \pi \leq u<2\pi }$:

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

## 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}]


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## File history

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

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