 # 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.

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Image:Klein bottle.svg

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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}]


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