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

Ofbyld:KleinBottle-01.svg is in fektorferzje fan dizze triem. As net fan sljochtere kwaliteit tsjinnet dizze brûkt te wurden yn plak fan dizze rasterôfbylding. File:KleinBottle-01.png → File:KleinBottle-01.svg Sjoch Help:SVG foar mear ynformaasje. In other languages ([//commons.wikimedia.org/w/index.php?title=Template:In_other_languages&action=edit translate this]) Alemannisch ∙ العربية ∙ беларуская (тарашкевіца) ∙ български ∙ বাংলা ∙ català ∙ нохчийн ∙ čeština ∙ dansk ∙ Deutsch ∙ Ελληνικά ∙ English ∙ British English ∙ Esperanto ∙ español ∙ eesti ∙ euskara ∙ فارسی ∙ suomi ∙ français ∙ Frysk ∙ galego ∙ Alemannisch ∙ עברית ∙ हिन्दी ∙ hrvatski ∙ magyar ∙ հայերեն ∙ Bahasa Indonesia ∙ Ido ∙ italiano ∙ 日本語 ∙ ქართული ∙ 한국어 ∙ lietuvių ∙ македонски ∙ മലയാളം ∙ Bahasa Melayu ∙ မြန်မာဘာသာ ∙ norsk bokmål ∙ Plattdüütsch ∙ Nederlands ∙ norsk nynorsk ∙ norsk ∙ occitan ∙ polski ∙ prūsiskan ∙ português ∙ português do Brasil ∙ română ∙ русский ∙ sicilianu ∙ Scots ∙ slovenčina ∙ slovenščina ∙ српски / srpski ∙ svenska ∙ தமிழ் ∙ ไทย ∙ Türkçe ∙ татарча / tatarça ∙ українська ∙ vèneto ∙ Tiếng Việt ∙ 中文 ∙ 中文（中国大陆） ∙ 中文（简体） ∙ 中文（繁體） ∙ 中文（马来西亚） ∙ 中文（新加坡） ∙ 中文（臺灣） ∙ +/−

 

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

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

${\displaystyle y=16\sin u\,}$

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

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}]