Gearfetting

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 ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

 

This is a featured picture on the Hebrew language Wikipedia (תמונות מומלצות) and is considered one of the finest images. See its nomination here. If you think this file should be featured on Wikimedia Commons as well, feel free to nominate it. If you have an image of similar quality that can be published under a suitable copyright license, be sure to upload it, tag it, and nominate it.

See also

Image:Klein bottle.svg

Lisinsje

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.

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}]