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This is a file from the Wikimedia Commons. The description on its description page there is shown below.Commons is a freely licensed media file repository. You can help.
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Description |
Figure-eight immersion of a Klein bottle into R3. Made with Mathematica. English: The "figure 8" immersion of the Klein bottle.
Italiano: L'immersione a "figura 8" della bottiglia di Klein.
Русский: Реализация бутылки Клейна в виде восьмерки
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Source |
Own drawing, Mathematica 5.1 |
Date |
05/08/06 |
Author |
Fropuff, Inductiveload |
Permission ( Reusing this image) |
The original image was released into the public domain by Fropuff:
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This image has been released into the public domain by its author, Fropuff. This applies worldwide.
In some countries this may not be legally possible; if so: Fropuff grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
العربية | Български | Deutsch | Ελληνικά | English | Español | Français | Italiano | 日本語 | 한국어 | Polski | Русский | Српски / Srpski | Svenska | தமிழ் | Українська | 中文(繁體) | 中文(简体) +/- |
The derived, redrawn, edited image was released into the public domain by Inductiveload:
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This image has been released into the public domain by its author, Inductiveload. This applies worldwide.
In some countries this may not be legally possible; if so: Inductiveload grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
العربية | Български | Deutsch | Ελληνικά | English | Español | Français | Italiano | 日本語 | 한국어 | Polski | Русский | Српски / Srpski | Svenska | தமிழ் | Українська | 中文(繁體) | 中文(简体) +/- |
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Other versions |
For a cut-away version see Image:KleinBottle-Figure8-02.png. |
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 some fixed positive constant.
Mathematica source
Klein8[r_:2] =
Function[{u, v},
{
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Cos[u],
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Sin[u],
Sin[u/2]Sin[v] + Cos[u/2]Sin[2v]
}
]
ParametricPlot3D[Evaluate[Klein8[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
PlotPoints -> 60, Boxed -> False, Axes -> False, ImageSize -> 800]
This image was then antialised with Chris Hill's code, made transparent around the surface and had stray pixels removed in an image editor.
File history
Click on a date/time to view the file as it appeared at that time.
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Date/Time |
Dimensions |
User |
Comment |
current |
06:09, 16 February 2007 |
560×400 (149 KB) |
Inductiveload |
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22:55, 4 March 2005 |
300×226 (75 KB) |
Dbenbenn |
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17:45, 3 March 2005 |
300×226 (91 KB) |
Fropuff |
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File links
The following pages on Schools Wikipedia link to this image (list may be incomplete):