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Difference between revisions of "Möbius strip"

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A non-orientable surface with [[Euler characteristic|Euler characteristic]] zero whose boundary is a closed curve. The Möbius strip can be obtained by identifying two opposite sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643102.png" /> of a rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643103.png" /> so that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643105.png" /> are matched with the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643107.png" />, respectively (see Fig.).
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A non-orientable surface with [[Euler characteristic|Euler characteristic]] zero whose boundary is a closed curve. The Möbius strip can be obtained by identifying two opposite sides $AB$ and $CD$ of a rectangle $ABCD$ so that the points $A$ and $B$ are matched with the points $C$ and $D$, respectively (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m064310a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m064310a.gif" />
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Figure: m064310a
 
Figure: m064310a
  
In the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064310/m0643108.png" /> the Möbius strip is a one-sided surface (see [[One-sided and two-sided surfaces|One-sided and two-sided surfaces]]).
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In the Euclidean space $E^3$ the Möbius strip is a one-sided surface (see [[One-sided and two-sided surfaces|One-sided and two-sided surfaces]]).
  
 
The Möbius strip was considered (in 1858–1865) independently by A. Möbius and I. Listing.
 
The Möbius strip was considered (in 1858–1865) independently by A. Möbius and I. Listing.

Latest revision as of 20:10, 11 April 2014

A non-orientable surface with Euler characteristic zero whose boundary is a closed curve. The Möbius strip can be obtained by identifying two opposite sides $AB$ and $CD$ of a rectangle $ABCD$ so that the points $A$ and $B$ are matched with the points $C$ and $D$, respectively (see Fig.).

Figure: m064310a

In the Euclidean space $E^3$ the Möbius strip is a one-sided surface (see One-sided and two-sided surfaces).

The Möbius strip was considered (in 1858–1865) independently by A. Möbius and I. Listing.

How to Cite This Entry:
Möbius strip. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_strip&oldid=23422
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article