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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]] 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]]).
  
 
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.
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Latest revision as of 08:20, 4 November 2023

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.


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How to Cite This Entry:
Möbius strip. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_strip&oldid=22819
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article