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Möbius strip (Definition)

A Möbius strip is a non-orientiable 2-dimensional surface with a 1-dimensional boundary. It can be embedded in $ \mathbb{R}^3$, but only has a single side.

We can parameterize the Möbius strip by

$\displaystyle x = r \cdot \cos{\theta}, \quad y = r \cdot \sin{\theta}, \quad z = (r-2)\tan{\frac{\theta}{2}}. $
The Möbius strip is therefore a subset of the solid torus.

Topologically, the Möbius strip is formed by taking a quotient space of $ I^2 = [0,1] \times [0,1] \subset \mathbb{R}^2$. We do this by first letting $ M$ be the partition of $ I^2$ formed by the equivalence relation:

$\displaystyle (1,x) \sim (0,1-x)$   where$\displaystyle \quad 0 \leq x \leq 1,$
and every other point in $ I^2$ is only related to itself.

By giving $ M$ the quotient topology given by the quotient map $ p: I^2 \to M$ we obtain the Möbius strip.

Schematically we can represent this identification as follows:

\includegraphics[scale=0.5]{mobius-2}
Diagram 1: The identifications made on $ I^2$ to make a Möbius strip.
We identify two opposite sides but with different orientations.

Since the Möbius strip is homotopy equivalent to a circle, it has $ \mathbb{Z}$ as its fundamental group. It is not however, homeomorphic to the circle, although its boundary is.




"Möbius strip" is owned by Mathprof. [ full author list (3) | owner history (3) ]
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See Also: Klein bottle, torus

Other names:  Möbius band

Attachments:
feature of Mobius strip (Feature) by juanman
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Cross-references: homeomorphic, fundamental group, circle, homotopy equivalent, orientations, opposite sides, represent, quotient map, quotient topology, point, equivalence relation, partition, quotient space, torus, solid, subset, boundary, surface
There are 9 references to this entry.

This is version 19 of Möbius strip, born on 2002-08-08, modified 2006-10-02.
Object id is 3278, canonical name is MobiusStrip.
Accessed 12275 times total.

Classification:
AMS MSC54B15 (General topology :: Basic constructions :: Quotient spaces, decompositions)

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