Möbius strip

A Möbius strip is a non-orientiable 2-dimensional surface with a 1-dimensional boundary. It can be embedded in 3, but only has a single .

We can parameterize the Möbius strip by


The Möbius strip is therefore a subset of the solid torus.

Topologically, the Möbius strip is formed by taking a quotient spaceMathworldPlanetmath of I2=[0,1]×[0,1]2. We do this by first letting M be the partition of I2 formed by the equivalence relationMathworldPlanetmath:


and every other point in I2 is only related to itself.

By giving M the quotient topology given by the quotient map p:I2M we obtain the Möbius strip.

Schematically we can represent this identification as follows:

Diagram 1: The identifications made on I2 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  as its fundamental groupPlanetmathPlanetmath. It is not however, homeomorphicMathworldPlanetmath to the circle, although its boundary is.

Title Möbius strip
Canonical name MobiusStrip
Date of creation 2013-03-22 12:55:28
Last modified on 2013-03-22 12:55:28
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 22
Author Mathprof (13753)
Entry type Definition
Classification msc 54B15
Synonym Möbius band
Related topic KleinBottle
Related topic Torus