Moebius Strip

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The Möbius strip or Möbius band (named after the German mathematician and astronomer August Ferdinand Möbius) is a topological object with only one surface and only one edge. It was co-discovered independently by Möbius and the German mathematician Johann Benedict Listing in 1858. A model can easily be created by taking a strip of material and giving it a half-twist, and then merging the ends of the strip together to form a single strip.

A Möbius strip made with a piece of paper and scotch tapeEnlarge

A Möbius strip made with a piece of paper and scotch tape

The Möbius strip has several curious properties. If you cut down the middle of the strip, instead of getting two separate strips, it becomes one long strip with two half-twists in it. If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along the strip, about a third of the way in from the edge, you will get two strips; one is a thinner Möbius strip, the other is a long strip with two half-twists in it. Other interesting combinations of strips can be obtained by making Möbius strips with two or more flips in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

The Möbius strip is a two-dimensional smooth manifold (a surface) which is not orientable.

A parametric graph of a Möbius stripEnlarge

A parametric graph of a Möbius strip

One way to represent the Möbius strip as a subset of R3 is using the parametrization:

This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the x-y plane and is centered at (0,0,0). The parameter u runs around the strip while v moves from one edge to the other.

(A similar parametrization was used to create the animation of the making of a Möbius strip that can be seen in the middle of the Visual Math page.)

In cylindrical polar coordinates (r,θ, z), an unbounded version of the Möbius strip can be represented by the equation:

Topologically, the Moebius strip can be defined as the square [0,1] × [0,1] with sides identified by the relation (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram: ----> | | | | <----

The Möbius strip has provided inspiration both for sculptures and for graphical art. Maurits C. Escher is one of the artists who was especially fond of it and based a great many of his lithographs on this mathematical object. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's The Wall of Darkness. Science fiction stories sometimes suggest that our universe might be some kind of generalised Möbius strip.

In the short story "A Subway Named Moebius", by A.J. Deutsch, the Boston subway authority builds a new line; the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear.

There have been technical applications; giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time).

Related objects

A closely related "strange" geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Möbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.

Another closely related manifold is the real projective plane. If a single hole is punctured in the real projective plane, what is left is a Möbius strip.

In terms of identifications of the sides of a square, as given above: the real projective plane is made by gluing the remaining two sides with 'consistent' orientation (arrows making an anti-clockwise loop); and the Klein bottle is made the other way.

A cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot be done in three dimensions without the surface intersecting itself.

The Möbius strip is a standard example used to illustrate the mathematical concept of a fiber bundle.