Möbius strip

The Möbius strip or Möbius band is a topological object with only one surface and only one edge. It was co-discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. A model can easily be created by taking a paper strip and giving it a half-twist, and then merging the ends of the strip together to form a single strip. There are in fact two types of Möbius strip depending on the direction of the half-twist: if the right hand twists the right end of the strip in a clockwise manner, the result is a right-handed Möbius strip. The Möbius strip therefore exhibits chirality.

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 (not a Möbius strip). If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along a Möbius 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 (not a Möbius strip). 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 often cited as the inspiration for the infinity symbol $\infty$ , since if one were to stand on a the surface of a Möbius strip, one could walk along it forever. However, this may be apocryphal since the symbol had been in use to represent infinity even before the Möbius strip was discovered.

Geometry and topology

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

x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos(u)

y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin(u)

z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}

where $0\leq u<2\pi$ and $-1\leq v\leq 1$ . 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.

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

\log(r)\sin(\theta /2)=z\cos(\theta /2).

Topologically, the Möbius 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 is a two-dimensional smooth manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S¹ with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z₂) bundle over S¹.

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. Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle. Such a figure is called a cross-cap (a cross-cap can also mean this figure with the disk glued in, i.e. an immersion of the projective plane in R³).

It is a common misconception that a cross-cap cannot be formed in three dimensions without the surface intersecting itself. In fact it is possible to embed a Möbius strip in R³ with boundary a perfect circle. Here is the idea: let C be the unit circle in the xy plane in R³. Now connect antipodal points on C, i.e., points at angles $\theta$ and $\theta +\pi$ , by an arc of a circle. For $\theta$ between $0$ and $\pi /2$ make the arc lie above the xy plane, and for other $\theta$ the arc below (with two places where the arc lies in the xy plane).

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.

Art and technology

The Möbius strip has provided inspiration both for sculptures and for graphical art. M. C. Escher is one of the artists who was especially fond of it and based several of his lithographs on this mathematical object. One famous one, Möbius Strip II, features ants crawling around the surface of a Möbius strip.

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 Möbius", 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).

A device called a Möbius resistor is a recently discovered electronic circuit element which has the property of cancelling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s, US#512,340 "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

External links

Visual Math has a nice animation of making a Möbius strip.
Weisstein, Eric W. "Möbius Strip". MathWorld.
Möbius strip — a web page with movies.