MÃ¶bius strip

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Image:Möbius strip.jpg The Möbius strip or Möbius band (pronounced Template:IPA) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. 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. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it is "handed".

1 Properties
2 Geometry and topology
3 Möbius strip with a circular boundary
4 Related objects
5 Art and technology
6 See also
7 References
8 External links

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Properties

The Möbius strip has several curious properties. If you try to split the strip in half by cutting it down the middle along a line parallel to its edge, 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.

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Geometry and topology

Image:MobiusStrip-01.png Image:MöbiusStripAsSquare.svg

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

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

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

where <math>0\leq u < 2\pi</math> and <math>-1\leq v\leq 1</math>. 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:

<math>\log(r)\sin\left(\frac{\theta}{2}\right)=z\cos\left(\frac{\theta}{2}\right).</math>

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the diagram on the right.

The Möbius strip is a two-dimensional compact 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.

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Möbius strip with a circular boundary

Topologically, the boundary of a Möbius strip is a circle. Under the usual embeddings of the strip in Euclidean space, as above, this boundary is not round. It is a common misconception that a Möbius strip cannot be embedded in three-dimensions so that the boundary is a round circle. In fact this is possible.

To see this, first consider such an embedding into the 3-sphere S³ regarded as a subset of R⁴. A parametrization for this embedding is given by

Here we have used complex notation and regarded R⁴ as C². The parameter <math>\eta</math> runs from <math>0</math> to <math>\pi</math> and <math>\phi</math> runs from <math>0</math> to <math>2\pi</math>. Since <math>|z_1|^2 + |z_2|^2 = 1</math> the embedded surface lies entirely on S³. The boundary of the strip is given by <math>|z_2| = 1</math> (corresponding to <math>\eta = 0,\pi</math>), which is clearly a circle on the 3-sphere.

To obtain an embedding of the Möbius strip in R³ one maps S³ to R³ via a stereographic projection. The projection point can be any point on S³ which does not lie on the embedded Möbius strip (this rules out all the usual projection points). Stereographic projections map circles to circles and will preserve the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R³ with a circular boundary and no self-intersections.

Image:MobiusSnail2B.png

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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 circular disk is cut out of 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 (see above). The projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

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.

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Art and technology

Image:OctopusFrontNew.jpg

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.

A popular limerick is often associated with this design which reads,

"A mathematician confided

That a Möbius band is one-sided,

And you'll get quite a laugh,

If you cut one in half,

For it stays in one piece when divided"

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). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, it allows the ribbon to be twice as wide as the printhead yet both half-edges are used evenly.

In "A. Botts and the Möbius Strip", a short story by William Hazlett Upson first published in 1945 in the Saturday Evening Post, the protagonist secretly restitches a conveyor belt to form a Möbius strip to frustrate a superior's attempt to "paint the outside, but not the inside" of the belt as a safety measure.

A device called a Möbius resistor is an 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.

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References

{{cite book

|    author = Clifford A. Pickover
|      year = 2006
|     month = March
|     title = The Möbius Strip : Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology 
| publisher = Thunder's Mouth Press
|        id = ISBN 1560258268 
}}

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