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A Mbius strip made with a piece of paper and tape.

The Mbius strip or Mbius band is a topological object with only one side (one-sided surface) and only one boundary component. It was co-discovered independently by the German mathematicians August Ferdinand Mbius 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 Mbius 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 Mbius strip. The Mbius strip therefore exhibits chirality.

The Mbius 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 Mbius strip). If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along a Mbius strip, about a third of the way in from the edge, you will get two strips; one is a thinner Mbius strip, the other is a long strip with two half-twists in it (not a Mbius strip). Other interesting combinations of strips can be obtained by making Mbius 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 Mbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

The Mbius strip is often cited as the inspiration for the infinity symbol [itex]\infty[itex], since if one were to stand on the surface of a Mbius 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 Mbius strip was discovered.

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## Geometry and topology

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A parametric plot of a Mbius strip

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

[itex]x(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\cos(u)[itex]
[itex]y(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\sin(u)[itex]
[itex]z(u,v)=\frac{v}{2}\sin\frac{u}{2}[itex]

where [itex]0\leq u < 2\pi[itex] and [itex]-1\leq v\leq 1[itex]. This creates a Mbius 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 Mbius strip can be represented by the equation:

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

Topologically, the Mbius 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 Mbius 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 Mbius 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 Mbius 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 Mbius 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 Mbius strip. Going in the other direction, if one glues a disk to a Mbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Mbius 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 Mbius 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 [itex]\theta[itex] and [itex]\theta + \pi[itex], by an arc of a circle. For [itex]\theta[itex] between [itex]0[itex] and [itex]\pi/2[itex] make the arc lie above the xy plane, and for other [itex]\theta[itex] the arc below (with two places where the arc lies in the xy plane). However, if a disk is glued in to the boundary circle, the self-intersection of the resulting projective plane is imminent.

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

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This Octopus card bears a stylized Mbius strip design.

The Mbius 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, Mbius Strip II (http://www.mcescher.com/Gallery/recogn-bmp/LW441.jpg), features ants crawling around the surface of a Mbius 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 Mbius strip. In the short story "A Subway Named Mbius" (http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf102), by A.J. Deutsch, the Boston subway authority builds a new line; the system becomes so tangled that it turns into a Mbius strip, and trains start to disappear.

There have been technical applications; giant Mbius 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 Mbius 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.

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