Möbius strip
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 codiscovered 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 halftwist, 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 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 halftwists 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 halftwists 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 halftwists, 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 twodimensional smooth manifold (a surface) which is not orientable.
One way to represent the Möbius strip as a subset of R^{3} is using the parametrization:
In cylindrical polar coordinates (r,θ, z), an unbounded version of the Möbius strip can be represented by the equation:
>     <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 continuousloop recording tapes (to double the playing time).
To date, no practical use for it (except possibly a bar trick) has been found.
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 threedimensional Euclidean space without creating selfintersections.
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 anticlockwise loop); and the Klein bottle is made the other way.
A crosscap is a twodimensional surface that is topologically equivalent to a Möbius strip. The term 'crosscap', 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.\n