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The Möbius strip or Möbius band (pronounced ) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. 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".
Properties The Möbius strip has several curious properties. A Möbius strip can be made by joining the ends of a strip of paper with a half-twist. A line drawn starting from the seam down the middle will meet back at the seam but at the "other side". If continued the line will meet the starting point and will double the length of the original strip of paper. This single contiguous curve demonstrates that the Möbius strip has only one boundary. If the strip is cut along the above line, instead of getting two separate strips, it becomes one long strip with two half-twists in it (not a Möbius strip). This happens because the original strip only has one edge which is twice as long as the original strip of paper. By cutting you have created a second independent edge, half of which was on each side of the knife or scissors. If you cut this new, longer strip 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 - it is the center third of the original strip. The other is a long strip with two half-twists in it (not a Möbius strip) - this is a neighborhood of the edge of the original 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. Geometry and topology
Sudanese Möbius Band 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 and the resulting figure is called the Sudanese Möbius Band. 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 runs from to and runs from to . Since the embedded surface lies entirely on S3. The boundary of the strip is given by (corresponding to ), 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. 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. In graph theory, the Möbius ladder is a cubic graph closely related to the Möbius strip. Art and technology
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