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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 was co-discovered independently by the German Mathematician s 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 . The Möbius strip is therefore '' Chiral '', which is to say that it is "handed". 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 Ring s. GEOMETRY AND TOPOLOGY One way to represent the Möbius strip as a subset of R3 is using the parametrization: : : : where and . 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: : Topologically , the Möbius strip can be defined as the Square × [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 ''S''1 with a fiber the Unit Interval , ''I'' = {Link without Title} . Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z2) bundle over ''S''1. 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 ''S''3 regarded as a subset of R4. A parametrization for this embedding is given by : : |
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