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The Möbius strip or Möbius band ( (non-rhotic) or ; ), Mobius or Moebius, is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any rectangle can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, and others cannot. A half-twist clockwise gives a different embedding of the Möbius strip than a half-twist counterclockwise – that is, as an embedded object in Euclidean space the Möbius strip is a chiral object with right- or left-handedness. However, the underlying topological spaces within the Möbius strip are homeomorphic in each case. There are an infinite number of topologically different embeddings of the same topological space into three-dimensional space, as the Möbius strip can also be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends. The complete open Möbius band is an example of a topological surface that is closely related to the standard Möbius strip but that is not homeomorphic to it. It is straightforward to find algebraic equations, the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface, having zero Gaussian curvature. A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution. The Euler characteristic of the Möbius strip is zero. ==Properties== The Möbius strip has several curious properties. A line drawn starting from the seam down the middle meets back at the seam but at the other side. If continued the line meets the starting point, and is double the length of the original strip. This single continuous curve demonstrates that the Möbius strip has only one boundary. Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists. If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip – it is the center third of the original strip, comprising 1/3 of the width and the same length as the original strip. The other is a longer but thin strip with two full twists in it – this is a neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip. Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. (If this knot is unravelled, the strip is made with eight half-twists in addition to an overhand knot.) A strip with ''N'' half-twists, when bisected, becomes a strip with ''N'' + 1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Möbius strip」の詳細全文を読む スポンサード リンク
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