The Möbius strip or Möbius band (UK /ˈmɜrbiəs/ or US /ˈmoʊbiəs/; German: [ˈmøːbi̯ʊs]), also Mobius orMoebius, is a surface with only one side and only one boundary component. The Möbius strip has themathematical 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.
A model can easily 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. In Euclidean space there are two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. That is to say, it is a chiral object with “handedness” (right-handed or left-handed).
The Möbius band (equally known as the Möbius strip) is not a surface of only one geometry (i.e., of only one exact size and shape), such as the half-twisted paper strip depicted in the illustration to the right. Rather, mathematicians refer to the (closed) Möbius band as any surface that is topologically equivalent to this strip. Its boundary is a simple closed curve, i.e., topologically 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 closed rectangle with length L and width W 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 3-dimensional space, and others cannot (see section Fattest rectangular Möbius strip in 3-space below). Yet another example is the complete open Möbius band (see sectionOpen Möbius band below). Topologically, this is slightly different from the more usual — closed — Möbius band, in that any open Möbius band has no boundary.
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