When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent to untwisted rings. In many cases these merely depict coiled ribbons as boundaries. However, it had been known long before, both as a physical object and in artistic depictions in particular, it can be seen in several Roman mosaics from the third century CE. The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858.
Many works of speculative fiction feature Möbius strips more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.Ĭhain pump with a Möbius drive chain, by Ismail al-Jazari (1206) Bach have been analyzed using Möbius strips. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. Performers including Harry Blackstone Sr. Many architectural concepts have been inspired by the Möbius strip, including the building design for the NASCAR Hall of Fame. Escher, Max Bill, and others, and in the design of the recycling symbol. In popular culture, Möbius strips appear in artworks by M. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in social choice theory. The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes appear opposite each other. Certain highly-symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip. A Möbius strip without its boundary, called an open Möbius strip, can form surfaces of constant curvature. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. The Sudanese Möbius strip is a minimal surface in a hypersphere, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a developable surface or be folded flat the flattened Möbius strips include the trihexaflexagon. It can be swept as a ruled surface by a line segment rotating in a rotating plane, with or without self-crossings. Several geometric constructions of the Möbius strip provide it with additional structure. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. Every non-orientable surface contains a Möbius strip.Īs an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. A Möbius strip made with paper and adhesive tape
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