Mobius Loop

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A ray-traced parametric plot of a Möbius strip.

A Möbius strip made with a piece of paper and tape. If its full length were crawled by an ant, the ant would return to its starting point having traversed both sides of the paper without ever crossing an edge.

A Möbius strip does not self-intersect but its projection in 2 dimensions does.

In mathematics, a Möbius strip, band, or loop (US: /ˈmoʊbiəs, ˈmeɪ-/ MOH-bee-əs, MAY-, UK: /ˈmɜːbiəs/;[1] German: [ˈmøːbi̯ʊs]), also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858,[2][3][4][5] though similar structures can be seen in Roman mosaics c. 200–250 AD.[6][7]

An example of a Möbius strip can be created by taking a paper strip, giving one end a half-twist, and then joining the ends to form a loop; its boundary is a simple closed curve which can be traced by single unknotted string. Any topological space homeomorphic to this example is also called a Möbius strip, allowing for a very wide variety of geometric realizations as surfaces with a definite size and shape. For example, any rectangle can be glued left-edge to right-edge with a reversal of orientation. Some, but not all, of these can be smoothly modeled as surfaces in Euclidean space. A closely related, but not homeomorphic, surface is the complete open Möbius band, a boundaryless surface in which the width of the strip is extended infinitely to become a Euclidean line.

A half-twist clockwise gives an embedding of the Möbius strip which cannot be moved or stretched to give the half-twist counterclockwise; thus, a Möbius strip embedded in Euclidean space is a chiral object with right- or left-handedness. The Möbius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends.

Finding algebraic equations cutting out a Möbius strip is straightforward, but these equations do not describe the same geometric shape as the twisted paper model above. Such paper models are developable surfaces having zero Gaussian curvature, and can be described by differential-algebraic equations.[8]

The Euler characteristic of the Möbius strip is zero.