math – update

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Day #8:: Hanging Around on a Möbius Stripe

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Little balls on the edge of a Möbius stripe change sides!

What do you really know about it??…

A: The Möbius strip or Möbius band (/ˈmɜːrbiəs/ (non-rhotic) or US /ˈmbiəs/; German:[ˈmøːbi̯ʊs]), 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 mathematiciansAugust Ferdinand Möbius and Johann Benedict Listing in 1858.[1][2][3]

Geometry and topology

A ray-traced parametric plot of a Möbius strip

 A parametric plot of a Möbius strip
To turn a rectangle into a Möbius strip, join the edges labelled A so that the directions of the arrows match.

One way to represent the Möbius strip as a subset of three-dimensional Euclidean space is using the parametrization:

  x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u
y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u
z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}

where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the xy plane and is centered at (0, 0, 0). The parameter u runs around the strip while v moves from one edge to the other.

Topology

Topologically, the Möbius strip can be defined as the square[0, 1] × [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.

A less used presentation of the Möbius strip is as the topological quotient of a torus.[7] A torus can be constructed as the square [0, 1] × [0, 1] with the edges identified as (0, y) ~ (1, y) (glue left to right) and (x, 0) ~ (x, 1) (glue bottom to top). If one then also identified (x, y) ~ (y, x), then one obtains the Möbius strip. The diagonal of the square (the points (x, x) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to “reflection” – geodesics (straight lines) in the Möbius strip reflect off the edge back into the strip. Notationally, this is written as T2/S2 – the 2-torus quotiented by the group action of the symmetric group on two letters (switching coordinates), and it can be thought of as the configuration space of two unordered points on the circle, possibly the same (the edge corresponds to the points being the same), with the torus corresponding to two ordered points on the circle.

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface that is not orientable. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface. As a result, any surface is non-orientable if and only if it contains a Möbius band as a subspace.

Source: EN: WikipediA, DE: WikipediA

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Author: Math - Update

Updating Math In Our Mind & Heart!!...

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