Mobius Strip

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Mobius Strip
Field: Topology
Image Created By: Wikipedia
Website: Wikipedia

Mobius Strip

A Mobius strip, also referred to as a Mobius band, is a surface with only one side and one edge.


Contents

Basic Description

If we take a rectangular strip of paper, then make a half twist and join the ends we come up with a Mobius strip. If we were to draw a line through the center of the strip without lifting the pencil off the paper, we would come back to the starting point but on the "opposite" side of the paper. Logically, this is only possible if the surface has only one side and only one boundary, meaning that while the Mobius strip appears to have two sides, it actually has one.


An orientable surface is one that has two sides, that is you can paint the two sides with two different colors. A sphere is an example of an orientable surface because you can paint the inside and the outside with two different colors. A Mobius strip, on the other hand, is a non-orientable surface because it has just one side.


Picture an ant walking along the center line of the Mobius strip; whereas on an orientable surface like a sphere it would have to go just once around the surface to arrive at the orginal starting point, on a Mobius strip it would have to go twice round the loop.

Strange Properties

This youtube video illustrates some of the interesting properties of the Mobius strip.

Experiment 1: Draw a line through the center of the strip. We would have to go round the loop twice to get back to the starting point. This is a key feature of the Mobius strip because it's what describes it as a non orientable surface.
Experiment 2: Cut through the center line. In general, if we cut a rectangular strip of paper lengthwise through the middle from end to end, we would expect to get two strips. This is not the case with the Mobius strip. Instead of getting two strips, we get one long strip with two full twists in it. If we cut the strip again through the center line, we come up with two strips wound around each other.
Experiment 3: Cut through the line about one third from the edge. (note: we have to go twice round the loop), we get two separate strips, one of which is thinner, but of the same length as the original strip. The other will be a long strip whose length is twice that of the original strip.
Experiment 4: Cut once round the loop through the line about one third from the edge, then cut through the center line of the resultant thicker strip. We get three strips wound around each other, one in the middle and one on either side.

A strip with an odd number of twists will behave the same way as a Mobius strip, that is, it will have one edge and one side. On the other hand, a strip with an even number of twists will have two boundaries and sides.

Parametrization

A parametric representation of a Mobius strip http://mathworld.wolfram.com/MoebiusStrip.html
A parametric representation of a Mobius strip http://mathworld.wolfram.com/MoebiusStrip.html
A Mobius strip of width 1 and whose middle circle has a radius 1, centered at (0,0,0) is represented by the following parametric equations:
x(u,v)= \textstyle \left(1+\frac{1}{2}v \cos \frac{1}{2}u\right)\cos u
y(u,v)= \textstyle \left(1+\frac{1}{2}v\cos\frac{1}{2}u\right)\sin u
z(u,v)= \textstyle \frac{1}{2}v\sin \frac{1}{2}u
where 0 ≤ u < 2π and −1 ≤ v ≤ 1. Parameter u runs around the strip, while v moves from one edge to another.



Applications

The universal recylcing sign is a Mobius strip http://chemistry.about.com/od/healthsafety/ig/Laboratory-Safety-Signs/Recycling-Sign.htm
The universal recylcing sign is a Mobius strip http://chemistry.about.com/od/healthsafety/ig/Laboratory-Safety-Signs/Recycling-Sign.htm
B.F.Goodrich Company manufactures Turnover Conveyor Belt System which has half twists in it to allow for equal wearing on both sides of the belt http://www.math.unh.edu/cgi-bin/generatePage.cgi?moebius
B.F.Goodrich Company manufactures Turnover Conveyor Belt System which has half twists in it to allow for equal wearing on both sides of the belt http://www.math.unh.edu/cgi-bin/generatePage.cgi?moebius

Mobius strips have been used as conveyor belts because their one sided nature allows equal wearing of "opposite sides" of the belt. This makes the belts last longer. The strip has also been used in recording tapes to double the playing time without having to manually take out the tape and change the side playing. It is also used in numerous electronic appliances especially those which have resistors and superconductors.

An artist like M.C. Escther who had recurring mathematical themes in his works used Mobius strip in some of his works. Escher's Parade of Ants is a lithograph which shows ants moving on the strip. Martin Gardner uses the idea of the Mobius strip in his amusing short story "No-Sided Professor," which can be found in the book Fantasia Mathematica.

The universal recycling sign is a Mobius strip.
















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