The mystery of the Mobius strip

Published: 19th February 2007
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The Mobius strip, also known as Mobius band, is a one sided surface with only one boundary component. It has the mathematical property of being impossible to orient. Also, it has a ruled surface. The Mobius strip was discovered in 1858 by two German mathematicians independently: August Ferdinand Mobius and Johann Benedict Listing.

A model of a this strip can be easily created by giving a half-twist to a strip of paper and then reconnecting the ends of the strip together to form a single strip. Two types of Mobius strips are found in Euclidean space clockwise and counter-clockwise depending of the direction of the half-twist.

No matter the direction of the half-twist, Mobius strips have several rather curious properties. If you take a pen and draw a line down the middle of a Mobius strip made of paper until you reach the starting point, you will end up with a line double the length of the original strip of paper on the whole surface of it. If you were to cut the strip along the line you have drawn, there won't be two strips obtained, as one would expect, but a longer strip with two half-twists that is not considered to be a Mobius strip. This amazing thing happens due to the fact that the original strip has only one edge that is twice as long as the strip of paper originally used.

Another combination can result from cutting such a strip about a third of the way in from the edge. This creates a thinner sample of the strip and another strip with two half-twists. Other combinations can be obtained when using two or more half-twists. If you divide lengthwise a strip that has three half-twists instead of one you obtain a strip that is tied in a trefoil knot. Cutting Mobius strips and giving them extra half-twists are operations that generate a lot of unexpected figures that are called paradromic rings.

There are closely related objects to the Mobius strip. One of them is the Klein bottle that can be produced by gluing two Mobius strips together along their edges, something that cannot be done in ordinary three-dimensional Euclidian space without creating intersections. A Klein bottle has only one surface, even though it looks like it has inside and outside surfaces. The real projective plane is also a related object to Mobius strips, due to the fact that, if you cut a circular disk out of the real projective plane, what you are left with is a Mobius strip. These two related objects are obtained by gluing together Mobius strips on different sides.

Mobius strips have been an inspiration for a lot of sculptures and graphical art. An example of this is M. C. Escher's representation of such a strip with ants crawling around its surface. Science fiction stories have been written using the Mobius strip as a recurrent feature like "The wall of darkness" by Arthur C. Clarke and "A subway named Mobius" by A. J. Deutsch. Also, there have been many devices designed on this pattern. Reaching recent days, movies have also a link with Mobius strips, such as some episodes from Stargate and also the Playstation 2 game Ace Combat 04: Shattered Skies has a fictional platoon named like that. To get a clear view of the popularity of Mobius strips, a lot of companies used this pattern to create their logos; the most important of them is the international symbol for recycling that is represented by a Mobius loop.

Moreover, theological meaning has been attributed to the Mobius strip. Due to the fact that there is no difference between the inside and the outside surfaces, it is considered to be an example of the creator which created everything that surrounds us from himself. The universe is also considered to be some kind of such a generalized strip.
Even though the Mobius strip is a mathematical pattern, it is surrounded by mystery. But everybody knows that mathematics is the universal language and that it stands as the basis for the universe. It has theological meaning attributed to it and it has been a source of inspiration for art and technology.

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