You may not have heard of a Möbius strip but you probably see one every day. The universal recycling symbol (the three bent arrows forming a triangle) is a form of this one-sided, chiral surface.
It has been suggested that the symbol for infinity (the lemniscate) was based on a Möbius strip because of its “endless” nature. But this is unlikely, because the lemniscate had been in use for around 200 years when German mathematicians August Möbius and Johann Listing discovered the Möbius strip in 1858.
A Möbius strip can be formed from a length of paper given a half twist before joining the ends together. The direction of the half twist determines the nature of the strip, hence its chiral nature. If an ant were to crawl along the length of this surface, it would return to its starting point having traversed every part of the strip without ever crossing an edge.
The Möbius strip is a non-orientable surface, which means that the concepts of left and right have no meaning. Not the obvious surface for a game of chess, you might think. But Möbius chess is played on a surface four squares wide and 16 squares deep, joined to make a Möbius strip.
This surface of a Möbius strip is more than simply a mathematician’s plaything, for it has a number of practical uses. Conveyor belts are made from them so that the entire surface of the belt gets the same amount of wear. And they are used in continuous loop recording tapes to double the playing time. They are also commonly used to make fabric computer printer and typewriter ribbons.
Science fiction writers also have been inspired by this topological curiosity. Arthur C. Clarke used the concept in his 1948 novel, ‘The wall of darkness’, in which a Möbian wall encircles a planet in a universe in which it and a sun are the only objects. In ‘A subway named Möbius’ by A. J. Deutsch, a subway line gets tangled into a Möbian strip causing trains to disappear.
Attaching two Möbius strips together along their edges forms an even stranger surface, the Klein bottle. This, however, cannot be done in three dimensions without creating self-intersections. The Klein bottle is a non-orientable surface with no boundaries and no identifiable inner or outer sides.
The Science Museum in London has a display of hand blown glass Klein bottles, illustrating a number of variations on the topological theme.
Mobius strips
Come on Prospector <GGG> having written such a good article, you must be aware that a Klein bottle only exists in four dimensions, and that any models are an approximation of the whole.
Martin Gardner gives a good treatment of the the whole Mobius phenomenon, in his book "Mathematical Puzzles and Diversions" "Pelican paperback)
You omitted to mention the fact that if you cut along the middle of a Mobius strip , then you get two interlocking bands.
I enjoyed this little diversion, and I hope you will produce more with a mathematical flavour. I can't count for toffee; the Lord only knows how I have not managed to poison anyone in my long pharmacy career, but recreational mathematics holds a strange charm for, even if the equations might as well be written in Urdu.
Regards
Bob Dunkley