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19 November 2018

Möbius and Impossible Objects

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One hundred and sixty years ago, August Möbius built a bridge to another reality, in which the rules are different from those of our three-dimensional world. His great discovery, today known as the “Möbius strip”, is an object that defies common sense and our prejudices about what is intuitive, and also has some curious mathematical properties that expanded knowledge and promoted the development of topology. In addition, the peculiarities of this strange way of visualizing the infinite have been converted into ingenious practical applications, most aimed at achieving more efficient and durable devices. It is not by chance that the global symbol of recycling is based on a Möbius strip.

Animation based on the work “Möbius Strip II”, by M.C. Escher Credit: romullus3d

It looks like a normal infinite circle, but it’s not. If we think of it like a wheel, it’s easy to imagine an ant walking on its outer surface without ever reaching the end. The Möbius strip takes this idea of ​​the infinite even further, and puts us in the difficult position of imagining an ant that in each complete lap passes along its exterior surface and its interior one, and also without crossing any of its edges, as imagined by the artist MC Escher. That is why, ever since the German mathematician August Ferdinand Möbius (17 November 1790 – 26 September 1868) described it in 1858, it has fascinated artists, engineers, environmentalists and scientists.

The Möbius strip fulfils the double paradox of being a single-sided strip and having only one edge. It is a two-dimensional object that has sneaked into our three-dimensional world and, what’s more, constructing one is within reach of anyone. Its simplest form is achieved by taking a strip of paper (which can be obtained by cutting in a straight line along a sheet of paper) and joining its ends, but giving one of them half a turn before gluing or taping them together.

This is how a Moebius strip is made. Source: brilliant.org

There are many other versions of the Möbius puzzle, which can be achieved with strips of any shape and size, provided that when joining their ends we perform an odd number of turns. And that idea inspired another German mathematician, Felix Klein, to imagine in 1882 what we now know as “Klein bottles.” They are four-dimensional objects that we cannot build in our three-dimensional reality, but if we manage to visualise them, they will confuse us even more: they are theoretical containers that cannot contain a liquid because the inside and outside are mixed up.

Möbius strips and Klein bottles share a curious mathematical property within the field of topological study. They are not orientable, something that, to simplify, can be explained by imagining that if we draw an arrow on them, it’s impossible to conclude if that arrow points up or down. In a non-orientable world, our image and the one we see in the mirror would be indistinguishable.

But returning to our world and leaving aside theoretical mathematics, the great idea of ​​Möbius has been applied to conveyor belts that last longer (because all its surface is worn down at equal speed) and magnetic tapes for recording sound that don’t have to change sides: they can be used twice as long without interruption and they are used to play music in an infinite loop. Its application has also been patented in electronic components (such as a resistor that doesn’t produce magnetic interference) and its use is being investigated to achieve superconductors of high transition temperatures, molecular motors and graphene structures with new electronic characteristics.

Central part of a mosaic of a Roman villa in Sentinum (in Italy). Source: Glyptothek

Such applications go far beyond what August Möbius imagined when he scientifically described this “impossible object” in 1858, although it should be acknowledged that this mathematician and theoretical astronomer was not the first to do so. Another German mathematician, Johann Benedict Listing, had come up with the same idea independently just a few months earlier. In fact, neither of them invented the single-sided strip: the concept is at least 1,600 years older, because a structure similar to the Möbius strip can be seen in Roman mosaics dating back to the third century.

However, the scientific weight of August Möbius—a student of the great mathematician Carl Friedrich Gauss and who went on to direct the astronomical observatory of the prestigious University of Göttingen—served to put his name to and to popularize this mathematical oddity whose greatest application, above all, has been to stimulate us to imagine beyond the space in which we live.

Francisco Doménech
 @fucolin

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