Disoriented world of the Möbius band – 11/24/2020 – Marcelo Viana

The Impa (Institute for Pure and Applied Mathematics) logo is a somewhat strange object. There are those who think it is the symbol of the infinite, but they are very different things. It is called the Möbius strip after the German mathematician and astronomer August Ferdinand Möbius (1790–1868), but the story is a bit complicated.

It’s easy to make a Möbius strip: take a narrow strip of paper and glue the two ends together like you’re making a clamp before turning one of the ends 180 degrees so that one side of the paper is on top of the other glued. It is the simplest example of a non-orientable surface, ie with only one surface: a normal clamp has the inner and outer surface, but in the Möbius area they are the same.

Another way to understand the non-orientation is that in the Möbius band it doesn’t make sense to speak right and left. A “resident” who leaves the house with a flag in one hand and turns the strip completely, returns home with the flag on the other, without ever changing hands. If you do not believe this, dear reader, do the experiment on the strip of paper (idealized that the paper has no thickness).

Möbius studied astronomy with Carl Friedrich Gauss (1777–1855) at the University of Göttingen. From there he went to Halle, where he received his doctorate under the direction of Johann Friedrich Pfaff (1765–1825), who had also been Gauss’ adviser.

He has done several works in astronomy and mathematics, particularly in the field of topology, which examines the properties of objects that are not affected when they are deformed. His most famous work, published in 1858, is the very article in which he described the title that now bears his name.

But it wasn’t the first: the same idea had been published a few months earlier by the German Johann Benedict Listing (1808–1882). Listing, who did his PhD under the direction of Gauss, was the first to use the word “topology” in the sense that I just stated in an 1847 article, and to do several work in this area.

It is curious that two mathematicians so close should have made the same discovery almost simultaneously. It was already suggested that the actual author would be Gauss, perhaps the greatest mathematician of all time, and that he was known for not publishing many of his results. We’ll probably never know.

What we do know is how the track became Impa’s logo: it was the initiative of the second director, Lindolpho de Carvalho Dias. “I thought we needed a brand. And the route not only has an important mathematical meaning, it is also very beautiful, ”he told me. You have it.

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