# How was the Möbius strip discovered?

Britannica is terse:"Möbius discovered this surface in 1858. The German mathematician Johann Benedict Listing had discovered it a few months earlier, but he did not publish his discovery until 1861". Möbius never published the discovery, it was found in his 1865 memoir and published posthumously after 1868. Derbyshire comments "It had previously been described by another mathematician, Johann Listing, also in 1858. Listing published, and Möbius didn’t, so according to the academic rules it should really be called “the Listing strip”. There is no justice in this world".

I am not sure how Britannica's "few months" are known, considering that publications came years after, but it is not unheard of for two people to make the same discovery almost simultaneously, when an idea is "in the air". Newton-Leibniz and Leverrier-Adams are famous examples. What seems to be missing is what exactly was in the air, how did the Möbius strip come up? MacTutor says only "this discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Académie". This is more or less repeated everywhere else. But the context is missing, and some strange fables are spun. The most outlandish one is that Möbius and Listing discovered the strip "together", and "decided" to give priority to Möbius. Another, from an old popular book that I can not recall, is that Möbius was inspired by the shape of a twisted stock garter dropped by his maid.

The Möbius strip is not exactly a polyhedron, what is the connection? Möbius worked in Leipzig, and Listing in Göttingen. Listing was not exactly unknown, he was recommended for the Göttingen position by Gauss himself. Did they both work on the Académie problem? were they aware of each other's work? did they correspond or meet? why did the name "Möbius strip" stuck? Can somebody fill in the context?

## 1 Answer

My answer is based on guesses, not on knowledge of original work. Those years were the years were polyhedra were studied in connection with the problem of understanding to what extent Euler's formula $V-L+F=2$ was valid. In particular it was clear that there were some exceptions to the rule that needed, in the prevalent opinion, not be considered as polyhedra (see Lakatos "Proofs and refutations" for a modern discussion, with historical references). Since Listing numbers, one of the things he is remembered for, are topological invariants of homological nature, it will not surprise me if he discovered the Möbius band while trying to understand characteristic of a triangulation that was violating Euler's formula.

Yes, it is not a polyhedra, but it is easily constructed from a triangulated polyhedra. But the original polyhedra with no identification satisfies Euler's formula and the triangulation on the strip does not. So this is my guess about the link between polyhedra and Möbius strip. But one should read original papers at this point...

ADDED: I finally received and read Tripodi's papers but didn't gat much out of them (at least no explicit reference to the Möbius strip). But I found a very nice paper by Cayley which corroborates my hypothesis. The paper refers, from the title, to Listing's work.

it concerns a generalization of Euler's formula for "a figure of any sort whatever"; it counts lines on the figure which may be curved, for example. So it clearly extend out of polyhedra. In fact, in reading, one can easily see homology theory in disguise (and I have to say, personally, it is thrilling to read some remarks from which you really can see topology in its childhood...). All examples from 13 starts from a rectangular piece of paper and says and construct a space after identifications of opposite side. Even though there is no explicit mention of the strip, if the work of Listing was exemplified by such constructions it would be quite reasonable that he found the twisted strip in connection to this problem. I cannot go any further from here, since I am not able to read German.

• Very interesting idea: It would be really awesome if you could hunt down some evidence for this in the original sources... – Danu Oct 11 '15 at 21:54
• MacTutor writes in Listing's biography:"In 1862 he published Der Census raumlicher Complexe oder Verallgemeinerung des Euler'schen Satzes von den Polyedern which discusses extensions of Euler's formula for the Euler characteristic of oriented three-dimensional polyhedra to the case of certain four-dimensional simplicial complexes", which is in line with your guess, including the timing. They do not mention connection to the strip unfortunately. This work is discussed in 1971 Tripodi's paper, but alas it is in Italian. www-history.mcs.st-and.ac.uk/Biographies/Listing.html – Conifold Oct 13 '15 at 0:24
• Well, well I am Italian, so I guess I'll try to have a look at Tripodi's paper; but not before next week... – Nicola Ciccoli Oct 16 '15 at 7:15
• I too would like to know of the Listing publication citation. – Gottfried William Oct 25 '15 at 2:59