Polishing Euler's Gem
Euler, you are awesome! |
Today is Euler's day 2/7/18. Every year on February 7th, we can celebrate it. But
this year is especially good. It was appropriate to talk about Euler at the
Math table. Here is the as distributed during the talk. I plan in the future to extend the text a bit, add references, illustrations and historical context. |
Polishing Euler's Gem : Just before Euler's day (2/7/18) it is appropriate to look at one of the most beautiful formulas in Mathematics, the so called Euler Gem v - e + f = 2. Less well known is that this formula has historically been proved wrong again and again, as counter examples have turned up, like Kepler solids. This was such an embarrassment for Mathematics, that Imre Lakatos suggested that theorems do "evolve" like species, maybe never really reaching their ultimate final form. In this talk we aim to refute Lakatos' hypothesis and give a crystal clear proof of the formula in arbitrary dimensions: the Euler characteristic of a discrete sphere is 1 + (-1)^{d}. The Euler Gem case is d = 2. Understanding the proof does not require any Euclidean space and is inductive. The key is to define in a precise way, and entirely combinatorially, what a discrete d-dimensional sphere is. The quest to do that combinatorially has only started in the 20th century with Hermann Weyl, and Euler plays also here an important role as he initiated linking topology with combinatorics. If time permits, the classification of all the Platonic solids in arbitrary dimension will be derived. While known in the case d = 2 already in ancient Greece, the higher dimensional case has been tackled first by rather unusual mathematicians: by Ludwig Schläfli, who was a high school teacher at first or by Alicia Boole Stott, the princess of Polytopia. |
Oil Painting by Emanuel Jakob Handman, from 1756. I got the picture from the book "Leonhard Euler", by Emil A. Fellmann which was published by Birkhäuser, Basel, Boston Berlin, 2007. Emanuel Jakob Handmann was a Swiss painter born in Basel 1718 and died in Bern 1781. (Wikipedia). He was a pupil of Schnetzler in Schaffhausen (1735-39) [ Schaffhausen has a traditional arts school there are many examples on rheinfall.com] and of Restout in Paris (1739 -42). (Source.) Handmann also painted the famous but grimm picture of Euler.