[Math] Did ancient mathematicians know Euler’s characteristic for convex polyhedra

Question

The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid proof). To have a concrete question: Is there any reference to this formula in ancient mathematics?

Answer 1

there is no doubt the answer to your question is "no"; for a wonderful and scholarly recent book on the whole story, see Euler's Gem: The Polyhedron Formula and the Birth of Topology by David Richeson.

They all missed it. The ancient Greeks -- mathematical luminaries such as Phythagoras, Theaetetus, Plato, Euclid, and Archimedes, who where infatuated with polyhedra -- missed it. Johannes Kepler, the great astronomer, so in awe of the beauty of polyhedra that he based an early model of the solar system on them, missed it. In his investigation of polyhedra the mathematician and philosopher René Descartes was but a few logical steps away from discovering it, yet he too missed it. These mathematicians, and so many others, missed a relationship that is so simple that it can be explained to any schoolchild, yet is so fundamental that it is part of the fabric of modern mathematics.

The great Swiss mathematician Leonhard Euler did not miss it. On November 14, 1750, in a letter to his friend, the number theorist Christian Goldbach, Euler wrote, "It astonishes me that these general properties of stereometry have not, as far as I know, been noticed by anyone else".

Centuries later, we remain astonished.