Euler’s Gem: The Polyhedron Formula and the Birth of Topology by David S. Richeson
I’ve always been fascinated by topology. The idea that certain simple properties constrain the shapes that can be constructed in the real world seems amazing to me. Writing a book around Euler’s polyhedron formula (F+V=E+2) and using it as a means of introducing lots of concepts of topology was a really good idea.
The book starts out by introducing the Greek mathematicians and their classification of the platonic solids, and then quickly moves on to Euler’s discovery of the formula and shows a number of proofs. It was the first time that I had seen Legendre’s proof using spherical geometry, and the author covers the proof really well. There is also a good discussion of the types of polyhedron that the formula applies to, and a discussion of how it was some time before the correct underlying concepts were discovered. The book uses the polyhedron formula as a way to introduce some graph theory, knot theory and more general topological theorems such as the hairy ball theorem.
I thought this was a really well written book that covers a lot of really interesting material.