Richeson begins with the Greeks, with Plato and Theaetetus, Euclid's proof that there are only five regular polyhedra (also known as Platonic solids, with identical regular polygonal faces) and Archimedes and the thirteen semi-regular solids (which can have different kinds of regular faces). Kepler mixed mysticism and science: here his most important observation was the notion of duality, where (for example) the faces and vertices of the octahedron can be mapped to the vertices and faces of the cube.
Euler's first important discovery in this area, Richeson argues, was the notion of an edge: "Giving a name to this obvious feature may seem to be a trivial point, but it is not. It was a crucial recognition that the 1-dimensional edge of a polyhedron is an essential concept." After presenting the central formula and Euler's proof of it, Richeson explores some applications of it (such as the reason soccer and golf balls must have exactly twelve pentagonal faces), whether Descartes pre-discovered (but didn't publish) it, and Legendre's proof, "the first to meet today's rigorous standards". Cauchy provided an alternative proof, using flattened polyhedra, and Lhuiler and others extended the result to some non-Eulerian polyhedra. Euler's formula also leads naturally to the famous Königsberg bridge problem, graph theory, and the four-colour theorem.
Richeson goes on to explore some other areas of topology: rubber sheet geometry and the classification of surfaces, knots and their classification (using Seifert surfaces), the concept of a vector field and the Poincaré-Hopf and Hairy Ball theorems, various results on angles and the way the topology of a polyhedron can constrain its geometry, and finally a glance at curved surfaces and the Gauss-Bonnet theorem. That may sound scary and there is some substantial mathematics in this — quite a lot for a newcomer to topology to get their head around — but readers should not be put off: the proofs are informal, there are no involved derivations, and while there are equations they are mostly pretty simple.
Euler's Gem assumes almost no previous mathematics: nothing an early high school student wouldn't have covered, and in particular no calculus. It should be accessible to a broad audience, but Richeson doesn't dumb things down ("I also wrote it for mathematicians") and proceeds at a fair clip: there were a few things that were entirely new to me in it and a good number I had forgotten, and it gives a nice overview of the field and how it all fits together.
One obvious audience is in schools. Despite the accessibility and fascination of the material it covers, almost nothing in Euler's Gem is included in any high school mathematics curriculum I have seen (or indeed in the early years of most university courses). Somehow topology has been entirely pushed out of the school system, even though there are surely a sizeable number of students who aren't inspired by algebra and find trigonometry and calculus arid, but might have a better intuition for topology (and perhaps geometry more generally). Every school library should have a copy of Euler's Gem and it would be a great present for an otherwise curious student under-challenged or unexcited by the mathematics they are doing.
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