Euler's Gem

[stoutfellow]

Euler's Gem, by David S. Richeson, is itself a gem, easily the most interesting piece of nonfiction I've read this year. It centers itself on a famous equation discovered by Leonhard Euler, but reaches back to the Pythagorean era and forward to the early twentieth century in tracing out its genealogy. Though there is a fair amount of mathematics (most of it accessible even to the mathematical dilettante), Richeson's primary interest is in the history of the idea, which intertwines fascinatingly with any number of major strands of mathematical thought.

The gem of the title is this. Consider a polyhedron: a tetrahedron, a cube, a pyramid, name your favorite. Count the number of vertices (V), of edges (E), and of faces (F). You'll find that the following formula holds: V - E + F = 2. (For the cube, for example, there are 8 vertices, 12 edges, and 6 faces; 8 - 12 + 6 = 2.)

The story of this equation roams in time from ancient Greece to the early twentieth century, and touches upon many important areas of mathematics: we visit the bridges of Königsberg, explore the four-color map theorem, and link Euler's result to subjects ranging from spherical trigonometry through differential geometry (my beloved Gauss-Bonnet theorem puts in an appearance) and knot theory to, ultimately, twentieth-century algebraic topology. (Euler would not recognize it, perhaps, but the core ideas of this recondite field rest squarely in his work, and specifically this equation.) Mathematicians from ancient Greece to modern-day Russia appear - the famous (Archimedes, Gauss, Poincaré), the obscure (Thomas Harriot, Simon Lhuilier, Johann Listing), and the simply odd (Charles Dodgson, AKA Lewis Carroll). It's a fascinating and mind-expanding history, showing the interconnections between seemingly disparate areas and the erratic way in which concepts develop. (The stories of the concepts "polyhedron" and "edge" are particularly interesting.) If you have any interest in the history of ideas, especially mathematical ideas, this is an excellent book.

Comments

[akatsukami.livejournal.com]

All polyhedra, or just regular polyhedra?

[stoutfellow.livejournal.com]

That's a major part of the story - determining exactly which polyhedra this applied to. The answer is: any polyhedron, regular or not, convex or not, which is topologically equivalent to a sphere satisfies this formula. (If you could mold it out of a lump of clay without poking holes through it, it's in.)

[countrycousin.livejournal.com]

Thanks for the rec. Looks good; I'll look for it.

simply odd surely not simply? :-)