# Euler's Gem: The Polyhedron Formula and the Birth of Topology

Euler’s Gem: The Polyhedron Formula and the Birth of Topology, David S. Richeson, 2008, 332 pp., \$27.95, ISBN: 978-0-691-12677-7, Princeton University Press, 41 William Street, Princeton, NJ 08540. 609-258-3897, www.press.princeton.edu.

Leonhard Euler is the main figure in Richeson’s “history and celebration of topology.”  By freeing geometry from measurement, Euler started the field of topology.  Topology’s other father figures would include Simon Lhuilier (whose last name can mean “the one who oils”, and thus he was another oiler), Solomon Lefschetz, Johann Listing, and Henri Poincaré.  It was Poincaré who inspired the growth of modern topology with his 1895 paper Analysis Situs and its sequels.  He and Euler are jointly recognized by the Euler-Poincaré characteristic function that is used to classify polyhedral and other topological surfaces.

Always looking for an interesting connection, Richeson presents dozens of people, mainly mathematicians, but also a few monarchs, briefly describing their personal stories as well as their influences on mathematics.  He covers major mathematical figures from Thales (6th century BCE) to Perelman (b. 1966), and visits many of the beautiful theorems of mathematics.  We even get coverage of how Stephen Smale managed to anger officials in President Johnson’s administration.

Euler’s gem is the formula, V – E + F = 2, relating the vertex, edge, and face counts of a polyhedron.  Richeson carefully describes Euler’s proof, as well as later more rigorous proofs.  He uses this formula to count the Platonic solids, describing the nature of soccer balls, including the 2006 World Cup soccer ball, and Fullerenes (a family of all-carbon molecules), and to explore elementary graph theory and round dice.

The text is nicely illustrated, with one or more figures on most pages.  There are templates for creating three-dimensional representations of Platonic solids, Möbius bands, Klein bottles, and more. There are numerous text references, but most illustrations are not credited.  One can read this book on different levels, choosing to work out, scan, or skip the occasional difficult or tedious section.

Clifford H. Wagner, Associate Professor of Mathematics and Computer Science,The Pennsylvania State University at Harrisburg, Middletown,PA