“Euler’s Gem: The Polyhedron Formula and the Birth of Topology” by David S. Richeson

Main Index / Reviews Index / Nonfiction Reviews Index / 2021 Reviews Index

Rating: 3/5 Stars

An interesting book that looks at Euler’s Formula, namely for a polyhedron, the number of Faces plus the number of Vertices (corner points) minus the number of Edges always equals 2, or F + V − E = 2.

The book starts with a look at the life of Leonhard Euler, from his early mathematical education to his life in both the Russian and Prussian academies of science and his death. It then gives the history of polyhedra and some early findings about them, before presenting Euler’s formula and some ways to prove it as devised by various mathematicians. It also provides a proof that there are only five regular polyhedra using the formula.

Other chapters show how the formula began to be used for other areas of mathematics. Graph theory is one area, as shown by the famous example of the Bridges of Königsberg problem. Attempts to provide a proper mathematical definition of a polyhedron using the formula would lead to the field of topology. Subsequent chapters show the application of the formula to various topological fields like knots, vectors on topological surfaces (via the Hairy Ball Theorem and the Brouwer Fixed Point Theorem).

Further applications of the theorem encompass other topics like the curvature of surfaces and how topology is now a field that has developed rigorous proofs, and its various parts combined into a unified whole.

The early chapters are light on mathematics, while some people may struggle with the later chapters on topology. But it should be possible to skip them and get the conclusion that Euler’s formula that once defined a relationship of polyhedra has now been applied to the larger area of topology, leading to new mathematical findings.

Book read from 2021/07/31 to 2021/08/09