Euler's Gem. The Polyhedron Formula and the Birth of Topology
The title of this book “Euler's Gem” refers to the well-known formula stating that the numbers of vertices, edges and faces of a convex polyhedron satisfy the relationship V-E+F=2. The author decided not only to trace the history of this simple and elegant theorem but also to present its generalisations and applications in different branches of mathematics, in particular in topology, graph theory and differential geometry. He covers an extremely wide range of topics, including the discovery of the five regular convex polyhedra, Descartes' version of the polyhedron formula, the graph-theoretic version of Euler's formula and its relation to the four colour theorem (with details on Kempe's attempt to prove the theorem), the topological classification of surfaces, knot theory, vector fields on surfaces and the Poincaré-Hopf theorem, the role of Euler's number in global results of the differential geometry of curves and surfaces, n-dimensional manifolds, the Poincaré conjecture and many more. The text requires no knowledge of higher mathematics. There are many proofs or sketches of proofs of simpler theorems and the author tries to provide explanations of more difficult theorems, omitting technical details. This book is a pleasure to read for professional mathematicians, students of mathematics or anyone with a general interest in mathematics.