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

Pure mathematics became highly abstract in the 20th century. Perhaps no branch of mathematics exemplifies this trend better than topology. The book definition of topology is the study of those properties of geometric objects that are preserved under reversible continuous transformations: if you have ever seen the famous illustration of how a coffee mug can gradually be transformed into a doughnut, then you have seen a proof that the two are topologically equivalent. Topologists study one-sided surfaces called Mobius bands, two of which can be fused together to produce a Klein bottle - a closed bottle with no interior. Never mind that Klein bottles cannot actually be constructed in the three-dimensional world of our sensory experience; we can still study them as mathematical abstractions.

David Richeson's book, Euler's Gem: The Polyhedron Formula and the Birth of Topology, is not a textbook of mathematics. Rather, it is a lively romp of mathematical exposition, organised around a historical narrative, in a style reminiscent of recent books by William Dunham, Simon Singh and Marcus du Sautoy. Richeson traces the origins of topology to the mid-18th century and a theorem discovered by the prodigious Swiss mathematician, Leonhard Euler (2007 was Euler's 300th birthday and Richeson's book is a welcome addition to the collection of popular and academic volumes that were published to celebrate that event).

Born and educated in Basel, Euler spent most of his adult life working in the academies of Berlin and St Petersburg, in the courts of Frederick the Great and Catherine the Great. He was perhaps the most prolific mathematician of all time, having produced more than 800 journal articles and about two dozen books, including important textbooks that educated the generation of mathematicians that followed him. Euler was equally important in the history of physics. Unlike his predecessors Newton and Huygens, whose discoveries were based on ad hoc geometrical arguments, Euler made classical physics into the subject we know today: a systematic study that reduces physical problems to differential equations, which are then solved analytically or by numerical methods.

Euler's mathematical work includes several exquisite theorems whose simplicity and elegance make them accessible and even attractive to non-mathematicians. The polyhedral formula may be the best example. Consider an ordinary three-dimensional figure, bounded by polygons. If we let F denote the number of faces on this polyhedron, E the number of edges and V the number of vertices, then the number V-E+F (now called the "Euler characteristic") must always be equal to 2. In the cube, for example, V-E+F = 8-12+6. In the sort of pyramid built by the ancient Egyptians, V-E+F = 5-8+5.

I see Richeson's book as dividing naturally into three parts. The first of these begins with a biography of Euler, followed by a history of polyhedra from Pythagoras to Kepler. Richeson then discusses Euler's work on the formula V-E+F = 2 in two papers written in 1751 or thereabouts, but published only in 1758 - even 18th-century journals had serious publication delays. Ever the great expositor, Euler published a preliminary discussion of his formula even before he could provide a formal proof of its validity. Within a few months, though, he had discovered a demonstration that the formula must hold for all convex polyhedra. Unfortunately the proof contained a subtle flaw that went unnoticed for many years, so the first part ends with a discussion of Adrien-Marie Legendre's 1794 proof of Euler's formula, the first proof to meet modern standards of rigour.

The middle portion of Euler's Gem concerns problems in graph theory, an active branch of modern mathematics with its roots in another of Euler's more accessible mathematical discoveries. By solving a problem about crossing the bridges in the city of Konigsberg, Euler established a new field, which today has applications to computer networks, utility grids and search engines. Richeson skilfully describes the relationship between Euler's polyhedral formula and various problems in graph theory, including the four-colour theorem: the assertion that in any map, a palette of only four colours suffices so that no two neighbouring land masses need have the same colour.

Topology is the focus of the final portion of Richeson's book. He gives a lucid account of the "rubber sheets" and "hollow doughnuts" of this subject, as well as the Klein bottles with no interior. All of these, as well as the geometry of curved space, can be better understood with the help of the Euler characteristic, which need not equal 2 in more exotic spaces.

Euler's Gem assumes that the reader has a keen interest in maths, but it requires remarkably little background. There is no calculus or trigonometry, for example. Of course, any reader with a phobia of equations will be made uncomfortable, but the equations here are usually quite simple and the variables within them take only whole numbers as values. Most of the reasoning is actually done by means of diagrams. Indeed, the book contains hundreds of diagrams, all beautifully rendered and seamlessly integrated into the narrative. Even without burdensome calculations, there are places where Richeson demands much from his readers, but he rewards them with a fast-paced and wide-ranging survey of some very fascinating facets of modern mathematics.