# Euler's Gem

The fact that the book is reprinted in its original version as a volume of the *Princeton Science Library* is a quality label as such. For completeness, I should here also mention the subtitle: "The polyhedron formula and the birth of topology". This rules out that by the "gem" is not meant the other famous Euler formula $e^{i\pi}+1=0$, but that it concerns the polyhedral formula *V*−*E*+*F*=2 (the letters stand for Vertices, Edges and Faces).

A short introduction serves as a teaser for the reader and explains the kind of problems that will be discussed in the sequel. A reasonable way to start the full story is to give a biography of Euler. He is recognized as one of the greatest mathematical minds of all times. Next, the reader is surprised by the fact that it is not so straightforward to define a polyhedron, or at least to identify the ones that one wants to focus on. So the reader is warped back to the Greek origin when the five Platonic solids (discussed in Book XIII of Euclid's Elements) and the thirteen Archimedean solids were studied. We then have to take a leap to the Renaissance of the fifteenth century before the polyhedra and the Greek knowledge was rediscovered. Kepler later built a whole world view and a solar system on polyhedral shapes. And then came Euler, who detected his above mentioned "gem". It is so simple an observation that it comes as a real surprise that, as far as we know, it was missed by everyone so far. Some explanation may be that previously one concentrated on the vertices and the faces (or the solid angles), but Euler also considered the edges as essential components of a polyhedron. Richeson reproduces Euler's proof (1750-51), but the formula does not hold for all (regular) polyhedra. When does it hold and when does it not? The reader is expecting to read the answer in the next chapter, but then Richeson surprises again by revealing that Descartes may have been the first one to have discovered the formula because a similar relation was found posthumously in his notes (notes that were miraculously saved from oblivion). However, a complete rigorous proof was only given by Legendre (in 1794), a proof that Richeson also explains later in the book.

From the story told so far, there are already many historical mathematicians involved and Richeson gives every time some short biographical sketch to situate him (so far only men) as a person who existed and lived a life of his own. It is not just some abstract name used to identify a result. Note also the way in which Richeson builds up his story. He takes the reader along to think about what polyhedra are, for what polyhedra does the formula hold, and how could it be adopted to hold in more general case? Euler and his proof are some kind of a climax, but then Descartes shows up as an unexpected twist, and Legendre's proof is not based on properties of planar faces but (another surprise) requires geometry on a sphere with geodesics. This shows how well the book is written and how Richeson manages to fascinate the reader, and make her curious about what is coming up next.

And next chapter is again some kind of a surprise because it introduces the problem of the Bridges of Köningsburg and how Euler solved the problem which is considered as the origin of graph theory. Not so surprising though if you know that a graph consists of vertices and edges. Cauchy uses this idea by projecting a polyhedron on a plane, giving a plane graph that can be analysed to prove Euler's formula. Now Richeson's story takes off into graph theory and applications: recreational mathematics (the game of sprouts and Brussels sprouts invented by Conway), the four colour problem for planar maps (and other graph colouring problems). Graph based proofs for the polyhedra formula and generalisations concludes the graph theory subject.

Next Richeson embarks on proper topology as the rubber sheet version of the usual geometry. This requires some new concepts and a classification of all surfaces. Therefore one needs to know when a surface is or is not homeomorphic to another and thus are topologically the same. For example, a torus is a sphere with a handle, a Möbius band is the same as a cross cap, and the projective plane is a sphere with a cross cap. Classification is connected to the definition of the Euler characteristic (or Euler number as Richeson calls it). Make a finite partition of the surface and count the "rubber versions" of vertices, edges and faces, then the Euler formula gives the characteristic *χ*

which is an invariant for the surface (2−2*g* for a sphere with *g* handles, and 2−*c* for a sphere with *c*

cross caps). This characteristic and the orientability of the surface allows some classification as started by Riemann but only completed in 1907 by Dehn and Heegaard.

I have to say that, although now Richeson is still explaining things at an introductory level of topology, (and continues to do so), it will take a more persistent and motivated topological layman to follow in pace and read on. We arrived now at about two thirds of the main text of the book and the mathematical level is not decreasing for the last part. It continues with knots and Seifert surfaces (whose boundary is a knot or link), the hairy ball theorem for vector fields on a sphere and more generally the Poincaré-Hopf theorem on surfaces with boundary, Brouwer fixed point theorem, the angle excess theorem for a surface, the Gauss-Bonnet theorems about the total curvature of an orientable surface, Betti numbers, and Richeson ends with an epilogue about the Poincaré conjecture. All of these are nicely presented in a smooth and logical succession by Richeson, but they are too technical to be discussed at the level of this review. However, for example an undergraduate mathematics student should not have a serious problem to read on.

Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious. Richeson also adds in an appendix building patterns that can be used to make paper models of polyhedra, of a (square and edgy, yet topologically a perfect) torus and even (the paper realization that looks like) a Klein bottle, or a projective plane. In the text he also gives advise on how to prepare the liquid to make soap-bubble models. These are aids to help visualising the surfaces if the graphics of the text do not suffice. There is a long list of papers referred to in the text, but also an appendix with an annotated survey of recommended literature.

Except for an additional preface by the author, the book is the unaltered reprint of the original version of 2009. Thus for example the facts of Martin Gardner passing away in 2010 and Perelman refusing the Millennium Prize for proving the Poincaré conjecture were still unknown in 2009. Although the latter was to be expected since he had already declined the Fields Medal in 2006 and an EMS Prize.

The first half of the book can be considered as a popular science book on a mathematical subject written for everyone. Depending on the motivation or knowledge of the reader this might or might not include the part on graph theory. Once Richeson dives deeper into topology, it becomes more a popular science book for the mathematics student of at least an amateur mathematician. People who are interested in this book may also be interested in a more recent book by Richeson that has also been reviewed here *Tales of Impossibility*.

**Submitted by Adhemar Bultheel |

**31 / Jan / 2020