3-D frontierland

October 10, 1997

Popular books on mathematics ued to be a rarity. The forbiddingly dry nature of the subject, the prevalence of equations and the perceived aversion of the public were enough to discourage the potential author. Lancelot Hogben's book Mathematics for the Million and Rouse Ball's Mathematical Recreations and Essays were notable exceptions. But Stephen Hawking's A Brief History of Time and Ian Stewart's numerous writings have shown that the barriers can be broken down. True, Hawking eschews equations and his subject strictly speaking is physics, while Stewart concentrates on the applications of mathematics in various areas of science. The lesson is that, to interest the intelligent reader, it is desirable to relate the mathematics to other more tangible elements of our experience. Mathematicians may distil the pure spirit but to sell it to the public it has to be adulterated.

Richard Cromwell's book on polyhedra passes this test. The topic itself, concerned with simple three-dimensional shapes, is easily visualised and it has a long history as part of our culture, interacting with philosophy, art and chemistry. Cromwell's treatment admirably exploits all these opportunities, with many interesting digressions and a lively historical commentary. Naturally the Greeks are given pride of place but the Egyptians and Chinese also figure prominently. Illustrations are copious and include works of art as well as mathematical models with examples of Escher to combine the two.

An elementary mathematical treatment may reach a wide public but this may entail a shallow approach that avoids or skirts the interesting deeper questions. We mathematicians know that our subject has depth and we recoil from the over-simplification that would disguise this fact. An intelligent reader might also be put off by superficial gloss.

Fortunately the subject of polyhedra combines simplicity with depth in a way that lends itself to popular exposition at a high level and this is what Cromwell achieves.

An excellent example is provided by the problem of calculating the volume of a pyramid. The corresponding problem of finding the area of a triangle is elementary and the answer is one-half the base times the height. The three-dimensional analogue has an equally simple answer, namely one-third the base area times the height, but the proof is much more subtle and involves intricate philosophical and mathematical questions. Surprisingly, it turns out that the formula for the volume of a pyramid requires an argument involving an infinite or limiting process. This was essentially known to the Greeks and its significance was fully recognised by Archimedes. While the infinitesimal calculus is a powerful tool for calculating volumes in general, it is remarkable that it should be needed for a polyhedron, a figure made up of flat pieces. Of course, the Greeks did not have access to the calculus of Newton and Leibniz, but they had a rudimentary version.

Although infinite processes seemed to be necessary for the calculation of the volume of a pyramid, there was no proof of its inevitability. Conceivably, someone might come along with a new and ingenious alternative that required only finite processes. It was Gauss who highlighted this fact and it was Hilbert who explicitly included, in his famous list of problems, enunciated in 1900, the challenge of proving that no elementary proof was possible. In fact, shortly afterwards, Max Dehn solved the problem by exhibiting two polyhedra which have the same volume but which cannot be dissected so that one can be reconstructed from the other by reassembling the pieces.

Dehn's argument does more. He assigns a number (now called Dehn's invariant) to each polyhedron and shows that this is unchanged by dissection and reconstruction. He then calculates this number for his two polyhedra (of the same volume) and shows that he gets different answers. On the other hand he demonstrates that two polyhedra with the same volume and the same Dehn invariant can always be dissected and reconstructed from each other. This answers Hilbert's question in the most definitive manner possible.

Although Cromwell does not pursue the story further it has turned out that Dehn's invariant is the first step in a more elaborate theory for higher dimensions that is at the forefront of research. It is one of the fascinations of mathematics that a topic studied by Archimedes can still stimulate new and significant work at the end of the 20th century.

Another topic with a long and continuing history is that of the regular polyhedra, starting with the five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron. Not only are they inherently beautiful but they occur in nature in various crystalline forms, most recently in the famous carbon C60 whose discovery by Sir Harold Kroto, Robert Curl and Richard Smalley led to the Nobel prize.

The symmetry of the Platonic solids, the fact that they can be rotated about various axes through appropriate angles and end up looking the same, has been one of the great impulses in the abstract mathematical treatment of symmetry: the theory of groups. This is now all-pervasive in modern mathematics and it has found remarkable applications in elementary particle physics. Cromwell's extensive treatment of the Platonic solids, and their various generalisations, takes us again to the threshold of current research.

One of the most famous formulae in mathematics relates the number V of vertices, E of edges and F of faces of a polyhedron. The formula, first enunciated by Euler in 1750, asserts that: V - E + F = 2.

For example, for a cube V = 8, E = 12, F = 6 while for an icosahedron V = 12, E = 30, F = 20. This formula does not require any regularity on the polyhedron and holds under very general conditions. This universality indicates that Euler's formula has some deep significance, taking it well beyond anything previously known. Subsequent developments over the past two and half centuries amply justify this statement. Euler's formula is now known to be a cornerstone of the new branch of geometry, called topology, which has come to dominate much of 20th-century mathematics. Topology is what is left of geometry when all measurements (distances, angles, etc) are ignored. Rather like the smile on the face of Lewis Carroll's Cheshire Cat this makes it slightly intangible. But when Christopher Columbus sailed west in search of the Indies, putting his faith in the roundness of earth, he was making a topological statement. The precise size of earth was immaterial, the important point was that by sailing west one would eventually get to the East.

Cromwell's account of Euler's formula again takes the reader to modern mathematics' frontier, the beginning of topology, though he can do no more than whet the appetite.

One of the advantages of a subject like polyhedra is that it is usually possible to provide reasonably simple proofs of the mathematical results. Pictures, in this subject, are a great help and replace, to a great extent, the use of symbols and formulae. Cromwell is meticulous in giving key proofs, wherever possible, but this is done in a leisurely fashion that does not make excessive demands on a diligent and willing reader. On the other hand, the general exposition, providing motivation and background, means that the detailed proofs can be skipped or taken on trust while preserving the story's outline.

As Cromwell himself recognises, the choice of material is very individual, reflecting the author's own interests. Thus we have many chapters devoted to the variety of regular polyhedra (including the stellated versions that stick out in various directions). This will delight readers who enjoy making models, and it gives a fairly definitive account of the topic. But there will be those who regret certain omissions. The combinatorial side is briefly treated, including an account of the famous four-colour theorem (asserting that four colours are sufficient to colour any map so that adjacent countries are distinguished).

Duality is omitted altogether on the grounds that a little is confusing and there is no room for more. Duality does deserve a book to itself but I regret that such an important and beautiful aspect was omitted. Despite these quibbles the book should have a wide and appreciative audience of all ages.

Sir Michael Atiyah has just retired as master, Trinity College, Cambridge.


Author - Peter R. Cromwell
ISBN - 0 521 55432 2
Publisher - Cambridge University Press
Price - £30.00
Pages - 451

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