Symmetry is of enormous importance in the physical sciences, for its deployment leads to great insight. It also helps to reduce the labour of calculations and, most deeply of all, underlies the classification of the fundamental particles and the forces through which they interact. Actually, that is not quite right, for it is the breaking of symmetry - such as the slight displacement of the heart to the left in the human body - that accounts for much of the richness of the world. Thus, the different properties of the four fundamental forces (electromagnetic, weak, strong and gravitational) can be ascribed to the breaking of an underlying symmetry that became apparent as the universe cooled from its initial very high temperature. At another level, the mathematical theory of symmetry, group theory, can be regarded as a component of the gradually emerging but still far-off theory of elegance and beauty.
Symmetry 2000 consists of the proceedings of a symposium held at the Wenner-Gren Centre, Stockholm, in September 2000. As such, it is a motley collection of articles featuring gems and pebbles. A high proportion of the authors give private addresses, which raises suspicions that symmetry is a romping ground for amateurs. To some extent that is true; but so was The Beagle , and look what came of that. The book interprets the title much more broadly than mathematical group theory, and a wide range of readers will be able to dig into it without too much effort and be rewarded by a number of surprises.
The first part (volume one) lies at the more rigorous end of the spectrum of presentations, with a great deal of space devoted to tiling, packing, tessellations and polytopes. M. C. Escher is an abundant and ever-agreeable source of illustrations, but there is a lot of material on applications, with much of it as wide ranging as one would expect for such an all-embracing concept. For instance, a discussion of n -dimensional semi-regular chiral polytopes rubs shoulders with an examination of the Sona tradition - pictographs and ideographs drawn in the sand by storytellers of the Chokwe peoples of eastern Angola. Then a study of the optimum packing of circles within a circle is the near-neighbour of a discussion of electroweak parity violation and bimolecular homochirality. You begin to glimpse the eclectic character of the articles.
Volume two lies at the more qualitative end of the spectrum. With self-organisation out of the way, it turns to such surprising and at first sight incongruous and whimsical inclusions as Luis Martin-Santos's engagement with symmetry and history in Franco's Spain and an examination of symmetry compositions in Pushkin's works, presented as an experience of natural scientific reading. The varied character of the collection is now brilliantly illuminated as though by a searchlight. Those who regard symmetry as the domain of quantitative argument will probably wish to spend their half of £110 at the other end of the spectrum; those of a more visionary and romantic nature will find it better to spend their half on the second volume. Armchair explorers and jackdaw collectors of the unexpected should consider buying both.
A few examples give the full flavour of symmetry as it is interpreted here: symmetry plays a role in architectural structures, including those stumbled into by nature in the course of evolution. A lattice structure is stabilised by axial forces acting between nodes, whereas a plate structure is stabilised by shear forces transferred over the intersecting line between two plates. Sea urchins have used this principle for millions of years, for this creature surrounds its soft parts with calcite plates with shear-resistant edges. Symmetry arguments show that an arbitrary plate structure can be transformed unambiguously into its dual lattice structure, which greatly facilitates the analysis of the strengths of these structures and leads to a deeper understanding of structures as diverse as radiolarians and the Eiffel Tower.
Then basket weaving. The crystallographic groups - the permitted symmetries of periodic solids - have been applied to the analysis of symmetries on Benin bronzes, smoking pipes of Ghana and cloths of Nigeria. And a study of why particular symmetries appear in African craft and art shows how five-fold symmetry emerged naturally when artisans were solving problems in basket weaving. The examples range from the weaving of fish traps and baskets to the fabrication of brooms.
I have already noted circle packing within circles. One application is to the winding of cables for suspension bridges so they can be packed closely to reduce voids between sub-cables (to minimise corrosion) and to reduce the diameter of the cross-section of a main cable. This was applied for the recently opened Akashi Bridge, at Akashi near Kobe, Japan - the main cable was made of 36,830 circles of 5 mm diameter packed into a circle. For similar reasons, solution of the circle-packing problem is relevant to cartilage-replacement surgery.
Quasi-crystals are of great interest in solid-state physics and chemistry. Conventional crystals are periodic - they fall into the same pattern of unit cells from one face of the crystal to another. Quasi-crystals can also be built out of a replicating pattern - but they are not periodic. The three-dimensional analogue of Penrose tiling is an example. One application of quasi-crystals is in alloys based on steel, the strength of which can be enhanced by incorporating a quasi-crystalline phase. Quasi-crystalline alloys show a number of interesting properties, including anomalous electrical resistance and a non-stick behaviour that makes them attractive as surface coatings. Even the mechanical properties of ordinary steel are an aspect of symmetry because a typical steel consists of a mixture of different crystalline phases (austenite, with a face-centred cubic structure, and ferrite, with body-centred structure) and the interplay between these phases renders it malleable.
Symmetry, symmetry breaking and asymmetry occur everywhere in nature, from the fundamental particles to the daisy, in which the flower head is composite and in the form of spirals based on the Fibonacci sequence (2, 3, 5, 8...). Here symmetry aids evolution, for leaf growth based on Fibonacci numbers ensures as much as possible that successive growth of leaves does not cast a shadow on the previous growth. Such patterns have a beauty of their own, being a step between the absolute periodicity of crystals and the fractal symmetry of more complex structures, such as trees and our vascular system.
People, too, especially tycoons and film stars, are living manifestations of symmetry, for the perception of symmetry is a possible indicator of quality and a sign, perhaps, that the potential mate is well nourished, healthy and used to comfortable, symmetrical environments. Animal observations suggest that individuals tend to select more symmetrical partners as mates and pollinating insects tend to prefer larger, more symmetrical flowers. Male barn swallows with symmetrical outermost tail feathers acquire a mate more rapidly than males with less symmetrical feathers. Human attractiveness is related to the symmetry of a person's face, and the effect can even be detected in monozygotic, almost but not quite symmetrical, twins.
Studies have also shown that individuals with symmetrical features engage in sexual intercourse at an earlier age than the average slightly distorted polloi . And if the inegalitarian bran-tub of genetic symmetry is not already unfair enough, it seems that the scent of more symmetrical men is more attractive to women than that of an average man, especially when a woman is ovulating. This conclusion is based on one of the more engaging experiments described in these volumes, in which T-shirts worn for a standard period by men of varying degrees of symmetry were sniffed by women at various stages of their menstrual cycle. There is no evidence that men prefer the scent of more symmetrical women.
Literary criticism is not immune from symmetry - Pushkin moves in these pages almost as much as Escher pervades the illustrations. It seems that Pushkin's The Poet is an example of the golden section, for the poem is divided into two stanzas, with the numbers of lines in approximately the ratio of the golden section (1:0.618033989...): one (the larger) describing the inspired state of the poet and the other (the lesser) the uninspired. The structure of From Pindemonte is based on the Fibonacci numbers, which ensures its relation to the golden section; but it is also fractal, of a sort, with self-similarity in its semantic organisation.
Who knows, although we physical scientists may blanche somewhat at the importation of our ideas to the arts, we may be seeing the first vestiges of what some of us hope will emerge in the future: a scientific understanding of aesthetics. To understand why a picture, poem or piece of music is a delight to the eye, ear and brain will add to our enjoyment, not diminish it. Understanding, called into action when judged appropriate, is always an enhancement of pleasure. Symmetry may be the bridge between inarticulate wonder and deep appreciation.
As should be clear, these two volumes provide an eclectic and engaging collection of articles. They will not have quite the spontaneous appeal of the first book on the mathematics of symmetry for a general readership, Sir David Brewster's The Kaleidoscope: Its History, Theory, and Construction with Application to the Fine and Useful Arts , which he probably wrote not just to educate the public but also to prevent patent infringements on his invention. Nevertheless, like the kaleidoscope itself, there is much here to delight and surprise.
Peter Atkins is professor of chemistry, University of Oxford.
Editor - I. Hargittai and T. C. Laurent
ISBN - 1 85578 149 2 (two volumes)
Publisher - Portland Press
Price - £110.00
Pages - 6