Symmetry: a Unifying Concept

by Istvan Hargittai and Magdolna Hargittai
Shelter Publications Inc., Bolinas, CA., 1994. 222 pp., Illus., $18. ISBN 0-936070-17-X.

Reviewed by Michele Emmer

Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect. This phrase is quoted from a little and well-known book entirely dedicated by the mathematician Hermann Weyl to the theme of symmetry [1]. The volume was first published in 1952 and it was a revised version of the series of lectures that Weyl gave in the previous year at Princeton University. One of the authors of the volume Symmetry: a Unifying Concept, Istvan Hargittai, has edited two large volumes, more than 1.000 pages each, dedicated to the same topic, with almost the same title. The first one in 1986, Symmetry: Unifying Human Understanding, [2] the second one in 1989 by the title Symmetry 2: Unifying Human Understanding [3].

Ervin Y. Rodin in his foreword to the first volume explains that: "The project Symmetry presented here was an ambitious one. Its scope is tremendous, with subjects from fractals through court dances to crystallography and literature. Symmetry is really a vast subject!" In his preface to the same volume, Hargittai, editor of the volume, quotes from the letter of invitation he sent to prospective contributors: "It is, of course, commonplace today that not only is symmetry one of the fundamental concepts in science, but it is also possibly the best bridging idea crossing various branches of sciences, the arts and many other human activities. Whereas symmetry has been considered important for centuries, primarily for its aesthetic appeal, this century has witnessed a dramatic enhancement in its recognition as a cornerstone scientific principle. In addition to traditionally symmetry-oriented fields such as spectroscopy or crystallography, the concept has made headway in fields as varied as reaction chemistry, nuclear physics or the study of the origin of the Universe. At the same time, in its traditional fields its meaning and utility have greatly expanded. It may be sufficient to refer to antisymmetry, dynamical symmetry, generalized crystallography, or the symmetry analysis of music and of artistic design." Istvan and Magdolna Hargittai wrote together another volume, Symmetry through the Eyes of a Chemist, in the introduction of which they wrote the following: "Fundamental phenomena and laws of nature are related to symmetry and, accordingly, symmetry is one of science's basic concepts. Perhaps it is so important in human creations because it is omnipresent in the natural world. Symmetry is beautiful although alone it may not be enough for beauty, and absolute perfection may be even irritating. Usefulness and function and aesthetic appeal are the origins of symmetry in the worlds of technology and the arts."

In his review of Symmetry 2: Unifying Human Understanding, Roger Malina wrote that "the editor of this 1,000-page opus has accomplished an amazing task. He has found authors with new things to say about symmetry. . . . This second volume extends the discussion to new areas (robotics, economics, medical sciences) and deepens it in well-known areas (crystallography, art, physics, etc.)."

I have started my review of the new volume by Istvan and Magdolna Hargittai quoting from the prefaces and introductions to other volumes they dedicated to the theme of symmetry. Perhaps the first question after this large selection of quotations could be: is it still possible to write something new on the theme of symmetry?

Well, it is possible, of course. SYMMETRY is really such a vast subject, so many are the possible meanings of this word and its use is so different depending on the area in which it is used. This implies the danger of being vague and obvious, to put together too much different information without a common denominator. It is even difficult to answer the question: what is symmetry? This was the title of one the papers in the Hargittai's volume written by Alan L. Mackay, a crystallographer. The answer was: Symmetry is the classical Greek word YM-METPIA, the same measure, due proportion. Proportion means equal division and due implies that there is some higher moral criterion. In Greek culture due proportion in everything was the ideal.

This of course is only one of the possible answers to the question. So, why make a review of a new book on symmetry? Why have the authors written another book on symmetry? My answer to the question is that the volume adds something new to the subject. Of course the new volume is part of a large project on the theme of symmetry (titles are almost the same!) but this new book is different from the previous ones of the same authors and from other publications on the same topic. I will try to explain why. If I am not mistaken the word symmetry -- wrote Herman Weyl [1, page 3] -- is used in our everyday language in two meanings. In the one sense symmetric means something like well-proportioned, well-balanced; and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry in the objects; the synonym harmony points more toward its acoustical and musical than its geometric applications. Harmony and proportion, in particular beauty, is the main topic of the new book by the Hargittais. Much of what you will see in this book has to do with the beauty and harmony we have discovered in our travels throughout the world. We have taken photos and utilized graphic material that not only conform to one or more defined symmetry principles, but that have often appealed to our aesthetic sensibilities as well. In this phrase is the key of the book; it is a visual book, a very personal reportage of two chemists traveling around the world, making pictures and drawings, looking for symmetry in a very large sense. It is a very personal book. It is divided in two sections: the first half, chapters I-IX, deals with point-group symmetries, while the second half (chapters XI-XV) covers space-group symmetries. Chapter X is dedicated to the symmetry of opposites or antisymmetry.

All concepts are visually illustrated with very few written explanations. Quotations from literature are also included, like this poem from William Blake:

Tyger! Tyger! burning bright In the forests of the night, Tyger! Tyger! burning bright In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry?

Titles of chapters include "Bilateral Symmetry, Shape & Movement," "Right Hand, Left Hand," "Pinwheels & Windmills," "Reflection & Rotation," "Snowflakes," "Building from Above," "Cubes & Other Polyhedra," "Balloons," "Walnuts & Molecules," "Antisymmetry," "Repeating Everything," "Helix & Spiral," "Bees & Engineering," "Rhythm on the Wall," "Diamonds & Glass."

I conclude with the words of the authors: "One can only marvel at the richness and diversity in the works of symmetry. Yet what we have seen here, in our mostly visual journey, is merely the tip of the iceberg. We have just scratched the surface. All of the subjects introduced, all the photos and drawings, all the roads embarked upon here can lead in many directions and may serve to introduce you to further discoveries and newer insights." A real visual book for everybody.

P.S.: if you wish to contribute to a future book on symmetry, write to: Istvan & Magdolna Hargittai.

References:

[1] Herman Weyl, Symmetry (Princeton, NJ: Princeton University Press, 1952), p. 145.

[2] Istvan Hargittai, ed., Symmetry: Unifying Human Understanding (Oxford, U.K.: Pergamon Press, 1986); reviewed by M. Emmer, The Mathematical Intelligencer, vol.12, n.4 (1990) pp. 75-78.

[3] Istvan Hargittai, ed., Symmetry 2: Unifying Human Understanding (Oxford, U.K.: Pergamon Press, 1989); reviewed by R. Malina, Leonardo, Vol. 23, n. 2/3 (1990) p. 329.

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