Symmetry

This complete and beautifully presented book reminded me of enjoying The Curves of Life, Theodore A. Cook’s classic 1914 book that explored instances of logarithmic spirals in nature and featured beautiful representations of seashell curves. Like that text, this relies little on mathematics but offers a bevy of vivid examples and eye catching illustrations.

Said by the publisher to be “[t]he first comprehensive book on the topic in half a century,” this deep exploration on the fascinating subject of symmetry takes a historical overview. It ranges from the pottery motifs of the misty past to more recent discoveries and applications in particle and high-energy physics, structural and biochemistry chemistry, genetics, brain research, business decision making, and more. The author starts from the very basic definitions of symmetry: reflection, rotation, and translation. From there the mathematical concepts never get beyond anything more complex than Euler’s Polyhedral Formula. There is also some basic linear algebra, as when matrices are multiplied to model rotations in three-dimensional space.

It is in symmetry concepts that more exploration is made, such as related ideas of dissymmetry (the minor breaking of symmetry) and antisymmetry, where symmetry is preserved in something’s opposite such as on a chessboard. Thus, this book is a march of ideas: symmetry clarified and exemplified ontologically.

The cost of entry into understanding all that this book has to offer is low. Not only enthusiasts of mathematics, but artists, engineers and others can find much to appreciate and learn here. I found it worthwhile to share sections on Penrose and non-periodic tilings with my wife, who likes to create mosaics from broken tiles but also recoils from equations of any form. Anyone that enjoys M. C. Escher will appreciate the many Escher works reproduced in here. It appears nearly every aspect of symmetry can be illustrated with an enlightening example of one of Escher’s many works.

Among the liberal excursions into photographic and illustrated examples of symmetry, many are reproduced in full color in a dedicated final section to the book. Other final sections include a long bibliography of related works and indexes by both name and subject. As a result, this book serves not only as enlightening and entertaining reading that can be entered into at any point, but a good reference work on symmetry. Admittedly, at some points one wishes that author would offer some more detail. Consider the figure that examines the structure of Bartók’s Sonata for Two Pianos and Percussion through the lens of the golden ratio. So much more could be said here to clarify the ideas of inversion, and positive and negative in the realm of musical composition. However, at over 500 pages the work is already long enough and in covering so many diverse topics touched on by symmetry, many must be left as points of departure for further study.

Tom Schulte is a PhD candidate in applied mathematics at Oakland University. He enjoys the music of Bartók and has an Escher tattoo.