topics: |  math | chaos | shape | pattern

sixteen short chapters organized into three parts: Principles and Patterns,
The Mathematical World, and Simplicity and Complexity.  Includes “Order in
Chaos”, a twenty-page précis of Does God Play Dice.  Each chapter is in turn
divided into 1–3 page segments, with text and extensively captioned color and
line illustrations intermingled in a busy fashion.  The result is that the
story proceeds on two levels: in the main text and in the pictures and
captions (like a Scientific American article).  One-,two-, and
three-dimensional patterns; crystal lattices; spots and stripes on animal
coats; waves in sea, sand, and cloud; scales in animal size and music;
seashell patterns; spirals of sunflower seeds, chemical reactions, and
hurricanes; space-time footfall patterns of animal gaits (a topic that
Stewart, Golubitsky, and Jim Collins have studied in detail); fractals, image
compression, and seashells (again); chaos and cosmology follow in dizzying
succession.

A rich and wonderful series of snapshots is presented in an engaging
manner, and we are told over and over that the patterns “are a consequence of
simple mathematical rules” (caption, p. 125).  Names are dropped (Turing’s
reaction-diffusion systems, p. 164), but things move so fast that specific
“rules” are rarely vouchsafed. The discussions of symmetry breaking on pages
152–5, of fractals on pages 158–63, of intrinsic geometry and the universe in

   Chapter 15, and the closing explanation of snowflake forms of
   Chapter 16 are
notable exceptions, but the blur of images tends to obscure the main message:
that fundamental mathematical principles (the symmetries of Euclidean space,
for example) determine a catalog of what we expect to see, while physical
laws and the mathematical models encoding them determine what we actually do
see. Stewart stresses that mathematics helps us idealize and hence better
understand the world, but this could have been a deeper and stronger book had
he applied his considerable talents to explaining some elements of his own
professional interests—normal form theory and bifurcation with
symmetry—thereby revealing more of the “underlying mathematics.”

Not only does the mathematics remain largely implicit, but many of the
images that appear in the illustrations are incompletely identified, although
some are acknowledged. For example, Constable’s painting The White Horse (now
in the Frick Collection, New York), which is reproduced unnamed on page 101
in connection with a discussion on mathematics and beauty, is mysteriously
credited to “Geoffrey Clements”.
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Ian Stewart is Professor of Mathematics at the University of Warwick, and has
written articles for Nature, New Scientist, Scientific American and many
other periodicals. He is the author of Does God Play Dice? (1990), Fearful
Symmetry (1992), The Magical Maze (1997), Life's Other Secret (1998) and
Nature's Numbers (1995), which was shortlisted for the 1996 Rhone-Poulenc
Science Book Prize. In 1995 he was awarded the MIchael Faraday Medal by the
Royal Society for the year's most significant contribution to the public
understanding of science.