The Self-Made Tapestry: Pattern Formation in Nature

  • Philip Ball
Oxford University Press: 1998. 287pp. £18.99, $37.50
In formation: “self-made” patterns induced in a bacterial colony by chemotaxis. Credit: ELENA BUDRENE, HARVARD UNIV.

One of the great and unexpected pleasures of the play Breaking the Code by Hugh Whitemore occurred when Derek Jacobi, playing Alan Turing, faced the audience and talked about spatial patterns in fir cones, and how they can be described by the Fibonacci series. When the play first opened at the Kennedy Center, my wife and I immediately bought tickets and took the train to Washington, figuring it would close after one or two performances. We were wrong. The audience was spellbound by Jacobi's soliloquies on mathematical logic, computing and pattern formation in animals and plants. The fact that Breaking the Code then went on to Broadway and enjoyed great success is still a bit of a puzzle — are there really enough biologists and mathematicians around to sustain a good Broadway run?

Part of the key to understanding the play's success, I think, lies in everyone's interest in the origins of symmetries and patterns in nature. It may even be that our visual system, with its remarkable abilities as a pattern detector, is at the root of this interest. It goes quite deep in another sense too, a need to discover first causes. We look for patterns in space and time, of course, but we also try to understand where they come from — how radial symmetries emerge from a spherical egg, why honeycombs are hexagonal, how radiolarians create beautifully patterned exoskeletons, how snowflakes form, the universal patterns apparent in mountainscapes and why soap bubbles pack the way they do.

In his new book, Philip Ball talks about these questions at engaging length and with a balance, thoroughness and simplicity that I found remarkable. In ten chapters he moves from bubble and foam structure, subjects that can be understood by appeal to simple geometry and physics, to travelling and standing waves used by living systems to set up coordinates, and then on to fluid dynamics, community ecology, and finally a chapter on physical principles, the best introductory chapter on this subject I have read.

In a sense, the first part of the book is divided into topics inspired by D'Arcy Thompson's On Growth and Form and by Turing's remarkable paper, “The chemical basis of morphogenesis”. Although both authors searched for physical explanations, Thompson's took the form of mathematical description and mechanical principle, while for Turing the language of mathematics was used to describe the emergence of patterns from apparently featureless initial conditions. Later in the book, Ball turns to contemporary discussions of branching patterns (snowflakes, crack propagation in solids, viscous fingering, diffusion-limited aggregation), fluid flow, sand-piles and self-organized criticality.

There are very few patterns in biology, physics and chemistry that Ball fails to examine carefully, illustrating along the way principles of self-organization and thermodynamics that work at all levels and length scales. There are particularly fine expositions of bubble and foam structure, which he introduces by discussing D'Arcy Thompson's views on honeycomb symmetries.

Ball's style is historical and comparative, illustrating the twists and turns that lead from observation to understanding. This makes for fascinating reading — just when I thought I understood why bees are such excellent geometers, I was given a new insight which, it turned out, also had its problems. Perhaps the true explanation has now arrived with the work of D. Weaire and R. Phelan on bubble packing (Nature 367, 123; 1994).

Ball describes the essential features of Turing's contribution to the genesis of spatial patterns, which has proved to be a very rich source of ideas in chemistry and biology, illustrating by example how local autocatalysis and long-range inhibition can yield such amazing variety. His examples are all aptly chosen: from biology we have sea-shell and animal coat patterns, the eyespots on butterfly wings, spiral waves in Dictyostelium territories and chemotaxing bacteria on agar surfaces; and from physics, wave propagation on catalytic surfaces, Liesegang rings in tubes and flow patterns in cylindrical flames. He gives the reader a good feeling for why order can emerge from these disparate systems, which vary enormously in scale and physical mechanism.

At times the going is tough. The Belousov-Zhabotinsky reaction can finally only be understood by quantitative arguments, I think. No amount of appeal to metaphor seems to work really well. Still, as one who has failed many times at teaching this reaction, I sympathize, and see this as a generic problem when discussing complex systems. When all is said and done, in some systems and circumstances, the interested reader must model complexity by coming to grips with the mathematics of the system. Still, it is remarkable, I think, that Ball manages more often than not to do so well without formal arguments.

Many authors who cover grand and complicated topics do so by sacrificing accuracy in their search for simple language. Ball's approach is to write carefully, be sceptical and non-polemical about data and their interpretation, and give credit to heterodox views where credit is due. In sum, he is critical of the literature and its interpretation, perhaps, as he says in the introduction, because his “years at Nature magazine have exposed me to too many amazing discoveries that vanish like morning mist under close scrutiny”.

Since the appeal of patterns in nature has such a strong aesthetic component, it is a pleasure to report that this book is handsomely and accurately printed and produced. It contains many wonderful illustrations, both in colour and in black and white, and very few errors, factual or typographical.