Beehives, butterfly and shell markings, tree branches, snowflakes and cloud formations: the natural world is filled with patterns. Many of these patterns may appear to be unrelated, yet hints of similarities between them suggest that we ought to be able to trace their origins to a few common sources.

In some cases, the source of the pattern may be a purely physical process, one that can be reproduced in a lab under controlled conditions and analysed in clean, mathematical terms. But for structures formed in biological and social systems, pattern-generating mechanisms are harder to pinpoint. Natural selection (or its analogue in social or economic systems) may have so many tricks up its sleeve that it can create patterns of arbitrary form. Still, if natural selection favours particular types of complex structures, then there may be general laws at work that we can discern by applying techniques similar to those developed for physical systems. Much current interdisciplinary research focuses on the extent to which observed, selected patterns can be understood as resulting from relatively simple physical processes.

In Nature’s Patterns Philip Ball describes how scientists have approached the pattern-formation problem and explains a number of their successes. In a series of case studies he illustrates how physical and chemical processes can account for certain observed patterns in geological and biological systems. The examples he selects come from a wide range of fields, including condensed-matter physics, physical chemistry, fluid dynamics, biophysics, physiology, geology and even social-network theory. The result – though not always an easy read – is a fascinating window onto the surprisingly complex structures that emerge from simple physical principles.

Nature’s Patterns is presented as a trilogy, with separate volumes entitled Shapes, Flow and Branches each concentrating on a different aspect of pattern formation. Taken together, the three books constitute a substantially revised and expanded edition of Ball’s earlier book The Self-made Tapestry. In the preface to Shapes, part one of the trilogy, Ball states that the new format does the subject more justice, but does not say how. I agree that organizing the material into three broad categories was a good move. However, there would have been a distinct advantage to collecting all three in a single volume: the communities working on these problems overlap substantially, and the results are more satisfying when viewed as a coherent body of work.

As a science journalist, Ball’s approach is to develop historical storylines that anchor each scientific explanation to a particular researcher. His writing inspires confidence that he is getting the science right, and he works hard to show how the modern explanations were motivated, enabled and sometimes anticipated by previous work. This does serve a pedagogical purpose: in many cases the older work is easier to grasp and therefore provides a useful entree to the issue at hand. So, for those who want a general sense of how far science has come in explaining patterns and what kinds of questions arise – and who are not ready to tackle any of the mathematics – the stories that Ball presents may be just the right vehicle.

For me, however, the effect was a bit overwhelming. Each book reads more like a history of ideas than a set of scientific results, and I found it hard to keep the relevant science in mind while following long tangents of primarily historical interest. To get the full benefit, readers must be prepared to pause frequently to refresh their memory about how the current discussion fits into the developing storyline. Ball suggests that readers also take time to perform some experiments of their own. I heartily agree, and I particularly recommend making the soap-film structures. There is no substitute for watching the real process by which patterns form, seeing them in 3D, and viewing how robust or fragile they are – all of which help clarify the science that Ball is trying to explain.

While experts may find a few points where they would have chosen to emphasize different features or draw finer distinctions between systems Ball takes to be analogous, he has on the whole done an excellent job of including relevant caveats and avoiding false conceptual impressions. He does miss some important opportunities, however. It struck me as odd, for example, that Shapes includes no mention of the ordering of atoms in solids or of molecules in liquid crystals, or of the faceted shapes of crystals. In all three cases, the physics is well understood, and any of them could have provided a clean illustration of fundamental concepts like symmetry breaking and long-range order. Including them would also have strengthened the message that conceptual approaches developed in the context of traditional physics have contributed greatly to the broader field. My suggestion would have been to cut out half of the 80 pages on bubbles and beehives to make room for crystals.

Ball’s choices of which researchers to highlight are reasonable, though not comprehensive. The fact is that the broad scientific themes have been pieced together by many researchers rather than investigated definitively by one person, so Ball’s story-based approach demands that he pick from among many alternatives. He has done a good job of balancing theory and experiment, covering work done around the world and representing the relevant scientific disciplines. The net effect, however, can be a bit disorienting to someone familiar with the science. The actual development of ideas about patterns involved multiple storylines playing out in parallel, and it would be interesting to know how Ball settled on the results he chose to highlight. Given his attention to many of the historical aspects of the science, a brief discussion of his own methodology would have made a nice addition, perhaps as an expanded preface.

A deeper problem is that it is hard to articulate a clear take-home message after reading one of the books. Following Ball’s train of thought requires great powers of concentration, as one must keep hold of not only the scientific thread but also the names and contributions of numerous individuals. In the end, this makes it hard for either experts or novices to locate the big scientific themes. One way to alleviate this problem might be to read the epilogue at the end of Branches (the series’ final book) before starting out. There, Ball provides a useful guide to the overarching scientific threads that tie the topics together – competition, symmetry breaking, non-equilibrium processes, dissipation, instabilities, correlations and defects.

In the past, I have used The Self-made Tapestry as one of several textbooks for a first-year undergraduate course on self-organization and the emergence of complex structure. In a seminar course for 15 students with varying backgrounds in mathematics and a teacher there to guide them, Ball’s earlier book worked well, and I was happy to see that he has added a considerable amount of material in Nature’s Patterns to flesh it out and bring it up to date.

I would like to think that there are a number of readers who would find the trilogy an exciting and rewarding project to tackle on their own, even outside a classroom setting. This is not the type of work, however, that will be devoured by teenagers who later cite it as their motivation to pursue a career in science. Such books are often inspirational precisely because they are not really comprehensible. Ball’s trilogy is written instead for sophisticated readers who wish to expand their scientific horizons and learn new ways of appreciating the structures they encounter in everyday life.