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The Algorithmic Beauty of Sea Shells

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When I sat down in my recliner with the third edition of Hans Meinhardt's The Algorithmic Beauty of Sea Shells, I thought I was in for an enjoyably light but informative look at sea shell design.  I soon found myself seated at the kitchen table with scratch paper and pencil in hand. The presentation of this sleek, coffee table style book with its glossy pages full of stunning photos of sea shells and computer generated images belies the fact that this is a serious volume on the mathematical modeling of sea shell patterns.

To put it in a sea shell (as opposed to a nut shell), this book provides a rigorous study of the mathematical models developed by the author to model the patterns of pigmentation found on numerous sea shells. A very informative introductory chapter gives the reader an understanding of patterns in biology and their relation to dynamical systems. In particular, patterns that vary over time verses patterns that vary in space are discussed. This chapter also presents interesting connections between biology, physiology and predator/prey models.

The following eight chapters progress from models (based on an activator-inhibitor system of pigmentation release) that predict simple patterns on shells to models that incorporate several activator-inhibitor genes that operate over time and location with complicated feedback processes. Each chapter provides detailed descriptions of the interactions between the processes at work, the parameterized differential equations that describe the sought-for pattern and color computer-generated patterns alongside color images of the shells themselves.

Some of the noteworthy natural shell patterns that are modeled show elaborate branching designs and Sierpinsky's triangles. The final chapters address models that depict the formation of shells in three dimensions, details of the computer programs used and pattern formation in higher organisms such as plants, simple animals and the veins and nerves in higher animals. This volume also has a CD-ROM to accompany the text which provides more detail and time-lapse graphs of each model.

This volume takes a rigorous approach, progressing systematically through the increasingly more complex models. The style, as might be expected, is dry, but it is usually (but not always) clear. It is also repetitive: by the fifth chapter, the reader can pretty much predict what is going to be said about each model. Finally, I would have liked the abundant detail about the models to be supplemented by more detail about the actual shells: where are they found, their size, how the shells are grouped (family, location, patterns).

The text raised many questions for me, many of which could be addressed by further researchers, such as modeling of patterns in other organisms. (There are some articles in math and science journals that make a start towards this field of research.) Some questions could have been addressed briefly in the volume to add depth and completeness. In particular, what are the mechanics for how pigmentation is laid down on the sea shells, what is the chemistry of the pigmentation? Though these questions could fill whole volumes on their own, a short discussion would have given me a feeling of closure.

The CD-ROM provides additional information and could almost be studied on its own. However, it was not interactive and did not show enough images of the completed models. I would have liked to be able to change the parameters and see the results. In general, more side information on the shells and their biology would make this a more well rounded book, one that would be of interest to a wider audience. As it is written, one would need to be keenly interested in the details of shell pigmentation or dynamical systems to stay interested throughout the entire work.

This book is accessible to anyone with a knowledge of dynamical systems and differential equations. It is not meant for the well educated lay reader. It is too technical and terse for a casual read. It would make a great text for either a course at the upper undergraduate or graduate level in mathematics or biology designed around it. It would also be a good supplemental text for a course in dynamical systems, differential equations, or a mathematically based biology course.

Overall, this is an intriguing book for the right audience, one that raises many questions and quite adequately answers several of them. It has left me with a new appreciation, wonder and curiosity about the myriad patterns on shells and how they develop. I now find myself stopping to contemplate any shell I come across.

Amy Shell-Gellasch is currently a freelance math historian living in Grafenwoehr Germany while her husband is on a three year tour of duty in Germany.  She received her bachelor's degree from the University of Michigan in 1989, her master's degree from Oakland University in Rochester, Michigan in 1995, and her doctor of arts degree from the University of Illinois at Chicago in 2000.  Her dissertation was a biographical piece on mathematician Mina Rees. Most recently, she conducted research with V. Fredrick Rickey on the history of the Department of Mathematical Sciences at the United States Military Academy, where she was an Assistant Professor. 

Date Received: 
Friday, June 6, 2003
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Hans Meinhardt
The Virtual Laboratory
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Amy Shell-Gellasch

Shell patterns as dynamic systems.- Pattern formation.- Oscillation and travelling waves.- Superposition of stable and periodic patterns.- Meshwork of oblique lines and staggered dots.- Branch initiation by global control.- The big problem: two or more time-dependent patterns.- Triangles.- Parallel lines with tongues.- Shell models in three dimensions.- The computer program.- Appendix: Pattern formation in the development of higher-level organisms.

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Thursday, October 11, 2007