The Algorithmic Beauty of Sea Shells

1. Front Matter

   Pages i-xii

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2. Shell patterns - a natural picture book to study dynamic systems and biological pattern formation

   • Hans Meinhardt

   Pages 1-17

3. Pattern formation by local self-enhancement and long range inhibition

   • Hans Meinhardt

   Pages 18-39

4. Oscillations and traveling waves

   • Hans Meinhardt

   Pages 40-51

5. Superposition of stable and periodic patterns

   • Hans Meinhardt

   Pages 52-69

6. Crossings, meshwork of oblique lines and staggered dots: the combined action of two antagonists

   • Hans Meinhardt

   Pages 70-89

7. Branch initiation by global control

   • Hans Meinhardt

   Pages 90-103

8. The big problem: two or more time-dependent patterns that interfere with each other

   • Hans Meinhardt

   Pages 104-129

9. Triangles

   • Hans Meinhardt

   Pages 130-145

10. Parallel lines with tongues

    • Hans Meinhardt

    Pages 146-165

11. Shell models in three dimensions

    • Hans Meinhardt

    Pages 166-185

12. The computer programs

    • Hans Meinhardt

    Pages 186-203

13. Pattern formation in the development of higher organisms

    • Hans Meinhardt

    Pages 204-229

14. Pattern formation in development in which shell-related mechanisms are implicated

    • Hans Meinhardt

    Pages 230-253

15. Back Matter

    Pages 1-14

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The pigment patterns on tropical shells are of great beauty and diversity. Their mixture of regularity and irregularity is fascinating. A particular pattern seems to follow particular rules but these rules allow variations. No two shells are identical. The motionless patterns appear to be static, and, indeed, they consist of calci?ed material. However, as will be shown in this book, the underlying mechanism that generates this beauty is eminently dynamic. It has much in common with other dynamic systems that generate patterns, such as a wind-sand system that forms large dunes, or rain and erosion that form complex rami?ed river systems. On other shells the underlying mechanism has much in common with waves such as those commonly observed in the spread of an epidemic. A mollusk can only enlarge its shell at the shell margin. In most cases, only at this margin are new elements of the pigmentation pattern added. Therefore, the shell pattern preserves the record of a process that took place over time in a narrow zone at the growing edge. A certain point on the shell represents a certain moment in its history. Like a time machine one can go into the past or the future just by turning the shell back and forth. Having this complete historical record opens the possibility of decoding the generic principles behind this beauty.

• algorithms
• artificial life
• computer graphics
• computer simulation
• dynamic processes
• growth processes
• natural phenomena
• simulation
• visualization

Review of the 4th edition: It is a strikingly beautiful book, with more than a hundred color illustrations of patterns on mollusk shells, side by side with the computer simulations that reproduce them. This bestiary of pattern formation includes stripes (straight or produced by traveling waves), dots, drops, triangles, and meshwork combined in any imaginable way. It is truly remarkable that this diversity of patterns seems to have arisen in the absence of any selective pressure, as the animals that express them spend their lives buried in mud or sand. Thus, these patterns are in essence a manifestation of Nature at play. Meinhardt has a great knack for explaining pattern-forming mechanisms in a non-technical way. In the first few chapters of his book he introduces us to the basic elements of his model in accessible and vivid terms. He shows how local non-linear self-activation combined with a rapidly diffusible repressor (or limiting substrate) leads to the formation of dots and stripes from homogeneous initial conditions, a phenomenon first described by Turing in 1952 as diffusion-driven instability. He then goes on to explain how oscillating patterns and traveling waves can be produced by the same basic model with different decay and diffusion rates. Here, the author and the book are at their best. Johannes Jaeger in: Bioessays 32: 638–640, 2010 WILEY Periodicals, Inc., DOI 10.1002/bies.201000031 From the reviews of the third edition:

This is a serious volume on the mathematical modeling of sea shell patterns. … This volume takes a rigorous approach, progressing systematically … . 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 … . It has left me with a new appreciation, wonder and curiosity about the myriad patterns on shells … ." (Amy Shell-Gellasch,MAA online, January, 2005)

• MPI für Entwicklungsbiologie, Tübingen, Germany

  Hans Meinhardt

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