The Cook Book

Recipe for the week of April 15 - 21

The Virtual SeaShells of Fowler, Meinhardt & Prusinkiewicz

Nearly ten years ago, Stephen Wolfram told me a tall tale about certain idyllic beaches populated by mollusks with CA rules on their shells. Cognoscenti could reach down, pick one up, and proclaim, "Aha! Rule 110." This week's soup, provided by Professor Przemek Prusinkiewicz of the University of Calgary's Department of Computer Science, substantiates the tale in the form of real and virtual Oliva porphyria. In his words,

The left shell is a real one, the right one is a simulation. Unfortunately, in spite of our best efforts, this is still easy to notice. The simulation was carried out by Deborah Fowler and myself in 1992, was first described in the SIGGRAPH '92 paper "Modeling seashells" by D. Fowler, H. Meinhardt, and myself, and is reproduced in Meinhardt's The Algorithmic Beauty of Sea Shells. The simulation uses a modified activator-inhibitor model, so it is not exactly CA-based (it is more like a numerical solution of a system of partial differential equations), but it is as close to a CA as our sea shell models get.

Download the 1024 by 768 version, bigshell.gif, for a closer look that makes a rather handsome desktop wallpaper. Readers also might want to revisit our Recipe for Jan. 1 - 7, Langton's 1d CA Applet, to see the qualitative similarities with certain Class IV one-dimensional CA rules, or perhaps the Recipe for Jan. 8 - 14, Rudy Rucker's CAPOW!, which generates dynamics closer in spirit to the activator-inhibitor system alluded to above.

So why do various mollusks paint CA-like patterns on their shells? The explanation is provided in an excellent cover story from the May-June 1995 issue of American Scientist. It was a pleasure to discover that Brian Hayes' article, in its entirety, is available online. Lacking are only the beautiful full-size cover graphic and 11 additional cover illustrations. To see those and more, and to dig deeper, consult Meinhardt's remarkable 1995 book from Springer-Verlag. Those interested in practicing their Web shopping can start here.

Professor Prusinkiewicz has been studying the formation of biological structure in plants and animals for several years now. With A. Lindenmayer, he wrote the 1990 Springer volume, The Algorithmic Beauty of Plants, to which Meinhardt's book is a sequel. The theoretical framework for both the plant and shell research is described in Prusinkiewicz' terrific hypertext notebook with Mark Hammel, Visual Models of Morphogenesis: A Guided Tour. Divided into 13 Chapters, the document includes 31 color plates, 25 Quicktime animations, and extensive references. The discussion of modeling paradigms is incisive and stimulating, while the movies of various blooming flowers are particularly captivating. This excellent resource has earned a permanent place in our Kitchen Sink.

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