The Sierpinski Seashell, or, Identify This Mollusk

Factual Questions
1

Part the first: The pattern

There is a well-known fractal called the Sierpinski triangle, consisting of a large number of nested equilateral triangles. One method for generating such a pattern is a simple cellular automaton, on a staggered grid. A cell is colored black if exactly one of the two cells above it is black, and white otherwise. Starting with a single black cell in the middle, you get something like


         X
        X X
       X   X
      X X X X
     X       X
    X X     X X
   X   X   X   X
  X X X X X X X X
 X               X
X X             X X

and so on. Basically, this is what you get if you color every odd number on Pascal’s triangle black, and every even number white.

Part the second: The shell

One of the seashells from my Mom’s shell collection has a pattern on it very reminiscent of the Sierpinsky triangle. In fact, it looks exactly like what you would get if you ran that cellular automaton, but occasionally randomly flipped a bit (i.e., put black where you should put white, or vice versa). The shell is a univalve, which is to say that it’s rather conchlike. It has a rounded shape, and is mostly convex (it doesn’t have spiky projections or a “lip” curved out, like some conchs do). It’s three inches in length, and about an inch and a half in diameter at its largest cross-section. There are a couple of smooth grooves just inside the shell, at one end of the opening. The predominant color is a sort of off-white, with the pattern showing up in brown. There are other shells in her collection of the same shape which show a vaguely similar pattern, but this was the only one which really looked like a Sierpinski pattern. It may have come from Florida, but Mom’s not sure about that, and given her travels, it could have been from almost anywhere on the coast of the United States (including Hawaii). Unfortunately, I do not have a picture of it, as I haven’t yet gotten the knack of focusing on small detailed objects.

Part the Third: I’m reasonably certain that it is not an opal.

Part the Fourth: The general questions

First of all, does anyone know what species would have produced this shell? And second of all, how is this pattern formed? Is it indeed, as I suspect, due to some sort of naturally-occuring cellular automaton running, with the color of each “pixel” of the shell being determined by the color of the two “pixels” immediately preceding?

2

I think we’re talking about a cone shell, like the one on this page:

http://www.gbrmpa.gov.au/corp_site/info_services/library/resources/reef_snapshots/molluscs.html

3

Conus textile

4

And apparently, you aren’t the first to have noticed this:

Wolfram’s 1D Cellular Automata

5

Those cone shells do look interesting, but I don’t think that they’re the same thing I’m seeing. The cone shell pattern looks more like overlapping scales, whereas the shell I have here has much more of the nested triangles property characteristic of a Sierpinsky fractal. The shape of the shell also looks like it might be a little different, but I’ll have to check again when I get home to make sure. Thanks for the effort, though, Demo!

6

Link to Wolfram’s 1983 paper:

Cellular Automata

7

Cymbiola innexa? This is the Sierpinski example cited in Hans Meinhardt’s book The Algorithmic Beauty of Sea Shells. Various workers have spotted these natural CAs on shells and theorized similarly (though as cited here - Waddington, Cowe, Meinhardt and others - Wolfram isn’t exactly forthcoming with credits). Here’s Meinhardt’s own page on Pigmentation patterns of shells of mollusks.

8

The pattern in that C. innexa seems to be a lot more precise than in my specimen, but the shape and color scheme look like a match. That might be it, if the size is right. Thanks!