A Pattern Painted One Line at a Time
A shell grows only at its lip. Each new strip of shell keeps the pigment the lip carried at that moment. Turn after turn, the surface becomes a picture of time.
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Growing Only at the Edge
A shell does not grow all over. New shell is added only at the rim of its opening, the lip: the edge of the mantle lays down calcium carbonate and protein, and the lip moves forward by a narrow strip.
The colour goes in at the same time. Along the mantle edge, some cells are putting out pigment at that moment and some are not. Once made, shell does not change, so the pattern can only be painted at the lip, and what is painted stays.
Model: a textile cone, about 10 cm long, its last turn not yet complete
A Picture of Time
Think of the lip as a single row of cells. Each time the shell grows a strip, the row’s state is recorded once more, just behind the last.
So the pattern on a shell is a diagram: along the lip is space, along the growth is time. Laid flat on paper and read from top to bottom, it shows the pigment along the lip changing from moment to moment.
The paper shows the same pattern laid flat: across, along the lip; down, the order of growth
Boost Yourself, Hold Back Your Neighbours
In the model, two substances run along the lip. The activator promotes its own production and also makes an inhibitor; the inhibitor in turn slows the activator down. Both spread along the lip, and both slowly break down.
Where there is a little more activator, it grows and grows, until the inhibitor it makes holds it in check. How fast the inhibitor spreads and how long it lasts decide what pattern appears. Pigment is laid where the activator is high.
Waves That Vanish When They Meet
If the inhibitor hardly spreads and lasts a long time, activation in one cell pushes into the next and leaves inhibitor behind. That is a wave running along the lip, and on the shell it draws a slanting line.
When two waves meet head on, neither can enter the inhibited wake of the other, so both die out and draw a point. All the active cells together make one signal; when fewer cells are active, the inhibitor breaks down faster, and somewhere a new wave starts and spreads both ways: the tip of a triangle.
One Set of Equations, Several Patterns
Change a few numbers and the same equations paint different patterns. When the inhibitor spreads far and breaks down fast, peaks of activation stay put and draw stripes along the growth; peaks that blink on and off draw rows of dots.
The triangles of many sizes on the textile cone, and the crossing lines of Oliva porphyria, come from waves that travel and vanish when they meet. The Oliva values are from the model author’s published program; the other three were tuned on this page in the same equations.
Like a Cellular Automaton, but Not One
A cellular automaton is also worked out one row at a time: a row of squares, each black or white, and each square’s next state depends only on itself and its two neighbours. The picture drawn by rule 30 is often set beside the textile cone.
The likeness is real: both build row after row, and each place looks only at its neighbours. But it is a likeness of appearance. Rule 30 has two states and a rule that is a table of eight entries; the cells of the lip hold concentrations that vary smoothly, and the pattern comes from activation and inhibition.
Left of the paper, rule 30; right, the textile pattern from this page’s model
Every Turn Grows by the Same Factor
The lip moves along a spiral, and with each turn its distance from the tip of the shell is multiplied by the same factor. Such a curve is a logarithmic, or equiangular, spiral: it always meets the line to the tip at the same angle. So the shell grows larger without changing shape.
In 1966 the palaeontologist David Raup described coiled shells with a few numbers: W, how much the opening grows in one turn; D, how far the opening sits from the axis it turns round; T, how fast it moves along that axis; and the shape of the opening itself. Change them and the same rule gives an ammonite, a snail or a clam. The cone here grows 3.2 times per turn, its opening hugs the axis and moves slowly along it: each turn almost wholly wraps the last, leaving only a low stepped spire on top.
The Bench
The growth lever grows the shell from very small to complete, and the pattern appears at the lip line by line. Switch between four patterns, or set how fast the activator is made and how fast the inhibitor breaks down; a change affects only the shell grown after it. “Flatten” draws the pattern on the paper beside it.
Try thisPick “Textile cone”, turn “Inhibitor breakdown” all the way down and grow it again: do the triangles get bigger or smaller?
What the model leaves out
The pattern is a one-dimensional activator–inhibitor system with 512 cells along the lip, stepped explicitly: 12 steps per line of growth for the travelling waves, 6 for the standing peaks. A real lip gets longer as the shell grows and new cells are added; here the number of cells is fixed, so the pattern grows with the shell. A real textile cone also has yellow-brown bands and fine dark lines; the model paints a single pigment. The shell is one fixed outline swept round an axis and enlarged along a logarithmic spiral; a real shell changes shape a little over its life, and its first, larval shell is different. The faint glow at the lip only marks the newest shell; it is not a real light. The stand holds the spire so the shell has a fixed point while it grows.
Easy to get wrong
Rule 30 looks like the textile cone, but the shell’s pattern is not computed by that rule; the likeness is one of appearance. Activator–inhibitor models reproduce many shell patterns in fine detail, but they are still models: the molecules involved have not been found. Another model treats the mantle edge as a net of nerve cells; it rests on the same idea, excitation nearby and inhibition further off, and paints similar patterns.
Sources
- Meinhardt: The Algorithmic Beauty of Sea Shells (Springer, 4th ed., 2009)
- Meinhardt & Klingler (1987): A model for pattern formation on the shells of molluscs, J. Theor. Biol. 126
- Max Planck Institute for Biology: a BASIC program for the patterns of Oliva porphyria
- Max Planck Institute for Biology: Pigmentation patterns of shells of mollusks
- Wolfram: A New Kind of Science (2002), chapter 8, Biological Pigmentation Patterns
- Coombes: The Geometry and Pigmentation of Seashells (University of Nottingham lecture notes)
- Phillips (AMS Feature Column, 2001): seashells, the equiangular spiral and Raup’s parameters
- Boettiger, Ermentrout & Oster (2009): The neural origins of shell structure and pattern in aquatic mollusks, PNAS
- World Register of Marine Species: Conus textile Linnaeus, 1758
Data
The pattern model is the activator–inhibitor system of Hans Meinhardt and Martin Klingler (J. Theor. Biol. 126, 1987), as set out in Meinhardt’s The Algorithmic Beauty of Sea Shells (Springer; 1st ed. 1995, 4th ed. 2009, with contributions by Przemyslaw Prusinkiewicz and Deborah R. Fowler). The Oliva porphyria values are those of Meinhardt’s published BASIC program on the Max Planck Institute for Biology Tübingen website; the other presets are the site’s own, in the same equations. The shell’s geometry follows David M. Raup’s coiling parameters (Geometric analysis of shell coiling: general problems, J. Paleontology 40, 1966), as explained in Tony Phillips’s AMS Feature Column (2001). The comparison with cellular automata is discussed in Stephen Wolfram’s A New Kind of Science (2002), chapter 8; the popular pairing of Conus textile with rule 30 appears, among others, in Stephen Coombes’s lecture notes The Geometry and Pigmentation of Seashells (University of Nottingham). The nerve-net alternative is that of G. Bard Ermentrout, John Campbell and George Oster (The Veliger 28, 1986) and Alistair Boettiger, Bard Ermentrout and George Oster (PNAS, 2009). The shell model, its outline and stand, the simulation code and the words are the site’s own.
Follow the question
Plate 06The Arithmetic of a SunflowerCount the spirals its seeds make, and you will arrive at a famous sequence of numbers.
Plate 16Where Spots and Stripes Come FromA leopard’s spots and a fish’s stripes may come from the same simple recipe: two ingredients, one fast and one slow.
