Chains of triangles From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company

Triangles: a permanent change between spreading and breakdown Chains of triangles result from a quick succession of enlargements of the pigment-producing regions and abrupt breakdowns. The latter form the lower edges of the triangles. Frequently, the pigmentation survives only at one side, the side that points to the region that was for a longer period inactive. Since the lower edges of the triangles are straight, the breakdown cannot be caused by a spreading reaction. As a first approximation, it is assumed that a global oscillation is involved that removes the activator of the pigment system, shifting the system from a spreading steady-state to a system with pulse-like activations…

Triangles: a permanent change between spreading and breakdown The pigmentation (black) spreads into a region of high substrate concentration (green). The oscillation (red) shortens the activator half life. The figure at right is the same simulation in a space-time plot

A pattern close to the (fractal) Sierpinsky triangles

The patterns on closely related shell can display either triangles or branching lines. This is highly non-trivial since triangles require an abrupt large-scale breakdown of a steady state situation. In contrast, as mentioned in the last chapter, branching requires the opposite, the temporary transition from a pulse-like into a steady state mode of activation, allowing the trigger of a backwards wave. This issue can be resolved by the assumption that the activating and the antagonistic components of a global oscillator have opposite effects on the pigment-forming reaction…

Connected triangles versus lines with branches Triangle-formation: the pigment-forming system is in a steady state. The activator (red) of the global oscillation system destroys the activator, terminating the steady state. Pigmentation production stops abruptly almost everywhere, but can survive at the outer margin, where the activation occurred anyway in a pulse-like fashion Oblique lines with branches: the pigment-producing system forms traveling waves. The inhibitor of the global oscillation system causes a stabilization of the pigment activator, shifting the pigment system transiently into a steady-state mode. This allows branch formation.

Evidence for three antagonistic reactions: a first and fast antagonistic reaction is responsible for the width of the oblique lines. On many shells, oblique lines terminate without a preceding collision (red arrows). This suggest that an additional long-ranging antagonist is involved (pink in the simulation), causing wave termination before a collision could take place. New activations appear frequently at the position of such gaps (green arrows), indicating that there is some memory in the system that there was for a long time no pigment production (white regions). This requires a third antagonist that does not diffuse (in this simulation an inhibitor, plotted in blue).

Conclusion: Both the activator and the antagonist of a modifying reaction could change the parameters of the pigment-forming reaction. This occurs in an antagonistic way, i.e., by shifting away from or by stabilizing a steady-state mode. During a burst, these actions occur in a given sequence. The formation of triangles and of fine branching lines is understandable on the basis of small parameter changes. Details on several shell patterns indicate that even more components are involved, causing, for instance, wave termination before a collision took place. The combinatorial possibilities account for an enormous diversity of patterns.