The Algorithmic Beauty of Sea Shells

Everyday we are confronted with systems that have an inherent tendency to change. The weather, the stock market, or the economic situation are examples. Dramatic changes can be initiated by relatively small perturbations. In the stock market, for instance, even a rumour may be sufficient to trigger sales, lowering quotations and causing panic reactions in other shareholders.

A very important class of shell patterning is caused by pigment productions that occur only during a short time interval, followed by an inactive period without pigment production. Stripes parallel to the growing edge and oblique lines belong to this class of patterns. Oscillations can occur if the antagonist reacts too slowly. A change in activator concentration cannot be immediately regulated again causing activation to proceed in a burst-like manner. Only after a sufficient accumulation of the inhibitor, or after a severe depletion of the substrate, will activator production collapse. A refractory period will follow with very low activator production in which either the excess inhibitor will degrade or the substrate will accumulate until a new activation becomes possible. The condition for oscillatory activations is the reverse of that given for stable patterns. In an activator-inhibitor scheme, oscillations occur if the decay rate of the inhibitor is smaller than that of the activator i.e., if the condition r _b > r _a in Equation 2.1 (page 23) is satisfied.

A widely distributed subgroup of shell patterns result from the superposition of a stable and a periodic pattern. The upper shell in Figure 4.1 shows two sets of parallel relief-like lines. One set is oriented parallel to the growing edge and results from a thickening of the shell at periodic time intervals. The other set is oriented parallel to the direction of growth and results from a permanently enhanced deposition of shell material at regularly spaced positions. In this example, the two patterns do not interfere with each other, a situation that is more the exception than the rule, but it shows that the assumption of two superimposed systems is reasonable.

Many shells display simple periodic patterns that cannot be accounted for with the elementary mechanisms described so far. Patterns of staggered dots and mesh-works belong in this class (Figure 5.1). These patterns are characterized by a periodicity along the time coordinate as well as along the space coordinate. This suggests that two antagonists are involved: a non-diffusible one that is responsible for the periodicity in time, and a second highly diffusible one that causes the pattern through space. The interactions described by the equations in this chapter are possible extensions of the activator-substrate and the activator-inhibitor model (see boxes). An important property of such mechanisms is that travelling waves can penetrate each other without annihilation. In other words, crossings of oblique lines can occur.

In the mechanisms discussed so far, only information exchange between neighbouring cells has been considered. However, several patterns indicate that particular events occur simultaneously at very distant positions. For instance, the shell of Oliva porphyria (Figure 6.1) shows oblique lines with branching. A branch along an oblique line indicates the sudden formation of a backward wave. Many branches are initiated at the same time at distant locations. The dashed line in Figure 6.2b indicates such a moment. There would not be enough time for a signal to travel over such a long distance by diffusion. On the other hand, not all lines branch at this critical moment. This indicates that from time to time a signal is generated everywhere in the mantle gland that greatly enhances the probability of wave splitting. Such a signal must be the result of a global control.

Many shells show patterns far more complex than those simulated so far. Figure 7.1 contains a collection of typical complex shell patterns. To show their inherent similarities, they are arranged such that each subsequent pattern contains elements of the preceding pattern as well as new features. Conus marmoreus (Figure 7.1a) shows white drop-like regions on a dark pigmented background. In Conus nobilis marchionatus (Figure 7.1b) the white drops are enlarged at the expense of the pigmented regions. The pattern is reminiscent of staggered wine glasses. Conus pennaceus (Figure 7.1c) shows, in addition, dark lines on a pigmented background, occasionally interrupted by small white drops. In Conus auratus (Figure 7.1d) the dark lines are maintained without the white drops. Instead, a periodic large-scale transition to oblique lines with crossings occurs. Shortly before this transition the continuous background resolves into narrow lines parallel to the growing edge (arrow). Such axially oriented parallel lines on top of a pigmented background are a characteristic pattern element in Conus textile (Figure 7.1e). Unpigmented regions with a drop-like shape occasionally appear. Their lower borders are formed by a dark pigmented line. In regions without pigmented background the pattern displays fine lines with wine glass shape as mentioned above. Similar parallel lines with occasional loops (tongues) are characteristic of Clithon (or Neritina) oualaniensis (Figure 7.1f). Here, however, the regions with fine lines are missing.

Several molluscs display triangles as their basic pattern element. The triangles may be connected to each other to form oblique lines with a triangular substructure. If both corners of the lower edge give rise to new triangles, the white regions in between also have a triangular shape although with opposite orientation. The triangles may cover different portions of the shell. If they are densely packed, it appears as if white triangles are arranged on a black background. The triangles can also be of very different sizes. On some shells they are a prominent pattern element, on others they appear more as a roughness in the oblique lines but are clearly visible on closer inspection. The triangles themselves may have a fine structure of lines parallel to the growing edge or they may resolve into bundles of lines parallel to the direction of growth. On some shells an almost continuous transition from triangle to branch formation can be recognized. The occurrence of triangles on very different molluscs, on bivalved mussels and on snails, indicates that the possibility of forming triangles is a basic feature of shell patterning. Figure 8.1 gives some examples. In this chapter, an attempt will be made to find a unified explanation for this diversity. I will begin with the basic features and how they can be modelled within the framework of the theory. Discrepancies with natural patterns will be used as guides to develop more complex models.

The upper shell in Figure 9.1 is decorated with many fine parallel lines. This pattern suggests the same synchronous oscillations as described earlier (see Figure 3.4). However, at particular positions, the parallel lines are deformed into U- or V-shaped gaps. The pattern on the lower shell is based on the same principle; only the size and regularity of the gaps are different. On the upper shell the gaps are restricted to particular positions. On the lower shell two broad bands are nearly free of parallel lines while smaller gaps appear at more scattered positions. The shells belong to the species Clithon oualaniensis (in older literature also termed Neritina or Theodoxus oualaniensis). These small brackwater snails are frequent on shores around India and Sri Lanka and display an incredible richness of patterns. Grüneberg (1976) made a careful study of the polymorphism of these shells. He termed the deviation from parallel straight lines “tongues”. His article contains many examples of different types of patterns, transitions from one type to another, and pattern regulation after perturbation.

The program supplied is a modified version of one of my own working programs. Originally written in FORTRAN, it has been translated into BASIC and compiled with Microsoft Professional Basic PDS 7.1 and Power Basic 3.0. It should also be possible to recompile it using Microsoft Visual Basic for DOS. Do not expect the program to be as perfect as a commercial product. Consider it as an extension of the book and a tool to discover more intuitive inroads into the complex events connected with non-linear interactions. I am sure that the program is not free from errors and I cannot exclude the fact that it contains awkward parts left over from earlier versions. However, since the source code is provided, it should be possible to make corrections or improvements if desired. Conversion from BASIC to other programming languages should not be too difficult.