Chapter 8: Implications for Everyday Systems

Section 6: Growth of Plants and Animals

[Mollusc] shell model

The center of the opening of a shell is taken to trace out a helix whose {x,y,z} coordinates are given as a function of the total angle of revolution t by a^t {Cos[t], Sin[t], b}. On row (a) of page 415 the parameter a varies from 1.05 to 1.65, while on row (b) b varies from 0 to 6. The complete surface of the shell is obtained by varying both t and θ in

a^t {Cos[t] (1 + c (Cos[e] Cos[θ] + d Sin[e] Sin[θ])), Sin[t] (1 + c (Cos[e] Cos[θ] + d Sin[e] Sin[θ])), b + c (Cos[θ] Sin[e] - d Cos[e] Sin[θ])}

where c varies from 0.4 to 1.6 on row (c), d from 1 to 4 on row (d) and e from 0 to 1.2 on row (e). For many values of parameters the surface defined by this formula intersects itself. However, in an actual shell material can only be added on the outside of what already exists, and this can be represented by restricting θ to run over only part of the range -π to π. The effect of this on internal structure can be seen in the slice of the cone shell on row (b) of page 414. Most real shells follow the model described here with remarkable accuracy. There are, however, deviations in some species, most often as a result of gradual changes in parameters during the life of the organism. As the pictures in the main text show, shells of actual molluscs (both current and fossil) exist throughout a large region of parameter space. And in fact it appears that the only parameter values that are not covered are ones where the shell could not easily have been secreted by an animal because its shape is degenerate and leaves little useful room for the animal. Some regions of parameter space are more common than others, and this may be a consequence either of natural selection or of the detailed molecular biology of mollusc growth. Shells where successive whorls do not touch (as in the first picture on row (c) of page 415) appear to be significantly less common than others, perhaps because they have lower mechanical rigidity. They do however occur, though sometimes as internal rather than external shells.

From Stephen Wolfram: A New Kind of Science [citation]