Optimal designs of mollusk shells from bivalves to snails.

Bivalve, ammonite and snail shells are described by a small number of geometrical parameters. Raup noted that the vast majority of theoretically possible shell forms do not occur in nature. The constraint factors that regulate the biased distribution of natural form have long since been an open problem in evolution. The problem of whether natural shell form is a result of optimization remains unsolved despite previous attempts. Here we solve this problem by considering the scaling exponent of shell thickness as a morphological parameter. The scaling exponent has a drastic effect on the optimal design of shell shapes. The observed characteristic shapes of natural shells are explained in a unified manner as a result of optimal utilization of shell material resources, while isometric growth in thickness leads to impossibly tight coiling.