The Mathematical Study of Mollusk Shells
4. Things get complicated in three dimensions,
The Molluskville model owed some of its simplicity to being in Flatland; when the same kind of zoning laws are extended to 3-dimensional dwellings the implementation must be more complicated, because the height of the dwelling must be scaled up along with its width and length. This forces each dwelling to approximate an infinite sequence of joined chambers. In that way, when an extra room is added on, the renovation can be similar to the original house.
The model house has an infinite sequence of ever-smaller rooms.
but Calculus comes to the Rescue
Since mollusks add onto their shells in small increments (rather than the large rooms in the illusration above) we may approximate the discrete and small by the continuous and the infinitesimal and apply Calculus to determining the mathematical consequence of terminal growth and unchanging form.
This calculation shows how the conditions of terminal growth and unchanging form imply that any reference point on the shell must, during growth, trace out a 3-dimensional equiangular spiral. This matches the D'Arcy Thompson quote at the beginning of this article.
It should be noted that when c=0 the equiangular spiral degerates to the straight line F(t) = evt(x,y,z). The corresponding shells, organized as pure cones, do exist, for example in Foraminifers of the genus Dorothia and in the fossil ammonite Didymoceras on exhibit at the American Museum of Natural History.
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