Seashells? What Math Is There, There?

 I remember growing up by the beach.  We'd head out to the beach as often as we could.  You couldn't help but see seashells in various stages of being crushed.  Some were whole, some were not.

At that age, I didn't care about the mathematics of them, I just collected them complete with sand.  Its only been recently when I realized that mathematical equations explained everything we see that I wondered about the math of seashells.

Most shells can be described using some elementary mathematics which with a few changes and some programming it is possible to create 3 dimensional computer renderings of a variety of shells. Imagine rotating an expanding semi-circle upwards around a central point.

The most common equation is for an equiangular spiral or a spiral of equal angles also known as gnomonic. These are mostly the spiral shells that grow as the inhabiting animal grows. The shape is always the same as it grows larger and larger. Several mathematicians did an in depth study of shells and concluded shells were formed by expansion, rotation and twisting which are three simple processes in the mantle of the shell.  It is the opening that gets added to and grows in a spiral pattern so the shell looks curved.

Bernoulli described it as the wonder spiral due to the way the widths of the lines ran from the center to the points on the shells but the amplitudes of the angles formed by those lines and the tangents remained constant.  Descartes figured out the mathematical formula is r()=A e ^cot 
where A is the radius of Theta = 0.

If this were done in a cartesian coordinate system, the equations would appear as:
 x(θ) = r(θ) cos θ
 y(θ) = r(θ) sin θ
 with sometimes a third equation for the 3rd dimension.

This particular formula also explains the growth of animal horns, nails, corals, and snails.  Although they use basically the same formula, there are some free parameters which changes the shape so it might be a bivalve instead of a nautilus. 

There are tons of papers out there with all the actual mathematics describing every equation used to describe the growth of seashells along with those used by the computer to recreate shells.  Its quite fascinating and lots of fun to read.

Have a great day and let me now what you think, I'd love to hear.