Mathematics of Seashell Shapes

By Xah Lee. Date: 2006-05-30

Seashells are showcasing of spirals. There are great variety of spiral shapes. Suppose we start with a circle winding around a spiral.

The circle's shape changes periodically like a sine function, creating a corrugated shell somewhat emulate that of Paper Nautilus.
If instead of a circle we have a polygon, we can simulate that of Top Shell or Cone shell.
If the rounding shape periodically changes shape to have spikes, then we might emulate shells that have horns such as the Murex shell or Venus's Comb shell.
The periodic change might also emulate those shell having ribs such as the Harper shell.

For a illustration showing the variety of seashell shapes, see: Seashell icons.

A simple seashell can be modeled using the following parametric formula:

{
2*(1 - E^(u/(6 π)))*Cos[u]*Cos[v/2]^2,
2*(-1 + E^(u/(6 π)))*Cos[v/2]^2*Sin[u],
1 - E^(u/(3 π)) - Sin[v] + E^(u/(6 π))*Sin[v]
}

The above photos show a variety of spiral shapes of seashells. For larger photos and info on these shells, see: Seashell Gallery.

The Algorithmic Beauty of Sea Shells 1998, by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R Fowler. amazon

Mike Willams has sent me various formulas, see Math Parametric Equation for Seashell