Mathematics of Seashell Shapes

By Xah Lee. Date:

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

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

seashell model
A mathematical model of the simplest seashell shape.

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]

Some seashells in Mathematica: seashell_parametric.nb; seashell_wentletrap.nb;

Gallery of Shapes

Tightness of Spiral

04020052m 09140024m 04020013m

Outline Variations

garden snail-m 04020023m thatcher DSCN0187m

Ribs And Spikes

04180006m 04020033m angulate wentletrap 04200075m 09130013m 09130014m 09130071m spider scorpian spiral 09130032m lataixis mawae

Internal Structure

cowie cut episcopal miter104 lamp chank107 276 290 295 Image-06

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

References and Sources

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

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

Seashell surface .