Mathematics of Seashell Shapes

Xah Lee, 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]
}

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

Code in Graphing Calculator: shell_para.gcf; shell_para2.gcf; shell_para3.gcf; spindle.gcf; corrugated-shell.gcf; seashell-tops.gcf; seashell-wentletrap.gcf.

Gallery of Shapes

Tightness of Spiral

Outline Variations

Ribs And Spikes

Internal Structure

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 (1998), by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R Fowler. amazon

Mike Willams has sent me various formulas, see here: 20050120-mike_williams.txt.

Seashell surface.