Mathematics of Seashell Shapes

Seashells are a 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.

above: A mathematical model of the simplest seashell shape.

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

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

Some seashells in Mathematica and Graphing Calculator: Graphics code; seashell_wentletrap.nb; shell_para.gcf; shell_para2.gcf; shell_para3.gcf; spindle.gcf corrugated-shell.gcf seashell-tops.gcf seashell-wentletrap.gcf

Gallery of Shapes

Basic shape variations:

Some ribs and spikes:

View of internal spirals:

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.com↗)

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

“Representing Seashells Surface”, by G Lucca, Italy. http://www.mi.sanu.ac.yu/vismath/lucca/index.html (2008-02)

Wikipedia: Seashell surface↗.

© 1995-2008 by Xah Lee.