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:;;

Gallery of Shapes

Tightness of Spiral

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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

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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.