# 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.

• 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;

### Gallery of Shapes

#### 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 , by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R Fowler. Buy at amazon

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

If you have a question, put \$5 at patreon and message me.