Model I

First, let us observe a shell, like the one in the picture below, and let us analyse its shape (regarding it as a bidimensional object).

The curve that best approaches its shape is the equiangular spiral (also known as the logarithmic spiral). Its equation in polar coordinates \(r\) and \(\theta\) is given by \[r(\theta )=Ae^{\theta \cot (\alpha )},\,\,\theta \geq 0,\]


To see what happens to the spiral when we change the parameters \(\alpha\) and \(A\), click on the following applet.


In cartesian coordinates, this spiral, \(h(\theta) = (x(\theta), y(\theta))\), is given by \[\begin{cases} x(\theta) & =r(\theta)\cos(\theta)\\ y(\theta) & =r(\theta)\sin(\theta) \end{cases}\]

The same phenomena may be observed in the growth of many corals, snails and animal horns and nails.

And what might we conclude when we look to the shell as a tridimensional object?