Shells 
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 \[r(\theta )=Ae^{\theta \cot (\alpha )},\,\,\theta \geq 0,\]
where
- \(A\) is the radius associated to \(\theta=0\), that is, the distance from the origin to the initial point of the spiral (\(\theta= 0\));
- \(\alpha\) (\(0% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o}% %EndExpansion <\alpha <90% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o}% %EndExpansion \)) is the angle of the spiral (the angle, in each point \(P\) of the spiral, defined by the tangent to the spiral at \(P\) and the line from the origin to \(P\); this is a constant angle along the spiral, that explains the terminology "equiangular").
To see what happens to the spiral when we change the parameters \(\alpha\) and \(A\), click on the following app.
Observations:
- \(r(\theta _{0})\) indicates the distance of the curve point in \(\theta_{0}\) to the origin;
- If \(\alpha =90% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o}% %EndExpansion\) the equiangular spiral degenerates to a circle. Of course the animal would not be very satisfied with a circular shell, because he could not keep growing inside it; the case \(\alpha =0% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o}% %EndExpansion\) generates a line what would make the shell not a good hiding-place for its inhabitant.
- For \(\alpha\) between 0 and 90, a true spiral forms, which corresponds to an enlargement of the shell. This growth process keeps the shape of the shell and is called gnomonic. This growth pattern is so common that it is referred by many as a "law of nature".
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?
