**The Mathematical Study of Mollusk Shells**

The model house has an infinite sequence of ever-smaller rooms.

A typical house (the suite of smaller rooms is represented by
a single green chamber) and its first three renovations.

- We start by supposing that the shell is growing steadily and smoothly as
a function of a parameter
`t`, which is a monotonic function of time. If we take a reference point on the growing edge of the shell, it will describe a curve`F(t)`in 3-space as the organism grows. - Since the shell grows by accretion, the shell at any time
`T`will bear the entire image of the curve up to then. At a later time`T+h`the image will be the curve`F(t+h)`for the same`t`-values. The "unchanging form " requirement means in particular that the new curve must be geometrically similar to the old. Since the seed point, or center, of the shell stays fixed, there must be a 3-dimensional rotation`R`and a dilation_{h}`D`which together take the first curve to the second:_{h}

`F(t+h) = R`._{h}D_{h}F(t) - Unless
`F(t)`is a straight line, the similarity transformations`R`and_{h}`D`are uniquely determined by_{h}`h`and vary smoothly with`h`. If we compare`F(t)`with`F(t+h`,_{1})`F(t+h`, and_{2})`F(t+h`, we see that the dilation_{1}+h_{2})`D`must be the composition of the dilations_{(h1 + h2)}`D`and_{h1}`D`: first you do_{h2}`D`, then you do_{h1}`D`. To dilate a figure you multiply all its dimensions by a "dilation factor." The dilation factor of_{h2}`h`must therefore be the_{1}+ h_{2}*product*of the the dilation factors of`h`and_{1}`h`. Since_{2}`h=0`must correspond to dilation factor`1`, the only possibility is for there to exist a constant`v`such that`D`is equal to dilation by a factor_{h}`e`.^{vh} - Similarly the rotation corresponding to
`h`must be the composition of the two rotations_{1}+ h_{2}`R`and_{h1}`R`, for any_{h2}`h`and_{1}`h`. The_{2}`R`therefore form what is called a 1-parameter group of rotations, and there is only one way this can happen: there is a constant_{h}`c`, and a set`(x,y,z)`of coordinates in 3-space so that, for every`h`,`R`is the rotation about the_{h}`z`-axis by an angle`ch`:

See For further thought for some more details of this calculation.`R`._{h}(x,y,z) = (x cos(ch) - y sin(ch), x sin(ch) + y cos(ch), z) - Applying the equation
`F(t+h) = R`to_{h}D_{h}F(t)`h=t`and`t=0`yields`F(t) = R`, so if_{t}D_{t}F(0)`F(0) = (x,y,z)`, in coordinates chosen as above, then the curve`F(t)`traced out on the shell must be`F(t) = e`^{vt}(x cos(ct) - y sin(ct), x sin(ct) + y cos(ct), z).

It should be
noted that when `c=0` the equiangular spiral degerates
to the straight line `F(t) = e ^{vt}(x,y,z)`. The corresponding
shells, organized as pure cones, do exist, for example in Foraminifers of
the genus Dorothia and in the fossil ammonite

- 1. Terminal growth and unchanging form
- 2. Zoning laws in Molluskville
- 3.
*Nautilus*and the Ammonites - 4. Things get more complicated in 3 dimensions, but Calculus comes to the rescue.
- 5. Mathematical models of helical shells
- 6. Ray Gildner's Java applet for shell sketching
- 7. For further thought

@ Copyright 2001, American Mathematical Society.