Correspondence about the Logarithmic Spirals

One of the key mathematical ideas discussed in On Growth and Form is the logarithmic spiral (also known as the equiangular spiral). In this type of spiral, the distance between the whorls continually increases. This is due to twisting at a constant rate but growing at a constant acceleration. This happens in nature, for instance, in the Nautilus shell. In such cases the new, growing tissue is added to the end of an old, already hardened, one. The new tissue does not change the shape of the shell, the shape of it remains continually similar during all of its growth. This allows the shell to grow asymmetrically (it grows at one end only), while retaining its unchanging form.

The logarithmic spiral differs from the more commonly seen Archimedes spiral where there is a constant distance between the whorls. The difference can be seen in the diagrams below.

The logarithmic spiral was first discovered by the French scientist René Descartes in 1638. On Growth and Form concentrates on the natural occurrences of this spiral. It is seen in the form of animal horns, teeth and tusks, as well as in the shells of molluscs and spiders’ webs.

In his second edition chapter on the Equiangular Spiral (which Geoffrey Thomas Bennett suggested be renamed as such), D'Arcy discusses gnomons and their relationship to the Spiral. A gnomon is any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. A numerical example is the square numbers having the successive odd numbers for their gnomons. Adding 3 and 1 gives the square of 2, adding 5 and 4 gives the square of 3, etc.

D'Arcy describes that in any triangle, one part is always a gnomon to the other. As an example, in an isosceles triangle ABC with one angle 36 degrees and the other two 72 degrees, we can bisect one of the base angles. This leaves us with a subdivided large isosceles with two isosceles triangles as subdivisions, one of which is similar to the whole figure and the other being its gnomon.