Golden spirals everywhere?

There are many different types of spirals¹; these include Archimedean, hyperbolic, and logarithmic (or equiangular) spirals. I suspect that nearly everyone has heard of the “golden ratio,” the Fibonacci sequence, and the related golden rectangles and golden spirals (often referred to in a botanical context, especially in connection with the arrangement of seeds in sunflower heads).

It is often claimed that golden rectangles and spirals can be found in many places, fitting (suitably cropped) pictures of the Parthenon in Athens, the human body, works of art (such as the Vitruvian Man by da Vinci), shells, spiral galaxies, hurricanes, curling waves, and so on. Unfortunately, the vast majority of these are spurious, for the simple reasons that (i) there are many different types of spiral, and (ii) the golden spiral is only one example of a logarithmic (or equiangular) spiral. A nautilus shell (Fig. 1) is probably the best example of something close to a logarithmic spiral because it is approximately self-similar.

We will restrict ourselves to logarithmic spirals in this month’s column; a common mathematical form in polar coordinates is r = ae^bθ, where a > 0 and b > 0. This type of spiral exhibits the abovementioned property of self-similarity: zooming in or out results in a rescaled version of the original spiral. In a true logarithmic spiral, the so-called pitch angle p is constant (see Fig. 2); in fact, b = tan p. The equation may be rewritten based on the factor (δ) by which it grows outward for every rotation of π/2 rad, i.e., r = aδ^2θ/π [For a true “golden spiral,” δ is equal to the golden number $ϕ=(1+5)/2≈1.618$⁠.]

Question 1: Show that the pitch angle $p=arctan(2lnδπ)$⁠.

Question 2:

• Make measurements of the pitch angle at several different locations on (i) the nautilus shell (Fig. 1) and (ii) the ammonite-like stone (Fig. 3) (ignore its shadow!).

• Find the resulting values of δ. What are your conclusions?