21 August 2016

Debunking golden ratio shells (1)

Nautilus Pompilius is an octopus living quietly in the deep waters of the southwest Pacific. It floats around with tentacles trailing, which is from right to left in the still below. (Nice videos on Youtube.)

This living fossil is the last cephalopod with an external shell. If you carefully saw an empty shell in two, you see an intricate and most elegant spiral. Here is an artist's impression:

It's a very popular icon, and in the Sacred Geometry of the Universe department second only to da Vinci's Vitruvian Man. The Natural History Museum (here) refers to both at the same time. Of the nautilus shell in Sloane's collection we are told that
Its coil follows the golden ratio used by Leonardo da Vinci to create proportions pleasing to the human eye.
Since we know that da Vinci did not consider the golden ratio at all, there is every reason to distrust the golden claim for the nautilus shell. Did the Divine Architect really design his humble servant Nautilus—among all creatures great and small—with ruler and compasses? In order not to spoil your pleasure, we won't reveal the answer yet, just stay tuned.

The shell in its natural position. The beautiful shell of the Nautilus consists of successive chambers, up to 38 of them, coiled around an axis. We could begin by placing it in the natural upright position, like we would do with the skeleton of a fish, a sea horse or any living creature governed by gravity.

The animal lives in the lower and largest chamber of the many it has successively built while growing to maturity. The aperture of this last chamber deviates from the vertical by an angle α which is some 30° in rest, and oscillates from 15° to 45° in motion. The abandoned chambers in the upper part create buoyancy.

The logarithmic spiral. Most golden numberists consider the Nautilus shell placed in a strange position, and set in a rectangular frame. This is very artificial, because there is nothing rectilinear about the whole Nautilus. The natural framework to use here are polar coordinates: an angle θ (counted in degrees, radians or complete turns, called whorls) and the distance r to the center point. There is every good reason to suppose that the curve is (approximately and for the most part) a logarithmic spiral, given in polar coordinates by

 with a>0 and b>0 constant. 

There is nothing very mysterious about exponentials appearing in natural phenomena. If the rate of change of something is proportional to what there is, the differential equation modelling the phenomenon has the form y' = b y with b constant, and the solution has the form y=a ebx with a and b constant.

We'll consider a real shell in the next post, but first a few mathematical facts about the logarithmic spiral.

The only non-trivial statement is the fourth; for a proof, see here.

To describe the form of a logarithmic spiral, one can choose between

being: the constant angle between radius and tangent, the factor by which the radius grows on each quarter turn, half turn or complete turn.

(continued in part 2)