Alex's Adventures in Numberland - Alex Bellos [121]
The more Levin looked for phi, the more he found it. ‘I found so many coincidences, I started to wonder what it was all about.’ He opened his laptop and showed me a slideshow of images, each with the three points of the gauge showing exactly where the ratio was to be found. I saw pictures of butterfly wings, peacock feathers and animal colourings, the ECG reading of a healthy human heart, paintings by Mondrian and a car.
When a rectangle is constructed so that the ratio between its sides is phi, you get what is known as a ‘golden rectangle’. This rectangle has the convenient property that if we were to cut it vertically so that one side is a square, then the other side is also a golden rectangle. The mother gives birth to a baby daughter.
Golden rectangle and logarithmic spiral.
We can continue this process to create granddaughters, great-granddaughters, ad infinitum. Now, let’s draw a quarter-circle in the largest square by using a compass, placing the point at the bottom right corner and moving the pencil from one adjacent corner to the other. Repeat in the second-largest square with the compass point at the bottom left corner, with the pencil continuing the curve for another quarter-circle, and then carry on with the smaller squares. The curve is an approximation of a logarithmic spiral.
A true logarithmic spiral will pass through the same corners of the same squares, yet it will wind itself smoothly, unlike the curve in the diagram, which will have small jumps in curvature where the quarter-circle sections meet. In a logarithmic spiral, a straight line from the centre of the spiral – the ‘pole’ – will cut the spiral curve at the same angle at all points, which is why Descartes called the logarithmic spiral an ‘equiangular spiral’.
The logarithmic spiral is one of the most bewitching curves in maths. In the seventeenth century Jakob Bernoulli was the first mathematician to investigate its properties thoroughly. He called it the spira mirabilis, the wonderful spiral. He asked to have one engraved on his tombstone, but the sculptor engraved an Archimedean spiral by mistake.
The fundamental property of the logarithmic spiral is that it never changes shape the more it grows. Bernoulli expressed this on his tombstone with the epitaph: Eadem mutata resurgo, or ‘Although changed, I shall arise the same’. The spiral rotates an infinite number of times before reaching its pole. If you took a microscope and looked at the centre of a logarithmic spiral, you would see the same shape that you would see if the logarithmic spiral above was continued until it was as big as a galaxy and you were looking at it from a different solar system. In fact, many galaxies are in the shape of logarithmic spirals. Just like a fractal, a logarithmic spiral is self-similar, that is, any smaller piece of a larger spiral has identical shape to the larger piece.
The most stunning example of a logarithmic spiral in nature is the nautilus shell. As the shell grows, each successive chamber is larger, but has the same shape as the chamber before. The only spiral that can accommodate chambers of different sizes with the same relative dimensions is Bernoulli’s spira mirabilis/span>.
As Descartes noted, a straight line from the pole of a logarithmic spiral always cuts the curve at the same angle, and this feature explains why the spiral is used by Peregrine falcons when they attack their prey. Peregrines do not swoop in a straight line, but rather bear down on prey by spiralling around it. In 2000 Vance Tucker of Duke University figured out why this is so. Falcons have eyes at the sides of their head, which means that if they want to look in front of themselves, they need to turn their