Wednesday, 2 May 2007

The Golden Ratio

When I studied Philosophy at A-Level we were given a talk by a composer and he brought the golden ratio to my attention and it is one of those things, like pi for instance, that just blow your mind away. The ratio, 1:1.61803, is not just a mathematical constant but is a ratio that can be found in nature and art with such frequency that the Romans titled it 'sectio divina' or 'the divine section' and it is one of the supposed proofs put forwards for the teleological argument for the existence of God (otherwise known as the argument from design).

The golden ratio, also known as phi after the Ancient Greek mathematician who seems to have discovered it and applied it in the construction of the Parthenon, can be found everywhere. Take your fingers for instance, the ratio from the largest bone to the middle bone is phi, so is the ratio between the middle bone and the smallest bone. You also find phi in the arrangement of branches on the stem of a plant, in the growing points in a plant (the distance the shoot grows before it is strong enough to support another branch), in the replication of patterns in leaves, it has been found in the proportions of chemical compounds in crystals. It has even been found as the proportion of drones to the population of bees in a beehive.

The reason it was a composer who lectured on this subject was because phi has been found in Mozart. His piano sonatas which are often conveniently split into two parts can be found to contain phi in the proportion from from one movement to the other. Phi can also be found in the ratio between key changes in Debussy's 'Image, Reflections in Water'. More recently Shostakovich applied the golden ration to his music though from memory it seems to have not produced that pleasant a piece. Leonardo da Vinci incorporated phi into his work, the Mona Lisa being being a prime example of its exercise. The golden ratio, for many, is the key to beauty. Leon Battista Alberti, the fifteenth century Italian architect, believed that beauty was a matter of proportion and that if a body was divided up into 600 parts beauty would be ‘a Harmony of all the Parts, in whatsoever Subject it appears, fitted together with such proportion and connection, that nothing could be added, diminished or altered, but for the worse’. The proportion which he believed would secure a harmony of all the parts was phi.

We humans have an incredible ability to see patterns everywhere, phi really could just be another example of this talent at making order of chaos or it could be one of the keys to unlocking our understanding of the universe. As an aside, Pythagoras was evidently spooked by the discovery of the golden ratio because he worked to keep it secret, its discovery was punishable by death.


gary thomson said...

I didn't know that about the proportion of drones to bees in a beehive. The Wikipedia article on Fibonacci numbers explains it however. Phi and the Fibonaccis are intimately connected.

Ron Knott at Surrey has what look like some excellent pages on Fibonaccis and Phi in nature.

Why does nature like using Phi in so many plants?

The answer lies in packings - the best arrangement of objects to minimise wasted space.

Was it the golden ratio or the existence of irrational numbers that spooked Pythagoras? Well Phi is an irrational number anyway, of course.

Nice blog post Paolo:)

Paolo said...

You might be right, Pythagoras and his disciples focused much of their attention on rational numbers so it is difficult to understand their relationship with phi.

Pythagoras believed that infinity was a force for destruction in the universe and that mathematics was a war between the finite and the infinite so for him the whole numbers were the purest of numbers, I suppose 1 being the purest of numbers or the farthest from infinite. So you can begin to understand why he would be averse to irrational numbers.

The pentagram was the symbol of his society which of course exemplifies the golden ratio in that in any given straight line, the intersecting section is in the proportion of phi to the larger section of that line so he was obviously somewhat obsessed by the concept.

The more you read about the golden ratio the more it boggles the mind such as the relationship between phi and fractals.

Thanks for the links, it is a fascinating subject.