Friday, October 13, 2006

More on the dance

More from Michael Mayne's Learning to Dance. This time from the chapter entitled May. I just read a kid's book on the way home from town: Fibonacci's Cows, by Ray Galvin, published in NZ. Gives a very good explanation of what Fibonacci did.

We live in a world of patterns. Nightly the stars move in a circle round the sky; the perfectly shaped spiral of nautilus shell echoes the spiral of the curling, breaking wave; the patterns in the desert sand point to the laws governing the flow of sand and air. Once again we are in the world of mathematics, the way we recognise and classify the patterns that lie all about us. And nothing is more intriguing than the mathematical equations that link leaf and flower patterns with the exact proportions the Greeks used in their architecture.

Leonardo Fibonacci was born in Pisa in about 1170. When he was twenty he went to Algeria, where he learned Arabic methods of calculation. He wrote on number theory and recreational mathematics. His most famous discovery was what is known as the ‘Fibonacci sequence’, in which each number is the sum of the two previous ones: hence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144….and these turn out to be the numbers that dominate the natural world. The number of petals on most plants is a Fibonacci number: for example, 3 petals on lilies and iris, 5 on buttercups and the wild rose, 8 on delphiniums, 13 on ragwort and corn marigold, 21 on asters, 34, 55, or 89 on most members of the daisy family.
More strikingly, the Fibonacci sequence is found in the spiralling seed-heads of sunflowers, where the clockwise spirals 55, 89 or 144; on the diamond-shaped scales of pineapples, with 8 rose sloping to the left, 13 to the right; and on the spirally base of pine cones, or the spiralling florets of a cauliflower. If genetics can give a sunflower any number of seeds it likes, a daisy any number of petals, or a pine cone any number of scales, why is there such a dominance of Fibonacci numbers? The answer almost lies with the dynamics of plant growth.

Now go one stage further: take the ratio of two successive Fibonacci numbers, starting with 5, and divide each by the number before it: 5/3 = 1.666, 8/5 = 1.6, 13/8 = 1/625, 21/13 = 1/61538. The ratio settles down to a particular value, 1/618, and this is known as the Golden Number or the Golden Mean, the ideal proportion used by the ancient Greeks in building: an oblong with the proportion 1 to 1.618 found to be particularly pleasing to the human eye.
At the time of the Renaissance this proportion was seen to be the secret of what we find beautiful in the human face and body: in a well-proportioned body 1/1618 is the ratio from the top of the head to the navel, and from the navel to the ground, as it is of the finger (measuring the length of the knuckle to the end of the first joint against that of the middle joint) or that of the middle joint against the finger tip; as it is also if you measure the width of the mouth against that of the nose, or the width of the incisor against that of the adjoining tooth.

There is in mathematics what is known as the Logarithmic Spiral: if you draw within an oblong with these proportions further and ever-smaller oblongs, in which the shorter and longer sides are in this same 1/1.618 proportion, and then join the corners, the resulting spiral exactly matches that which is everywhere in nature: in the shell of the hermit crab, in the ammonite, the ram’s horn, the breaking wave.

It doesn’t stop there: 90 per cent of plants show the Fibonacci numbers in the arrangement of the leaves around their stems; and in the bee colony the number of female workers to male drones will be around the golden number of 1/1.618.
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