Monday, August 10, 2009
Sunflowers - maths, art and the pineapple
My apologies for such a long gap since my last blog - I can only plead pressure of work - some paid, some voluntary unpaid .... and a great deal unpaid and not particularly voluntary either.
But I just couldn't let this picture of sunflowers from our garden pass without a blog.
Why? .......not because it's one of the most beautiful flowers in cultivation or because it's one of the most economically valuable plants in the world.
No ..... what stirred me into action was the connection between the sunflower and a little known Italian mathematician from the thirteenth century, Renaissance artists and the pineapple.
Unlikely, I know, but let me explain.
When you look at the sunflower above, what's one of the first things to catch your eye once you've taken in the overall beauty of the flower and the purity of its yellow colour? It's the spiral patterns shown by the developing flower seeds in the centre of the flower.
It's obviously a result of the way the seeds are packed into the seed heads to get the maximum number into the space available - a triumph of evolutionary development creating such perfection? Maybe ... but there's a lot more to it than that.
Look closely at the flower and you'll see two sets of spirals radiating in opposite directions. In one direction there are 21 and in the other 34. And this is where the Italian mathematician comes in. Those two numbers are successive numbers in a sequence starting with 0 and 1, where the following number is the sum of the previous two. So the sequence goes:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
This sequence is known as the Fibonacci Numbers after Leonardo of Pisa also known as Fibonacci who lived from 1170 to 1250. The sequence of numbers was his attempt to mathematically define the growth of a population of rabbits under idealistic conditions. That may have been what he had in mind, but actually the sequence has been found very widely in nature - as in our sunflower. There they are - ninth and tenth in the sequence - 21 and 34.
Look at any sunflower and count the spirals. They will always be two successive numbers from the sequence. The huge sunflowers may have 55 and 89 spirals or even 89 and 144 - but they will always be two adjacent numbers from the sequence.
The reason is the golden ratio - and that's where the connection with art comes in.
Renaissance artists who studied classical art and architecture found that there was often a ratio (0.618 to 1) between key features. For example in Michaelangelo's statue of David many measuremnts between important features of the figure follow this ratio. Similarly on the Parthenon in Athens, the ratio between the measurements of key architectural features fitted the golden ratio. Artists felt that using the ratio in their paintings would capture the essence of classical art and used it to position figures in the canvas, or to divide it into land and sky and often in much more subtle ways.
The golden angle is the golden ratio expressed as degrees of a complete circle of 360 degrees. It works out as 137.52 degrees.
Are you keeping up at the back there?!
So, to return to our sunflower. The flower grows on the end of a shoot and if you blow up the shoot tip under a microscope the end is like a flattish circular dome. Each seed starts as a little bump just to one side of the very tip of the dome and moves down the dome as it develops and as new ones are formed after it. To get the maximum number of seeds on the flower head each new bump that becomes a seed forms at exactly 137.52 degrees round the head of the dome from the previous one and so on. It has to be accurate - just a small error and the whole thing would become chaotic - the spiral patterns would disappear and the exquisite packing of the seeds would be lost.
But nature doesn't get it wrong! It keeps producing those little bumps at exactly 137.52 degrees from the previous one hundreds and hundreds of times over to make a single flower head. And it repeats it millions of millions of times around the globe on every sunflower from the Andes to Ireland to Australia.
And the trick is repeated on lots more plants from daisies to pineapples - which makes our final connection.
Ever noticed that the blocks that make up the pineapple fruit are arranged in spirals? Count them up and they will usually be 8 one way and 13 the other. Numbers 7 and 8 in the Fibonacci sequence.
So that completes the connection between the sunflowers growing in our garden, an Italian mathematician, Renaissance art and the pineapple.