Did you know that yesterday was Fibonacci day? Neither did I until this morning, so my apologies for being a day late. If you are among the many people who are asking Who? or even What? I have to confess that until a few years ago I was one of them. But **Leonardo Fibonacci** is worthy of a celebratory day, not least because the awesome beauty of his findings are around us, all day and every day, in flowers and plants, in seashells and snails, in hurricanes and spiral galaxies.

Fibonacci first came to my attention as part of my daughter Lydia’s mathematics homework, and in helping her to research his work, I stumbled over many fascinating facts about this 13^{th} century Italian mathematician and the intriguing sequence of figures he discovered. I went on to write an article about it all for *The Tuscan Time*s. Since this is not regular reading for most of you, I’ve attached it below, because it also gives an opportunity to show some lovely pictures, particularly of sunflowers. Here’s *The Tuscan Times* aritcle, with some additional illustrations.

*The next time you look a sunflower in the face, don’t just marvel at its vibrant colours, nor its wondrous capacity to turn its head to follow the sun, but spare a thought for a 13th Century Italian mathematician, called Leonardo Fibonacci.*

*How many petals does your sunflower have? 34? 55? They’re ‘Fibonacci numbers. And when you look closely at the seed pods packed in its head, you’ll see that they are arranged in spirals, one winding clockwise and the other turning anticlockwise. You’ll notice that there are more spirals turning one way than the other. How many? To save you counting, let me tell you that the spirals generally come in sets: of 21 one way with 34 the other; or 34 and 55; or 55 and 89; or even 89 and 144. Fibonacci has something to say about that, too.*

*There are magnificent fields of sunflowers around Pisa, where Leonardo hailed from, but he’ll never have seen one, since these native flowers of North America were only brought to Europe by the Spanish in the 16 ^{th} Century. No, what Fibonacci was primarily interested in was rabbits!*

*Or rather, the growth of a population of rabbits. In his mathematically streamlined and biologically impractical world, Fibonacci dreamed up a pair of new-born rabbits, one male and one female, and put them into a field. They could mate when they were month old and, a month after that, the female would produce another pair of rabbits, one male and one female. In this surreal mathematical world, rabbits never die, and a mating pair always produces a new pair (one male, one female) every month.*

*So, how will the population grow? At the end of the first month there’s still one pair of rabbits: they mateand, at the end of the second month, produce a second pair. At the end of the third month the original female gives birth to another pair, so now there are three pairs. At the end of the fourth month, the original female has produced yet another new pair, and the female born two months earlier gives birth her first pair. Now there are five pairs of rabbits: the field is getting crowded. But next month there’ll be eight pairs, then 13, then 32, 34, 55, 89, 144, 233, 377… They’re breeding like rabbits!*

*This is called the Fibonacci sequence. It’s quite easy to work out: each number in the sequence is the sum of the two numbers before it: 0 in the beginning; then 1; then 0+1 =1; then 1+1=2; 1+2=3; 2+3=5; 3+5=8: 5+8=13 and on and on… Fibonacci wrote all about in a book published in 1202 (it’s in the Biblioteca Nationale in Florence), although there’s evidence that Indian mathematicians knew about these numbers centuries earlier.*

*Fibonacci numbers keep cropping up everywhere, in the abstruse equations of higher maths (there’s even a scientific journal dedicated to them) and, most important of all, in Nature, not just in sunflowers, but in other flower petals, in pine cones and pineapples, in the way trees branch and in the arrangement of leaves,*

*The reasons why are quite complicated, but in simple terms, doing things the Fibonacci way is the most efficient use of materials, space and sunlight, thus giving the biggest chance of survival and reproduction. Take those sunflower seeds: they’re round and, to pack the greatest number of seeds in the round seed head (and so increase the chances of more of them germinating), each seed, which grows from the centre of the head, must set off at a precise angle towards the edge. It’s called the Golden Angle and it is about 137.5°. Seeds pushing out one after another at that angle fill the space in the most efficient way, creating those Fibonacci spirals as they do so. The number of petals around the seed head reflects the number of spirals.*

*The Golden Angle that produced the Fibonacci numbers is closely related mathematically to the Golden Ratio*, regarded as aesthetically perfect by artists and architects for thousands of years. This ratio is the division of a line into two parts, so that the proportion of the whole line to the longer part is the same as the proportion of the longer part to the shorter part. Architects use the Golden Ratio to bring pleasing proportions to their buildings, in the Parthenon in Athens, for example, and in Filippo Brunelleschi’s innovative architecture in 15 ^{th} Century Florence, particularly the Pazzi chapel in Santa Croce. As ever, Art and Nature are closely related and mathematics (and Fibonacci’s rabbits) helps to explain why this is so.*

*As the numbers get larger, the quotient between each successive pair of Fibonacci ( …8/5, 13/8, 21/13, 34/21… and so on) numbers becomes nearer and nearer 1.618 (or its inverse 0.618). This proportion is known by many names: the Golden Ratio, the Golden Mean, and the Divine Proportion, among others.

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