Nature and mathematics combine in glorious colour

 Painting by Maggie Renner Hoffmann, a tutor at The Watermill at Posara painting holidays/vacations, Tuscany, Italy.

Happy Easter! It’s a little early for those glorious sunflowers, but to brighten our Easter Monday morning, I couldn’t resist this gloriously colourful bloom by Watermill tutor Maggie Renner Hellmann, which reminded me of a Facebook post and an article I wrote some time ago for The Florentine newspaper about the fascinating mathematics of their petals and seeds. Enjoy your day!

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 Leonardo Fibonacci, a 13th Century Italian mathematician.

How many petals does your sunflower have? 55? 89? They’re ‘Fibonacci Numbers’.

And when you look closely at the seed pods packed in the sunflower’s 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 specific sets: of 21 one way with 34 the other; or 34 and 55; or 55 and 89; or even 89 and 144. All Fibonacci Numbers.

Picture: Flavio Terenzoni

There are magnificent fields of sunflowers around Pisa, where Fibonacci lived, but he’ll never have seen one, since they are natives of North America and were only brought to Europe by the Spanish in the 16th Century.

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Sunflower fields Pisa. Picture: Andre Yakibu

No, what Fibonacci was primarily interested in was rabbits! Or rather, the growth of a population of rabbits. In his mathematically streamlined and biologically impossible 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 world, rabbits never die, and a mating pair always produces a new pair (one male, one female) every month. As month follows month the number of pairs increases faster and faster: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377… They’re breeding like, um, 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 in 1202 (it’s in the Biblioteca Nationale in Florence), although there’s evidence that Indian mathematicians knew about these numbers centuries earlier.

Divide any number in the sequence by the one before it, and as the numbers get bigger (233/144, 377/233…), the answer moves closer and closer to the Golden Ratio (about 1.618), a ‘divine number’ used for thousands of years by artists and, especially, by architects to bring pleasing proportions to their buildings. You can see it in Parthenon in Athens and in Filippo Brunelleschi’s innovative architecture in 15th Century Florence, particularly the Pazzi chapel in Santa Croce. By the way, the Golden Ratio is also intimately linked with the Golden Angle, about 137.5 degrees. But enough mathematics!

The glorious space of the Pazzi chapel in the cloisters of Santa Croce, Florence.

Fibonacci numbers keep cropping up everywhere, not just in maths but in nature. And not only in sunflowers, but in other flower petals (for example: irises: three petals; buttercups: five; corn marigolds:13; daisies: 34, 55 or even 89). You can see Fibonacci numbers in pine cones and pineapples, in the way trees branch out and in the arrangement of leaves on stem.  But why?

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 as an example: the more there are in the round seed head the better: it increases the chances of more of them germinating. To achieve maximum numbers, each seed (which grows from the centre of the head) must set off towards the edge at a precise angle. Yes, it’s 137.5°, the Golden Angle, and as they push out, they create those Fibonacci spirals — and the number of petals around the seed head reflects the number of spirals.

So, nature and art are intertwined and mathematics (and Fibonacci’s rabbits) help to explain why this is so. Something to contemplate as you pass those wonderful Tuscan fields of sunflowers.

Come and see them for yourself on one of our renowned Watermill creative courses. More about that can be found by clicking here.

The Watermill at Posara for painting, creative writing, knitting, and Italian language holidays/vacations/workshops, Tuscany, Italy.
Lois welcomes our guests with a handful of sunflowers.

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