Nature by numbers How the Fibonacci sequence of numbers occurs naturally in many plants and can be used in design

The Fibonacci sequence is a mathematical pattern that seems to crop up frequently in nature, as well as in architecture and garden design. TIM ENTWISLE studies the figures

What’s the connection between swirling spirals of seed in a sunflower head, the off-kilter symmetry of pine cones, and our fondness for odd-numbered plantings? Maybe none, as it turns out, but we can track our obsession to finding one back to at least the 13th century, to accountant and mathematician Leonardo of Pisa.

Fibonacci (as he was nicknamed a few centuries later) loved a good conundrum, including the number of rabbits you might expect from a single breeding pair. As we know in Australia, it doesn’t take long for a few bunnies to become a plague, but our friend from Pisa was interested in the detail of what happens next.

By Fibonacci’s reckoning, the first couple of rabbits will beget two more pairs, followed by 3, 5, 8, 13, 21 and so on. The ‘so on’ is the sum of the two previous numbers: the series starts with one breeding pair so the next number is also one (zero + one), then two (one + one), and we are off!

These are called Fibonacci numbers, and they seem to pop up all the time in nature, architecture and art. As does the golden ratio, also known as the divine proportion or phi, which is what you get when you divide a Fibonacci number by the one before it (see overpage).

Look at the seeds in a sunflower head or the scales on a pine cone and you will see they are arranged in rows that spiral out from the middle or apex. There is a clockwise spiral and an anticlockwise spiral. If you count the number of spirals in each direction it is often a Fibonacci number. The number of spirals in the opposite direction is usually the Fibonacci number before or after the other one.

You can also find Fibonacci numbers, or rough approximations to them, in a rosette succulent such as an agave, aloe or echeveria, or in a head of Romanesco broccoli. If you count the number of spirals, more often than not it will be a number from that series.

The reason for the pattern is rather prosaic. The most competitive and efficient place for a new leaf is to be offset by 137.5 or 222.5 degrees to the last leaf, rather than directly adjacent or opposite. This results in as many leaves as possible accessing full sunlight. Over time, this offset creates spirals of leaves, and in the most efficient packing there is often a Fibonacci number of them.

In architecture, the design of the Acropolis of Athens or the Parthenon in Rome are sometimes said to be indebted to the golden ratio. In the case of the Acropolis, the dimensions don’t quite match up and I gather there is no evidence the original design took that proportion into account. Whereas Renaissance buildings, such as the

Laurentian Library in Florence and the Notre-Dame de Paris, do have design elements based on the golden ratio, as have some Renaissance gardens.

In design generally, an odd number of something is considered more attractive than an even number if we want a relaxed and informal feel. In small gardens, this means we tend to use Fibonacci numbers like three and five when planting groups of trees. However, two and eight are in the sequence and a coppice of two or eight trees is unlikely to form a pleasing outcome. On the other hand, a mix of shrubs that reach 8m mixed with some that reach 5m – both Fibonacci numbers – is a planning tool used by some landscapers.

The golden ratio is sometimes used in gardening as well. A rectangular bed with proportions approaching 1:1.6, say 5m wide and 8m long, will look ‘about right’ to most people. Of course, if you have a long, narrow garden or a curvy space you’ll have to either break it up into little golden ratio units or create something with a different aesthetic.

The Fibonacci number and the golden ratio may well provide for an intrinsically beautiful object or collection of objects, but in most cases other practicalities and other equally attractive proportions prevail. In nature, Leonardo of Pisa’s calculations didn’t actually work in the rabbit burrow – he failed to allow for more than two offspring for a start – and they are only a rough approximation in flower and plant geometry. Still, the mathematics remains a beautiful thing.