What do plants know about maths?
THE Fibonacci is a famous sequence of numbers that begins with 1. Each subsequent number is obtained by adding together the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on. The sequence takes its name from Leonardo Bonacci of Pisa who discussed it in his Liber Abaci in 1202, but it appeared much earlier than this in Indian texts.
The Fibonacci numbers have a curious tendency to arise in nature and especially in botany, for example as number of petals on a flower, or the number of visible left or right winding spirals on a pineapple or pine cone. The reasons for this are still being explored by mathematicians, botanists and other scientists, and they are connected to a very special number known as the Golden Ratio.
From successive pairs of consecutive Fibonacci numbers, we can build the list of fractions 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, and so on. If you write these fractions as decimals, you will see that they are getting closer and closer together - after the first 12 or so, they all have 1.618 as their first four digits. In fact these fractions are getting progressively closer to the Golden Ratio which is denoted by the Greek letter φ (phi). This number is approximately 1.618 but it is an irrational number, which means that it can't be precisely written as a fraction involving whole numbers, and its decimal digits continue indefinitely with no repeating pattern.
When you make squares using the Fibonacci numbers, you get an attractive spiral (see illustration).
The Fibonacci numbers and the Golden Ratio arise in many areas of science, nowhere more prominently than in the study of phyllotaxis, which is the arrangement of leaves, seeds, petals and other visible features of plants.
The summer gives us a chance to inspect the fine structures of flowers, for example the spectacular arrangement of seeds in the giant sunflower.
Starting from the centre and moving outwards, we can observe prominent spiral patterns of seeds in both the clockwise and anticlockwise directions.
If you count the number of clockwise, and anticlockwise, spirals you will most likely encounter two consecutive terms of the Fibonacci sequence, such as 21 and 34, or 34 and 55 (another seasonal example of the same phenomenon is the pattern of spiral arrangements of seeds on the surface of a strawberry).
The angle that governs the phyllotaxis of the sunflower seed head is one that guarantees the most uniform possible distribution of the seeds over the available circular space, The mathematical basis for this uniformity is that even among irrational numbers, φ has a special property. This property of φ was identified and formulated mathematically around the end of the 19th century. Amazingly, the biological processes of evolution have been quietly discovering the same fact over millions of years, and reveal it in such wonders as sunflowers and strawberries.
Dr Rachel Quinlan is senior lecturer at the School of Mathematics, Statistics and Applied Mathematics, NUI Galway