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middle tip was on the middle bar of the E and the bottom tip was on the bottom bar. I had assumed that when I wrote a capital E I positioned the middle bar equidistant between the top and the bottom, but Levin’s gauge showed that I was subconsciously placing the bar slightly above halfway – in such a way that it divided the height of the letter into two sections with lengths of ratio 1 to 1.618. Although I had scribbled my name with gay abandon, I had adhered to the golden mean with uncanny precision.

Levin smiled and moved on to my S. He readjusted the gauge so that the side points touched the topmost and bottommost tips of the letter and, to my further amazement, the middle one coincided exactly with the S line as it curved.

‘Spot on,’ Levin said calmly. ‘Everybody’s handwriting is in the golden proportion.’

The golden mean is the number that describes the precise ratio when a line is cut into two sections in such a way that the proportion of the entire line to the larger section is equal to the proportion of the larger section to the smaller section. In other words, when the ratio of A + B to A is equal to the ratio of A to B:

A line divided into two by the golden ratio is known as a golden section, and phi, the ratio between larger and smaller sections, can be calculated as . This is an irrational number, whose decimal expansion begins:

1.61803 39887 49894 84820…

The Greeks were fascinated by phi. They discovered it in the five-pointed star, or pentagram, which was a revered symbol of the Pythagorean Brotherhood. Euclid called it the ‘extreme and mean ratio’ and he provided a method to construct it with compass and straightedge. Since at least the Renaissancehe number has intrigued artists as well as mathematicians. The major work on the golden ratio was Luca Pacioli’s The Divine Proportion in 1509, which listed the appearance of the number in many geometric constructions, and was illustrated by Leonardo da Vinci. Pacioli concluded that the ratio was a message from God, a source of secret knowledge about the inner beauty of things.

Mathematical interest in phi comes from how it is related to the most famous sequence in maths: the Fibonacci sequence, which is the sequence that starts with 0, 1 and each subsequent term is the sum of the two previous terms:

The pentagram, a mystical symbol since ancient times, contains the golden ratio.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…

Here is how the numbers are found:

Before I show how phi and Fibonacci are connected, let’s investigate the numbers in the sequence. The natural world has a predilection for Fibonacci numbers. If you look in the garden, you will discover that for most flowers the number of petals is a Fibonacci number:

3 petals

lily and iris

5 petals

pink and buttercup

8 petals

delphinium

13 petals

marigold and ragwort

21 petals

aster

55 petals/89 petals

daisy

The flowers may not always have these numbers of petals, but the average number of petals will be a Fibonacci number. For example, there are usually three leaves on a stem of clover, a Fibonacci number. Only seldom do clovers have four leaves, which is why we consider them special. Four-leaf clovers are rare because 4 is not a Fibonacci number.

Fibonacci numbers also occur in the spiral arrangements on the surfaces of pine cones, pineapples, cauliflower and sunflowers. As the picture below shows, you can count spirals clockwise and anticlockwise. The numbers of spirals you can count in both directions are consecutive Fibonacci numbers. Pineapples usually have 5 and 8 spirals, or 8 and 13 spirals. Spruce cones tend to have 8 and 13 spirals. Sunflowers can have 21 and 34, or 34 and 55 spirals – although examples as high as 144 and 233 have been found. The more seeds there are, the higher up the sequnce the spirals will go.

The Fibonacci sequence is so called because the terms appear in Fibonacci’s Liber Abaci, in a problem about rabbits. The sequence only gained the name, however, more than 600 years after the book was published