Fibonacci Numbers – and how they are related to flowers

These are not random numbers – they are members of the following sequence -

1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 etc

This sequence is known as the Fibonacci series, and is well known in mathematics. Each number is the sum of the previous two. The ratio of successive pairs tends to the so-called golden section (GS) – 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have

1/GS = 1 + GS

Plants do not know about this – they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.

This is well described in “Patterns in Nature” by Peter S Stevens (Peregrine Books) ISBN 0 14 055 114X, and, more recently, in “Nature’s Other Secrets” by Ian Stewart (Penguin Books) ISBN 0 14 025876 0. See also the older classic in this field, “On Growth and Form” by D’Arcy Thompson.

So nature isn’t trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect. The plant is responding to physical constraints, not to a mathematical rule.

The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .

If we call the golden section GS, then we have

1 / GS = GS / (1 – GS) = 1.618033989 . . . .

If we call the golden angle GA, then we have

360 / GA = GA / (360 -GA) = 1 / GS.

The life and numbers of Fibonacci

Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175AD. He was the son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence.

In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:

These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.

Sources: http://pass.maths.org.uk/issue3/fibonacci/index.html
http://www.branta.connectfree.co.uk/fibonacci.htm

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~ by trilochankaur on January 21, 2007.