Barefoot Contessa: March 2011

Named after the Leonardo of Pisa,the greatest European mathematician of the middle ages, more commonly known as Fibonacci, the Fibonacci Sequence is defined mathematically by the relation Fn=Fn-1+Fn-2 with seed values F0=0 and F1=1. In layperson's terms, the Fibonacci Sequence is a linearly recurring sequence in which the next number is produced by adding the previous two numbers together. Thus, the sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, ….

Although this is a seemingly simple sequence of numbers, its occurrence in nature and its relationship to the Golden Number have lured many mathematicians, scientists, and artists to its mysterious properties.

Fun Facts about the Fibonacci Sequence

• Before the Fibonacci Sequence was known in Europe, it was used in ancient India for the metrical sciences, also known as prosody (the study of poetic meter).

• If you divide a Fibonacci number by the number before it, (as n approaches infinity) the ratios produced settle into what is known as the Golden Ratio or the Golden Number (approximately 1.618034).

• It was used by Fibonacci to illustrate an idealization of rabbit population growth. It has also been applied to cow population and honey bee populations.

• Two consecutive Fibonacci numbers have been found in tree branches, the number of leaves on a stem, the structure of pineapples and artichokes, etc.

• Sometimes the Fibonacci numbers are called pine cone numbers because of their application to the structure of pine cones.

• If you construct a set of rectangles in a spiral formation using Fibonacci numbers as unit lengths, the resulting spiral is very similar to the spirals on snail, nautilus, and other shells.

• The Fibonacci Sequence has been used in the visual arts because it is believed to produce aesthetically appealing images. One of the most famous artists (and mathematicians!) who used the Fibonacci Sequence in his art is Leonardo da Vinci.

• It has also been used in music, most notably by Mozart.

Fibonacci and Nature

Plants do not know about this sequence - 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.

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 . . . .

It is quite amazing that the Fibonacci number patterns occur so frequently in nature
( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature". Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves The Fibonacci numbers are also found in the family tree of honeybees.

Nautilus shell - symbol of nautre's beauty