The following comprises a set of slides on "Fibonacci Numbers in Nature".
In the commentary, it is presumed that the student is familiar with the sequence.
[If not, visit Fibonacci Numbers and the Golden Section for a wealth of information.]
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Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, several things would become apparent. First, we would find that the number of petals on a flower is often one of the Fibonacci numbers. One-petalled ...
 
white calla lily
and two-petalled flowers are not common.
 
euphorbia
Three petals are more common.
 
trillium
There are hundreds of species, both wild and cultivated, with five petals.
 
columbine
Eight-petalled flowers are not so common as five-petalled, but there are quite a number of well-known species with eight.
 
bloodroot
Thirteen, ...
 
black-eyed susan
twenty-one and thirty-four petals are also quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13, 21, 34, 55 or 89 petals are quite common.
 
shasta daisy with 21 petals
Ordinary field daisies have 34 petals ... a fact to be taken in consideration when playing "she loves me, she loves me not". In saying that daisies have 34 petals, one is generalizing about the species - but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35.
The association of Fibonacci numbers and plants is not restricted to numbers of petals. Here we have a schematic diagram of a simple plant, the sneezewort. New shoots commonly grow out at an axil, a point where a leaf springs from the main stem of a plant.
If we draw horizontal lines through the axils, we can detect obvious stages of development in the plant. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, then produce new branch shoots at the beginning of each subsequent stage. The same law applies to all branches.
Since this pattern of development mirrors the growth of the rabbits in Fibonacci's classic problem, it is not surprising then that the number of branches at any stage of development is a Fibonacci number.
Furthermore, the number of leaves in any stage will also be a Fibonacci number.

The type of growth exhibited by the sneezewort occurs also in simple tree growth, each stage of development lasting one year.
The schematic diagrams of the sneezewort and tree have been presented as though the plants were flat. This illustrates the development which leads to Fibonacci numbers, but it suppresses the characteristic of a majority of plants that successive leaves or shoots spiral around the main stem as successive stages develop. Suppose we fix our attention on some leaf on the bottom of a stem on which there is a single leaf at any one point.
If we number that leaf "0" ...
and count the leaves up the stem until we come to the one which is directly above the starting one, the number we get is generally a term of the Fibonacci sequence.
Again as we work up the stem, let us count the number of times we revolve about it.
This number, too, is generally a term of the sequence.

The arrangement of leaves can then be expressed as a ratio.  The number of leaves in our sample plant is "8", ...
and the number of revolutions "5".
Our plant is said to have phyllotaxis 5/8. Each species is characterized by its own phyllotaxis. Almost always the ratios encountered are ratios of consecutive or alternate terms of the Fibonacci sequence.

With the scale patterns of pinecones, the seed patterns of sunflowers and even the bumps on pineapples we have something rather different.
The seed-bearing scales of a pinecone are really modified leaves, crowded together and in contact with a short stem. Here we do not find phyllotaxis as it occurs with true leaves and suchlike. However, we can detect two prominent arrangements of ascending spirals growing outward from the point where it is attached to the branch.
In the pinecone pictured, eight spirals can be seen to be ascending up the cone in a clockwise direction ... 
while thirteen spirals ascend more steeply in a counterclockwise direction.

In the close-packed arrangement of tiny florets in the core of a daisy blossom, ...
we can see the phenomenon in almost two-dimensional form.
The eye sees twenty-one counterclockwise ...
and thirty-four logarithmic or equiangular spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence.

Pineapple scales are also patterned into spirals and, ...
because they are roughly hexagonal in shape, three distinct sets of spirals may be observed.
One set of 5 parallel spirals ascends at a shallow angle to the right, ...
a second set of 8 parallel spirals ascends more steeply to the left, ...
and the third set of 13 parallel spirals ascends very steeply to the right.

The Fibonacci number patterns encountered herein occur so frequently in nature that we often hear the phenomenon referred to as a "law of nature". Not all four-petalled flowers are so rare as the four-leaf clover is reputed to be and deviations, sometimes even large ones, from Fibonacci patterns have been found. If this is at all disturbing to the modern botanist, it is not at all so to the Fibonacci devotee, for whom the whole phenomenon, if not a "law", is at least a fascinating prevalent tendency!


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20-June-2011
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