ϕ - A number worth a thousand pictures
There are many beautiful examples of how Fibonacci numbers are employed in nature. Here is explained why Fibonacci numbers and the related number ϕ are employed in nature. Their occurrence is not random or arbitrary. It is so because it must be so.
Examples in Nature
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0, if n = 0,
F(n) = 1, if n = 1,
F(n-1) + F(n-2), if n > 1
Fibonacci's numbers in Nature
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
The Sunflower
The sunflower, shown on the right, is a perfect example of the Fibonacci sequence and the corresponding "golden ratio" appearing in nature. Firstly, see how the florets are arranged in a spiral pattern both in a clockwise and counterclockwise fashion. There are 34 spirals that turn clockwise and 21 spirals that turn counterclockwise. The counter-clockwise spirals appear to grow according to the golden ratio. An approximate measure of this is that the radius of the spiral doubles with every 90° of rotation.
The advent of spiralling growth
Spiralling growth occurs on the stems and branches of plants. Lichen is one of the earliest plants to grow on land. It is the partnership (symbiosis) of fungus and an alga. The resulting growth does not involve a stem. Mosses were also early land dwellers but, like fungi, they do not use vascular structures (like veins) to grow. Ferns were the first land plant to use vascular structures. The fern branch uses a fractal pattern of growth, where each smaller branch looks like a copy of the whole. So fern branches do not have spiralling growth, but the branch growth from the trunk of a fern tree does. See here for a modern example. The picture makes it appear they are packed in a hexagonal formation, like equal-sized balls would stack if packed efficiently. The picture shows scars from dropped fronds forming horizontal lines as well as (it appears) five spirals in each direction. Fibonacci spiralling would be either 3 and 5 spirals or 5 and 8 spirals.
When you hold a hammer, everything looks like a nail
The terms Golden Ratio and Fibonacci have gotten over-worked at times and not all examples are deserving of the terms. Here is a note on this.
Copyright © 2007-2009 James Grant
