In the head of a sunflower, the tiny florets that turn into seeds are typically arranged in two intersecting families of spirals, one winding clockwise and the other winding counterclockwise. Count the number of florets along a spiral and you are likely to find 21, 34, 55, 89, or 144. Indeed, if 34 floret (or seed) rows curve in one direction, there will be either 21 or 55 rows curving in the other direction.
Fibonacci numbers (and the golden ratio) come up surprisingly often in nature, from the number of petals in various flowers to the number of scales along a spiral row in a pine cone. Pine cones and pineapples, for instance, have rows of diamond-shaped markings, or scales, which spiral around both clockwise and counterclockwise. If you count the number of scales in one of these spirals, you are likely to find 8, 13, or 21. How do these numbers and the golden ratio arise?
In the June Mathematics Magazine, Michael Naylor of Western Washington University described a simple mathematical model of how a sunflower produces its florets (and seeds). The idea underlying the model is that a sunflower produces florets one by one at the flower's center, and these push the other florets outward.
"Each seed settles into a location that turns out to have a specific constant angle of rotation relative to the previous seed," Naylor remarked. "It is this rotating seed placement that creates the spiraling patterns in the seed pod."
To simulate these spiraling patterns, Naylor described the location of any seed, k, using polar coordinates: r = √[k] and θ = k α, where r is the radial distance, α is the angle from the zero-degree line, k is seed number (starting with 1 at the center) and θ is the angle between any two seeds (which is constant).
Suppose that the seed angle is 45º (or 1/8 of a complete rotation). Seed 1 would be located at a distance of √1 and an angle of 45º. Seed 2 would be located 45º from the first seed, or 2 x 45º = 90º from the zero-degree line at a distance of √2 from the origin. Seed 3 would be located at 3 x 45º at a distance of √3, and so on. Note that seed 9 would fall on the same line as the first seed, starting a new cycle.
"Notice how close together the seeds become and how much space there is between rows of seeds," Naylor commented. "This is not a very even distribution of seeds."
You could try to get a better distribution by choosing a different seed angle, say 15º or 48º. However, if this angle is a rational fraction of one revolution, you would end up with distinct spokes, and the seed distribution would still be quite uneven.
What about a seed angle derived from the golden ratio, an irrational number? In this case, the angle would be about 0.618 revolutions or roughly 222.5º.
Why do Fibonacci numbers arise out of such a "golden" pattern?
If you number the seeds consecutively from the center, you find that the seeds closest to the zero-degree line are numbered 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so onâ€”all Fibonacci numbers. Indeed, the numbered seeds converge on the zero line, alternating above and below it. That's just how ratios of pairs of consecutive Fibonacci numbers converge to the golden ratio, alternately less than and greater than the golden ratio.
"The larger the Fibonacci numbers involved, the closer their ratio to [the golden ratio] and therefore the closer the seeds lie to the zero degree line," Naylor remarked. "It is for this reason that seeds in each spiral arm in a golden flower differ by multiples of a Fibonacci number."
What happens with other irrational numbers? Would they work just as well?
Naylor generated seed arrays in which the seed angle was derived from the irrational number pi, the ratio of a circle's circumference to its diameter. In this case, the seed angle is about 0.14159 revolutions, or roughly 50.97º.
Pi spirals: 500 seeds (left) and 2000 seeds (right).
Intriguingly, an angle of rotation related to the square root of 2 produces a remarkably even distribution of seeds. The resulting pattern also features a number of distinctive families of spiralsâ€”families that correspond to the numbers 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, and so on.
There's much more in Naylor's delightful article. It provides not only
a thought-provoking introduction to mathematical modeling in the service
of phyllotaxis but also a wonderful excuse for playing with patterns on
Copyright © 1992 by Ivars Peterson.