Golden Blossoms, Pi Flowers

In the head of a sunflower, the tiny florets that turn into seeds typically arrange themselves in intersecting families of spirals, one set winding clockwise and the other set winding counterclockwise. Count the number of spirals of a certain type and you are likely to get 21, 34, 55, 89, or 144. Indeed, if 34 spirals curve in one direction, there will be either 21 or 55 spirals curving in the other direction.

These numbers all belong to a sequence named for the 13th-century Italian mathematician Fibonacci. Each consecutive number is the sum of the two numbers that precede it. Thus, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, and so on. The ratios of successive terms of the Fibonacci sequence get closer and closer to a specific irrational number, often called the golden ratio. It can be calculated as (1 + √5)/2, or 1.6180339887… . For instance, the ratio 55/34 is 1.617647…, and the next ratio, 89/55, is 1.6181818… .

Fibonacci numbers (and the golden ratio) come up surprisingly often in nature, from the number of petals in various flowers to the number of spirals among the scales of 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 these spirals, you are likely to find 8, 13, or 21. How do these numbers and the golden ratio arise?

In the June 2002 Mathematics Magazine, Michael Naylor 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.

Forty-five-degree spiral.

When you plot these locations for 100 seeds, you can readily detect a spiral near the center, but a radial pattern of eight spokes becomes the dominant one farther away from the center.

"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.

Golden-ratio spiral.

"Notice how well distributed the seeds appear; there is no clumping of seeds and very little wasted space," Naylor observed. "Even though the pattern grows quite large, the distances between neighboring seeds appear to stay nearly constant."

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 on—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).

This time, the seed distribution is quite uneven. For relatively small numbers of seeds, seven spiral arms dominate the resulting pattern. Curiously, for patterns made up of thousands of seeds, an additional set of 113 spiral arms becomes apparent at longer distances from the center. Why these particular numbers arise is related to rational approximations of pi's value: 22/7 and 355/113.

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 spirals—families that correspond to the numbers 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, and so on.

Root-2 spiral.

The numbers of this sequence are the numbers in the so-called Columns of Pythagoras. Start with 1 at the top of each of two columns of integers. Given a row with the numbers a and b, the first entry in the next row is a + b = c and the second entry in that row is a + c = d. The ratio of the numbers in each row converges to the irrational, numerical value of the square root of 2.

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 a computer.

Originally posted September 2, 2002

References:
Naylor, M. 2002. Golden, √2, and π flowers: A spiral story. Mathematics Magazine 75(June):163-172.