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The Mathematical Lives of Plants
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Each yellow nub in the center of this daisy is actually its own miniature flower, complete with a full set of reproductive organs. The buds form interlocking clockwise and counterclockwise spirals.
Scott Hotton

The seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone all grow in whirling spiral patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand.

A surprising number of plants have spiral patterns in which each leaf, seed, or other structure follows the next at a particular angle called the golden angle. The golden angle is about 137.5º. Two radii of a circle C form the golden angle if they divide the circle into two areas A and B so that A/B = B/C.

The golden angle is closely related to the golden ratio, which the ancient Greeks studied extensively and some have believed to have divine, aesthetic or mystical properties.

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If C is the area of the whole circle and A/B = B/C, then the marked angle will be the golden angle.
J. Rehmeyer

Plants with spiral patterns related to the golden angle also display another curious mathematical property. The seeds of a flower head form interlocking spirals in both clockwise and counterclockwise directions. The number of clockwise spirals differs from the number of counterclockwise spirals, and these two numbers are called the plant's parastichy numbers (pronounced pi-RAS-tik-ee or PEHR-us-tik-ee).

These numbers have a remarkable consistency. They are almost always two consecutive Fibonacci numbers, which are another one of nature's mathematical favorites. The Fibonacci numbers form the sequence 1, 1, 2, 3, 5, 8, 13, 21 . . . , in which each number is the sum of the previous two.

The Fibonacci numbers tend to crop up wherever the golden ratio appears, because the ratio between two consecutive Fibonacci numbers happens to be close to the golden ratio. The larger the two Fibonacci numbers, the closer their ratio to the golden ratio. But this relationship doesn't fully explain why parastichy numbers end up being consecutive Fibonacci numbers.

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This sunflower has 21 clockwise and 34 counterclockwise spirals.
Scott Hotton

Scientists have puzzled over this pattern of plant growth for hundreds of years. Why would plants prefer the golden angle to any other? And how can plants possibly "know" anything about Fibonacci numbers?

Initially, researchers thought these patterns might provide an evolutionary advantage by somehow promoting plants' survival. But more recently, they have come to believe that the answer lies in the biochemistry of plants as they develop new leaves, flowers, or other structures. Scientists have not entirely solved the mystery, but a basic understanding of the process seems to be emerging. And the answers are sending botanists back to their electron microscopes to re-examine plants they thought they had already understood.

Mathematicians made the first contribution to the puzzle. In 1830, two brothers, Auguste and Louis Bravais, worked out a mathematical proof that spiral lattices generated by the golden angle have parastichy numbers that are consecutive Fibonacci numbers. But their proof still left the question of why the plants prefer the golden angle and Fibonacci numbers in the first place.

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Plants form new seeds or buds from the center. In this picture, the circle labeled 0 would be the most recent bud. The circle labeled 1 would have been formed just previously, and it forms the golden angle with bud 0. Similarly, bud 2 forms the golden angle with bud 1. Buds 0, 13, and 26 form a clockwise spiral and buds 0, 8, 16, 24, and 32 form a counterclockwise spiral.

The first suggestion that the biochemistry of plant development might provide the key came in 1868. German botanist Wilhelm Hofmeister was studying the growing tips of plants, which contain cells that haven't yet acquired a particular function in the plant. These unformed cells are called stem cells in plants and, derivatively, in animals as well. The stem cells form tiny bumps called primordia, which then turn into flowers, stems, or other plant structures.

The primordia form in a small region at the tip of a stem. Hofmeister proposed that the precise spot in which they form within that region is the spot that is furthest from older primordia. The primordia then move outward and downward along the stem as the tip continues to grow.

Images from electron microscopes have confirmed Hofmeister's theory. Furthermore, in 2000, Didier Reinhardt of the University of Fribourg worked out the biochemistry within a plant that creates this behavior. As a primordium forms, it absorbs a plant hormone called auxin that promotes growth. The most auxin is left in the area furthest from other primordia, so the primordium moves in that direction.

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An image of the tip of a Norway spruce branch, viewed through an electron microscope, shows small buds that are primordial. In this case, they will eventually turn into needles. The primordia form at the tip and then move outward and downward.
R. Rutishauser

But how does this explain the spiral patterns, golden angle, and Fibonacci numbers? Two physicists, Stéphane Douady and Yves Couder from the Laboratory for Statistical Physics in Paris, performed a compelling experiment in 1992 that tied these ideas together. They dropped magnetized drops of ferrofluid into a dish that was magnetized at its edge and filled with silicone oil. The droplets were simultaneously attracted to the edge of the dish and repelled from one another.

When the team dropped the oil in slowly, the droplets moved directly away from each other. But when they increased the speed, two older droplets would repel the new droplet simultaneously. So instead of simply marching to one side or the other, the droplet would move in a third direction—at the golden angle from the line connecting the drop's landing point with the previous droplet. The resulting pattern formed spirals.

 In this movie (click here or on the image, above, to watch), magnetized ferrofluid droplets fall into a pool of silicone oil. The droplets are attracted to the e*dge of the pool and repel one another. When they fall slowly, the droplets move in precisely opposite directions to one another. But when the speed increases, they move away from one another at the golden angle, ultimately forming spirals.Douady and Couder

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In this movie (click here or on the image, above, to watch), magnetized liquid iron droplets fall into a pool of silicon oil. The droplets are attracted to the edge of the pool and repel one another. When they fall slowly, the droplets move in precisely opposite directions to one another. But when the speed increases, they move away from one another at the golden angle, ultimately forming spirals.

Douady and Couder's result gave a beautiful analogy for plant growth, but Scott Hotton of Harvard University still wondered why the golden angle would emerge from this. He reduced Douady and Couder's experiment to a simple mathematical model, which showed that the forces Hofmeister described—outward, downward, and away from other primordia—produced golden angle spirals.

But Douady and Couder's work, along with Hotton's, had a surprising implication. Golden angle spirals weren't the only patterns that could emerge from Hofmeister's forces. The flowers could also produce their primordia at angles of approximately 99.5º. In that case, the numbers of spirals in each direction would not be Fibonacci numbers, but the closely related Lucas numbers, which begin with 1, 3, 4, 7 . . . , and continue with the sum of each two consecutive numbers forming the next number. Researchers have identified a few plants that grow in this pattern.

The researchers also found some even more peculiar possibilities. Instead of producing primordia at the same angle each time, plants could produce them at angles that vary but repeat. For instance, Hotton found that the angle could be 131, then 88, then 88 again, then 131, then 89, then 87, then 131, then 315, and then go back to 131 and start over.

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This cactus, a Mammilaria moellerana, has golden-angle spirals.
Eleanor Farrington

"What's interesting about this is that the pattern that actually forms would be hardly distinguishable from the one where the angle was the same," Hotton says. "You could actually see opposing pairs of spirals. You could count them and see that there were five in one direction and eight in the other. But the angles wouldn't be the same every time; it would be following this periodic sequence."

Do any plants show this peculiar growth pattern? Botanists are still working to find out. Some preliminary results suggest that such patterns exist, but no one has yet found any conclusive evidence.

Corrections: As originally published, this article contained several errors. Ferrofluid and silicone oil were originally misidentified, and a caption inaccurately identified the relationship among numbers in the diagram by Atela and Golé. The errors have been corrected in the text above.

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m9bnat m9bnat2
Jan. 10, 2010 at 3:30am
• For what it's worth, 30 seconds' examination of the quadratic form of the circle diagram gives the Golden Angle as pi(3 - sqrt(5)) radians exactly. A little bit more manipulation yields this as being (2 - phi) x 360 degrees, where phi is the Golden Ratio.

I believe Alan Turing did some work in this field, peripatetic intellect that he had.
David Gillies
Dec. 23, 2010 at 4:29pm
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