The seeds of a sunflower, the spines of a cactus, and the bracts of a pinecone all form whirling spiral patterns. Remarkable for their regularity and beauty, these natural structures also show some surprising mathematical properties.
In more than 90 percent of the spiral formations made by plants, the angle between successive elements of the spiral—the leaves on a stem, for example—is approximately the golden angle. That geometrical quantity, which is about 137.5°, is closely related to the famous “golden ratio” and was first studied by the ancient Greeks. Furthermore, hidden within the spirals of many plants is a pattern involving the celebrated Fibonacci sequence of numbers, which is closely related to the golden ratio. Spiral patterns involving the golden angle and the Fibonacci sequence pop up throughout the natural world, in objects as disparate as galaxies and seashells.
Scientists discovered these mathematical regularities in the spiral patterns of plant growth hundreds of years ago and have been puzzling over them ever since. Why would plants prefer the golden angle to any other? And how can plants know about Fibonacci numbers?
Recently, biologists, mathematicians, physicists, and computer scientists have made a start on explaining how and why plants accomplish their mathematical feats. Biologists now understand the basic biochemistry that drives patterns of plant growth, called phyllotaxis, and that knowledge has fed into mathematical and computer-based models of the process.
Some scientists are beginning to tackle an even more difficult problem: Why do some plants show peculiar phyllotactic patterns? In some spiral-growth patterns, the angle between successive elements is not the golden angle, but an angle of about 99.5°. There are even plants in which that angle varies systematically within a single spiral formation. And spiral structures in some plants transition from one pattern to another as the plants grow.
“Nature is playing a geometric game with phyllotaxis,” says Pau Atela, a mathematician at Smith College in Northampton, Mass. “We are now starting to figure out the geometric essence of what is going on.”
Do plants do math?
Two magnitudes, A and B, form the golden ratio if A/B = (A + B)/A. The golden angle is essentially the golden ratio applied to a circle: Two radii of a circle form the golden angle if they divide the circle into two areas, A and B, whose ratio is the golden ratio.
As a plant puts out leaves or seeds around some central structure, each seed or leaf spaced from the last by about the golden angle, interlocking spiral arms form in clockwise and counterclockwise directions. The number of clockwise arms never equals the number of counterclockwise arms; in fact, they are almost always two consecutive members of the Fibonacci sequence. That’s the sequence 1, 1, 2, 3, 5, 8, 13, 21, and so on, in which each number is the sum of the previous two.
The appearance of Fibonacci numbers isn’t entirely surprising, because they have an intimate connection to the golden ratio. As one proceeds along the Fibonacci sequence, the ratio of successive numbers gets ever closer to the golden ratio.
In 1837, crystallographer Auguste Bravais and his brother Louis, a botanist, used this relationship to prove that in golden-angle spirals, the numbers of clockwise and counterclockwise spiral arms must be consecutive Fibonacci numbers. But that doesn’t explain why plants prefer the golden-angle and Fibonacci numbers in the first place.
Researchers once thought that these patterns might provide an evolutionary advantage by somehow promoting plants’ survival. Mathematical models show, for example, that a golden-angle spiral packs a maximum number of leaves onto a minimum of stem space while also allowing each leaf sufficient access to light.
But because plants grow and bend in response to the irregular ways in which sunlight reaches them, the ideal packing of a golden-angle spiral is unlikely to significantly improve the efficiency of light absorption. Scientists have now come to believe that spiral phyllotaxis is a side effect of the biochemistry of growing plants.
Growth spiral
In 1868, German botanist Wilhelm Hofmeister suggested that the mechanisms of plant development might help explain spiral phyllotaxis. He was studying the growing tips of plants, which contain cells that haven’t yet acquired a specific function. These immature cells form tiny bumps called primordia, which eventually develop into flowers, leaves, or other plant structures.
Hofmeister proposed that each new primordium develops on the tip of the growing stem in the spot that is farthest from older primordia. As the tip continues to grow from its center, the primordia are pushed outward and form spiral patterns. In recent decades, electron microscope images have added support to the idea that primordia arrange themselves according to Hofmeister’s rule.
In 1992, physicists Stéphane Douady and Yves Couder of l’École Normale Supérieure in Paris performed a compelling experiment that showed how Hofmeister’s rule could explain spiral patterns. They let droplets of a magnetized liquid fall into a dish that was filled with silicone oil and magnetized along its outer edge. Magnetic forces attracted the droplets to the edge of the dish but made them repel one another.
When Douady and Couder added droplets slowly, each new droplet would move toward the side of the dish, directly opposite from the previously added drop. But when they added droplets faster, the two most recently added droplets would both strongly repel the new one. Instead of marching to one side or the other, the new droplet would move in a third direction—at the golden angle from the line connecting the drop’s landing point with the previous droplet. A stream of droplets added in this way formed a spiral pattern.
Douady and Couder’s results galvanized the study of phyllotaxis. The droplets in their experiment behaved like primordia, in that their attraction to the edge of the dish corresponded to the primordia’s outward march on the growing tip. But what biological mechanism could push the new primordia away from the older ones, in the same way that magnetization made the droplets repel one another?
In 2003, biologists Didier Reinhardt and Cris Kuhlemeier of the University of Bern in Switzerland and their collaborators published a paper in Nature fingering a plant-growth hormone called auxin as the critical factor.
A growing stem continually produces auxin, and a new primordium forms only when the concentration of auxin reaches a critical value. Once a primordium begins to form, more auxin flows into the primordium’s cells “like a river carving a canal in the sand,” says computer scientist Przemyslaw Prusinkiewicz of the University of Calgary in Alberta. This inflow not only stimulates the growth of the existing primordium but also depletes the surrounding stem of hormone and suppresses the formation of new primordia nearby.
Auxin is depleted least in the spot on the growing stem that is farthest from the older primordia. As auxin production across the stem tip continues, that farthest spot will be the first to reach the critical threshold to form a new primordium. In this way, Reinhardt and Kuhlemeier showed that the biochemistry of plant growth can explain Hofmeister’s rule that new primordia form farthest from older primordia.
The discovery of auxin’s role has led to an explosion of research unraveling the details of this growth process. Prusinkiewicz has worked with Kuhlemeier and his team to create a computer model that demonstrates how the hormone’s actions can create most common growth patterns in plants, including spirals. The group published its results in the Jan. 31, 2006 Proceedings of the National Academy of Sciences.
Mathematical meld
Meanwhile, mathematicians have been developing models of their own. Their scenarios are far less complex and biologically precise than Prusinkiewicz’, but that very simplicity has brought surprising insights into puzzling patterns of plant growth.
“I feel the biological models are so complicated that some of the beauty of phyllotaxis is lost in the process. It becomes purely computational,” says Jacques Dumais, a Harvard University botanist who works closely with mathematicians. “The mathematical models are so simple and yet so powerful, and they explain so much of what we see. What we’re looking for is, ‘What is the essence of phyllotaxis?'”
Harvard mathematician Scott Hotton produced a model based on Douady and Couder’s oil-drop experiment. The model shows how the forces in that experiment—designed to mimic the motion of primordia—indeed produce golden-angle spirals.
Hotton’s work also gave theoretical foundation to a surprising result of Douady and Couder’s experiment, which was that patterns other than golden-angle spirals can form. That finding corresponds with the observation that plants sometimes produce their primordia at angles of approximately 99.5°. In that case, the numbers of spirals in each direction would be members not of the Fibonacci sequence but of the closely related Lucas sequence. In both series, each number is the sum of the previous two, but the Lucas sequence begins with 1, 3, 4, 7.
Hotton has found some even more peculiar possibilities. Instead of producing primordia at the same angle each time, his model shows that some plants could produce primordia at a series of angles in a cycle. He found one pattern that goes 131°, 88°, 88°, 131°, 89°, 87°, 131°, 315° and then repeats.
“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 [for example] 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.”
Botanists believe that they have seen plants that grow in these periodic patterns, and Dumais is currently working to verify these observations.
Dethroning the golden angle
Hotton is now working with Atela and Christophe Golé, also of Smith College, on a more general and powerful model that can explain growth patterns the biologists can’t yet explain biochemically.
The team began by noting that in flowers with hundreds of seeds, such as sunflowers, so many primordia form so quickly that many of them must develop simultaneously. In that case, it doesn’t make sense to talk about the angle—golden or not—between successive primordia.
Indeed, Atela and Golé argue that the golden angle doesn’t really appear in the growth patterns of plants. “The golden angle comes up when you superimpose a spiral lattice on top of the plant and force it a bit,” Atela says. “It’s very close, but there’s wiggle room, especially when you have many, many primordia.”
Atela, Golé, and Hotton have replaced models involving the golden angle with a new model that they call the “coin game,” which allows many primordia to develop at once. They presented their new results in January at the Joint Mathematics Meetings in New Orleans.
The researchers imagine sticking pennies to the outer surface of a cylinder to represent primordia on a stem. The pennies must sit as close as possible to the base of the cylinder, but they must not overlap. Each new penny therefore goes into the lowest-possible vacant spot, and multiple pennies may be added simultaneously in multiple low spots. The rules of the game are a variation on Hofmeister’s observation that primordia form wherever the most space is available.
Pennies placed in this way can form a pattern of interlocking spirals. Atela, Golé, and Hotton have shown that in many such cases, the numbers of clockwise and counterclockwise spirals are in the Fibonacci sequence.
But the coin-game model can also account for a variety of more-complex patterns that appear in plants. One such example is when the number of spirals changes as the plant grows.
In addition, the model may resolve a puzzle about the early development of plants with spiral leaves. When the seed of such a plant sprouts, it puts out two primitive leaf structures 180° apart. As it grows, the plant somehow transitions to a pattern in which its leaves form a golden-angle spiral. Because the coin-game model produces Fibonacci spirals from a variety of initial penny placements, it may be able to explain the observed transition to spiral growth in plants.
The simplicity of the new model means that it doesn’t correlate in detail with the mechanisms that biologists are discovering, the mathematicians acknowledge.
“We’re digging on one side of the tunnel,” says Golé, while researchers who are creating detailed biological models “are digging on the other side of the tunnel. What we need to do at some point is to match their complicated models with our simple models and prove that our models are an abstraction of theirs in some sense.”
