The Mathematical Lives of Plants
By Julie J. Rehmeyer
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.
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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. |
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. |
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.
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This sunflower has 21 clockwise and 34 counterclockwise spirals. |
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.
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.
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An image of the tip of a Norway spruce branch, viewed through an electron microscope, shows small buds that are called primordia. In this case, they will eventually turn into needles. The primordia form at the tip and then move outward and downward. |
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.
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 a magnetized liquid 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.
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In this movie (click here or on the image, above, to watch), a magnetized fluid drops into a pool of silicone 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. |
"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.
References:
Douady, S. and Y. Couder. 2002. Phyllotaxis as a physical self-organized growth process. Physical Review Letters 68(March 30):2098-2101. Abstract available at http://dx.doi.org/10.1103/PhysRevLett.68.2098.
Hotton, S. 1999. Symmetry of Plants. Ph.D. Thesis, UC Santa Cruz. Available at http://maven.smith.edu/~phyllo/Assets/pdf/thesis.pdf.
Huntley, H.E. 1970. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover. See http://store.doverpublications.com/0486222543.html.
Peterson, I. 2006. Fibonacci's missing flowers. Science News Online (June 3). Available at http://sciencenews.org/articles/20060603/mathtrek.asp.
______. 2002. Golden blossoms, pi flowers. Science News Online (Aug. 31). Available at http://sciencenews.org/articles/20020831/mathtrek.asp.
______. 2001. Fibonacci's Chinese calendar. Science News Online (Feb. 3). Available at http://www.sciencenews.org/articles/20010203/mathtrek.asp.
Reinhardt, D., T. Mandel, and C. Kuhlemeier. 2000. Auxin regulates the initiation and radial position of plant lateral organs. Plant Cell 12(April):507-518. Available at http://www.plantcell.org/cgi/content/full/12/4/507.
Smith, R.S. . . . D. Reinhardt, et al. 2006. A plausible model of phyllotaxis. Proceedings of the National Academy of Sciences 103(Jan. 31):1301-1306. Available at http://www.pnas.org/cgi/content/full/103/5/1301.
For more information about plant growth patterns, go to http://maven.smith.edu/~phyllo/.
For more information about Fibonacci numbers and the golden ratio, go to http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html.
Comments
I find the fact that sunflowers , cones from pine trees and even cacti have patterns that can be worked out mathematically . I cannot understand though that you say that these plants evolved to have these patterns , thought went into this , A Divine thought pattern . All these patterns show design and thought behind it . Therefore I disagree with you when you say they have evolved . Nothing evolves , it adapts to the environment yes , but it does not evolve from a primitive life form into this complicated pattern that we see in these plants and in all plant life and animal life for that matter .
Posted by: Maria Albuquerque | May 5, 2007 04:46 AM
I'm no mathematician, but I believe these number sequences and the golden angle growth patterns could be used to measure human interference with the environment against various control subjects. Human interference in biological material tends to disrupt, distort and "disfigure" natural cell and growth patterns. With that in mind, aberations would be very easy to identify. The potential for this is increasing as solvents increase in atmospheric "strength". Since solvents are "disrupters" at the molecular level, and since their impact on living tissue has become so pervase, it seems possible to me that at some point, plant growth could become an obvious indicator.
To take the observations to the macro extreme, it would be interesting to see if there were points of comparison between the numbers as applied to floral characteristics could be applied to star clusters. Nature loves to repeat itself...
Posted by: John Newell | May 6, 2007 05:15 PM
I find it remarkable that D'Arcy Thompson's "On Growth an Form" is omitted.
Posted by: Zoltan Blum | May 7, 2007 02:19 PM
Just think about this for a minute...
"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."
If you believe in evolution, does anything exist that doesn't provide an evolutionary advantage?
Now the pattern doesn't cause the advantage the underlying chemistry does? But you can continue on that path infinitely. Next you could say, that which underlies the chemistry gives the evolutionary advantage.
Posted by: Dave | May 7, 2007 02:42 PM
Another intersting point is that this spiral pattern has also been observed in the arms of galaxies.
One would postulate that the outward, downward and away motion which produced the spirals in the Stéphane Douady and Yves Couder experiment in 1992 would also be similar to the motion involved in the evolution of galaxies.
Furthermore, the fact that such large-scale structures have similar geometries to that of small-scale structures hints towards a repetative cycle. The universe may very well grow in a pattern which is similar in nature to the way in which a plant grows...
I do not hold to the belief that "God" created the universe 6000 years ago. Instead, I believe this universe precedes our current existance by multitudes of magnitudes and is constantly evolving by a similar method in all directions.
Posted by: Aaron Skaley | May 7, 2007 03:00 PM
Um, errors?
A > B and B < C, so A/B can't equal B/C.
Sorry, the caption uses different letters from the text. A corresponds to b and B corresponds to a.
Isn't most recent bud in Atela and Golé figure numbered 0?
Yes, thanks. Fixed now.
What is actually moving: primoridial plant cells, or the location of maximum growth?
The primordial plant cells are moving with respect to the tip of the plant.
Iron melts at 1500C. Surely you mean ferrofluid rather than "liquid iron?"
I do mean ferrofluid. I certainly didn't mean melted iron. I thought liquid iron was a reasonable simplification for people unfamiliar with the term ferrofluid, but a bit of research shows that I was wrong. I've fixed it now.
Also, silicon is a crstalline semiconductor. Do you mean "silicone oil?"
Yes, thanks. I corrected the spelling. I appreciate the careful reading! --jjr
Thanks.
Posted by: Jonathan Foote | May 7, 2007 05:33 PM
There is an interactive demonstration of phyllotaxis spirals like the ones shown here at http://demonstrations.wolfram.com/PhyllotaxisSpirals/
Enjoy!
Posted by: Joe Bolte | May 8, 2007 02:35 PM
there is a company that has applied natural geometry found in plants and sea organisms to design more efficient fans, propellers, pumps and mixers.
A lot of their work is based on Fibonnaci and other convergence series.
A curious relationship also exists between such series and fractal space. This exists at all magnitudes, from repeating amino acid groups in DNA all the way to galaxy formations.
Based in California,
http://www.paxscientific.com
CEO is Jay Harman
Posted by: andre | May 9, 2007 01:10 AM
I found your article to be interesting and informative. But in your third paragraph, you say "...golden ratio, which the ancient Greeks... believed to have divine and mystical properties. Leonardo da Vinci believed that the human form displays the golden ratio." Unfortunately, to be blunt, that is spouting pseudoscience.
You might begin by looking at the Wikipedia article http://en.wikipedia.org/wiki/Golden_ratio .
(Of course, that article should be taken with a grain of salt!)
Concerning the ancient Greeks, in the section on architecture, Keith Devlin http://www.stanford.edu/~kdevlin/ is quoted as saying "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value."
Concerning Leonardo, it would _seem_ that the Wikipedia article supports your statement by saying, in the section on aesthetics, "Da Vinci himself maintained that the human body has proportions that approximate the golden ratio." But note, crucially, that that sentence is followed by "[citation needed]"! That's because there is no primary reference. Concerning Leonardo, I recommend George Markowsky's "Misconceptions about the Golden Ratio"
http://www.umcs.maine.edu/~markov/GoldenRatio.pdf, an excellent reference given in the Wikipedia article. (At the end of that article, the section on disputed sightings should be _much_ larger.)
Another reference of interest is
http://www.laputanlogic.com/articles/2005/04/14-1647-4601.html :
"The Golden Ratio, once a pristine jewel of geometrical truth and simplicity, has become a deity for a cult of hyperlinking headnodders whose chief devotional practice seems to be to handwave their way from one disconnected and unexamined falsehood to another." That is, very sadly, true. I hope that you will rewrite your third paragraph. Furthermore, I suggest that you might consider writing an article similar to Markowsky's but intended for the layperson, for _Science News_ or a similar publication.
Best regards,
David W. Cantrell
You might want to check out this MathTrek article that Ivars Peterson wrote somewhat along these lines: http://www.sciencenews.org/articles/20050402/mathtrek.asp. --jjr
Posted by: David W. Cantrell | May 9, 2007 12:36 PM
I have to assume that C is also defined as the Circumference of the circle C to make sense out of the two ratios that form the proportion A/B=B/C. Is that correct? Do not the subtended areas also form the same proportion? Or is that what you meant?
In the text, I'm talking about two areas A and B with A < B, and C is the area of the circle. Then A/B=B/C.
In the daigram, a and b are arcs, and a > b (my apologies for the reversal of the letters!). In that case, if c is the circumference of the circle, b/a=a/c. --jjr
Posted by: Alan Marcelius | May 11, 2007 08:10 PM
In the (ideal) sunflower, all sufficiently small Fibonacci numbers appear as spiral counts, in different bands.
Which florets are nearest neighbors depends on the trade-off between two numbers: the distance between successive coils and the length of each coil. As the latter increases with N, the fractional part of a small integer multiple of the golden angle gets bigger; as the former decreases with N, more distant multiples of the golden angle (always Fibonacci multiples), with smaller fractional parts, come closer.
My webpage includes an illustration showing the effect more clearly; see also.
Posted by: Anton Sherwood | May 18, 2007 02:39 PM
I am a chemist, and I have read many articles both inside and outside my area of expertise.
This has to be one of the most eye-opening and well-written articles I have ever read. It is concise and suspenseful and cross-disciplinary. It has everything: easy math, geometry, pictures, diagrams and a movie! Thanks so much.
Posted by: mark nicholas | May 19, 2007 03:59 AM
yes it´s a really interesting article. thx a lot
Posted by: roberto | May 20, 2007 03:07 PM
The answer would seem surprisingly simple to me, and I'd love to see a computer program to simulate and prove my theory.
Start with a blank slate and place a small dot there with a certain growth rate (enlarging at a given rate that changes over time the same way a plant seed grows then the growth rate slows as it enlarges.)
Now, with one dot there, attempt to add another small dot to the same spot. Inevitably, one of the dots will have to shift to make room. Let's assume that nature forces the older dot to move (although for the first iteration it may not matter, and they are likely to both shift a bit.)
With two adjacent dots, what happens when a 3rd dot forces its way through? They form a triangle.
But, when you add a 4th dot, you don't get a square, you get more of a triangle with a dot in the center, but now the first dot is significantly larger than the 4th so this 4th dot will be offset from the center.
Add a 5th into the center of the blob of dots and, because the 1st dot is dominating the center, this 5th dot will show up next to the first dot and start to push it away from the center.
Add a 6th and it will tend to push the 2nd dot away, which has since taken over most of the center.
As this continues, and all of the dots tend to stay in the center because they are contained inside a close shell (bud), this pattern continues and creates a pattern as observed in so many things in nature.
When this pattern extends and you view it from above, you will likely see a dual-spiral pattern so familiar in large flowers.
BTW--these things do not "know" the golden angle and they do not "know" the "Fibonacci" series. Rather, the golden angle and Fibonacci series are a side-effect of nature so it is natural that nature would follow its own laws in things that it creates.
Posted by: Rob Spahitz | July 22, 2007 01:38 PM