Can plants count?

Have you noticed that sampaguitas have five petals? Gumamelas and adelfas, too, have five petals. A Calla Lily has only one, some other lilies have three, a rose has many. But if you look at a rose closely, you would notice that there are groups of five petals spiraling outwards counterclockwise, with each petal partially overlapping the outer side of the previous one, and other groups of five or eight spiraling outwards clockwise, again with each petal partially overlapping the outer side of the previous one... and a daisy has 21 petals.

It’s curious, isn’t it, that there are very few flowers with four petals, or six, or 17 or 20, for that matter. Even more curious, perhaps, is the fact that the more commonly observed numbers of petals, as well as numbers associated with the petal arrangements of flowers with many petals, are all elements of a well-known mathematical sequence. The first few elements of the sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Each element of the sequence, except for the first two, is obtained by adding the previous two. Thus, the next element of the segment shown above is 55+89 = 144; the one after that is 89+144 = 233… and so on. It is called the Fibonacci sequence (e.g., Ref. 1) in honor of Leonardo Fibonacci, also known as Leonardo Pisano Bogollo, a medieval Italian mathematician, reputed to be the most eminent European mathematician of that era.

The sequence is not original to Fibonacci. It was known to Indian mathematicians as far back as the third century, BC, but it was Fibonacci who popularized it in Europe. His book, Liber Abaci (Book of Calculations), published in 1202, contained a description of the sequence together with other mathematical subjects that he learned while studying under the most prominent Arab mathematicians of the time.

The Fibonacci sequence has many properties that have occupied mathematicians for centuries. One of the most interesting is this: if you divide one element of the sequence by the element that preceded it, you get an approximation of what is known as the golden ratio, 1.61803…, usually designated by the capital Greek letter phi, _. The further along the sequence you go, the better the approximation is. The golden ratio has been observed in such things as the structure of the chambered nautilus, hurricanes, and galaxies. In fact, the golden ratio has long been considered to be the ideal proportion (hence, “golden”) and has figured prominently in art and architecture since antiquity.

But the most curious connection of the Fibonacci sequence is probably with plants (Ref. 1; see also Refs. 2 and 3).

The leaf arrangement in many plants follows a spiral. That is, as you go along a stalk, the leaves project from it in different directions, each leaf pointing at some angle relative to the previous one. If you start from a random leaf and count the number of leaves that you pass, as well as the number of turns that you make, until you get to the leaf directly above the leaf where you started, you would find that the number of leaves that you pass is an element of the Fibonacci sequence, and the number of turns is the previous element. If you reverse directions from the original leaf, and count the number of turns it takes to pass the same number of leaves as on the upward trip, it turns out that this is the Fibonacci number that precedes the first two. If on the upward trip it takes three turns and five leaves to get to the leaf that covers the leaf from where you started, then on the downward trip, it takes only two turns to pass five leaves.

The ratio of the number of leaves to the number of turns on the upward trip, or the number of leaves per turn, is an approximation of _. Turning it upside-down, the ratio of the number of turns to the number of leaves, or the fraction of a turn between successive leaves, is an approximation of the reciprocal of the golden ratio 1/_, and is also equal to _ -1; just some of the idiosyncrasies of the golden ratio.

These connections between plants and the Fibonacci numbers cannot be mere numerical accidents. Whenever related organisms share ubiquitous features, it is very likely that these features endowed the organisms with some evolutionary advantage.

It is easier to see this in the case of the leaves. Suppose that there are exactly two leaves per turn or, equivalently, there is a half-turn between leaves. Then, after one full turn, the third leaf exactly covers the first. After another turn, the fourth covers the second and leaves number one, three and five are all on top of each other. If there are six leaves per four turns, then after every four turns, a leaf exactly covers another. For the plant, this means that the leaves that are covered do not get as much sun as those on top, and only the topmost ones catch the rain that will eventually flow down the stalk to water the plant. 

To make sure that all the leaves get as much sun and rain as possible, no leaf should exactly cover another. That is, the number of turns per leaf or, equivalently, the number of leaves per turn should not be an exact ratio of two integers — it must be an irrational number. The ratios of successive Fibonacci numbers are rational numbers that approximate the irrational golden ratio. And the approximation gets better as you go up the stalk. Even the distribution of branches around a trunk often follows a Fibonacci pattern.

But plants don’t stop there. While the steady acquisition of basic nutrients (sunlight and water among others in the case of plants) is essential for the maintenance of life, reproduction is the key to the survival of the species (seed dispersal in the case of many plants). In this instance also, plants make use of the Fibonacci numbers and the golden ratio.

Take the sunflower, for example. The spiral arrangement of the seeds in the sunflower maximizes the number of seeds that could be packed in (essentially) a plane. In plants where the seeds (when provided with edible flesh to attract seed-dispersal vectors like birds, rats, and man) are packed in a volume like cones, Fibonacci spirals are observed. This is seen in pine cones, strawberries, and pineapples. One wonders if we would see Fibonacci spirals also in atis, langka, and guayabano.

Some physicists have actually achieved Fibonacci spirals in microstructures that they had constrained into conical shapes (Ref. 4). They conjecture that Fibonacci spirals represent the least energy configuration in cones. But they could not provide a mathematical proof. Maybe, they should just ask a pineapple.

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References:

1. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/

2. http://jwilson.coe.uga.edu/EMAT6680/Parveen/Fib_nature.htm

3. http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

4. C. Li, X. Zhang, Z. Cao. Triangular and Fibonacci Number Patterns Driven by Stress on Core/Shell Microstructures. Science 2005, 309:909-911; http://www.crystalinks.com/fibonacci.html

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Alfonso M. Albano is Marion Reilly Professor Emeritus in the Department of Physics, Bryn Mawr College, Bryn Mawr, Pennsylvania, USA. He may be contacted at [email protected]. Eduardo A. Padlan is an adjunct professor in the Marine Science Institute, College of Science, University of the Philippines Diliman. He may be contacted at [email protected].