(Cross-posted at Transistions, the Evolution of Life)

I spend a lot of time in forests. As an ornithologist, I spend a lot of time looking *up* in forests. With luck, I see the bird I am searching for. If not, my eye will wander the canopy, appreciating the play of light through the leaves. One day, my mind, as well as my eye, wandered. Was there a pattern to this seemingly chaotic riot of green? Nature, I know, is a most efficient master. It seemed reasonable that leaves, as food factories designed to carry out photosynthesis, should probably be positioned in order to maximize their exposure to sunlight.

This is, in fact, the case. It may not always be easy to see, because environmental conditions, physical constraints, injuries, etc. obscure the patterns, but the method of leaf arrangement, or phyllotaxis, on plants is both precise and quite astounding.

There are three basic ways that leaves are arranged on the stems of plants or trees. One is whorled, with three or more leaves arranged in a whorl around the stem. This is found on catalpa trees, as well as many herbaceous plants. A quick look will verify that the leaves of each whorl are placed so that they do not block the light of the previous whorl.

Another is opposite. Among tree species featuring opposite leaves are maples, ashes, dogwoods, and horsechestnuts — you can remember these genera by the acronym “MAD Horse”. Each rank of leaves will emerge at right angles to their successors, thereby not interfering with light transmission.

The third and most common leaf arrangement is alternate, which is found on nearly every other deciduous tree and many plants. In this array, leaves are ordered up a stem in an alternating pattern. The leaves don’t just alternate, they actually spiral around the stem so that each leaf gets maximum light exposure. Nor are these just ordinary spiral patterns. They are organized with mathematical precision.

Each leaf is positioned a partial turn around the stem from its successor. In each species of tree, this angle remains constant throughout the tree: every branch around the trunk, every twig around each branch, and every leaf around each twig is at the same angle. The pattern of any given species can be written using a fraction. Although this is easier said than done, is accomplished as follows.

Start with a leaf. Count the leaves going down the stem until you reach another leaf directly below the leaf you started with (in other words, located in the same vertical position on the stem). Also note how many turns around the stem it took to reach that leaf. A pattern of five turns consisting of eight leaves is written as the fraction 5/8, shown in the illustration (click to enlarge; image source Jill Britton’s Investigating Patterns page).

Many grasses have a fraction of 1/2 while beech trees come in at 1/3, and oaks, like many hardwoods, are 2/5. Holly leaves are arranged in a 3/8 pattern, and willows have the 5/13 phyllotaxy. If you have any mathematical prowess (I don’t), you are getting a creepy feeling here. The numerators and denominators of phyllotaxic fractions are nearly always numbers in the following series:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on, where each number is the sum of the two numbers preceding it. This is known as a Fibonacci sequence, named for an Italian mathematician.

Fibonacci numbers abound in nature. If some enlightened teacher had pointed this out to me in grade school, I would have been inspired by numbers rather than bored and intimidated by them.

One of the most frequent and easy to observe examples of Fibonacci numbers in the natural world are flower petal numbers. Go count some. And the preponderance of botanical examples of Fibonacci numbers leads me to believe this is why four-leaved clovers are so rare. But I digress. Let me continue to dwell on those phyllotaxic fractions.

Take a look at the head of a sunflower*, packed with seeds. The seeds are arranged in collapsed spirals, one winding clockwise, and the other counterclockwise. For example, 21 counterclockwise spirals crossing clockwise spirals creates the fraction 21/34, another fraction in the Fibonacci sequence. Why would florets, and therefore seeds, need to be in such an exact pattern? Rather than the most efficient use of light, in this case it is the most efficient use of space, resulting in the maximum number of seeds.

Then there are pine cones*. A fraction of 8/13 is found in a pine cone where it takes five circuits around the axis of the cone touching 13 scales to reach a scale directly above the first. Not only an efficient use of space and increased structural stability, but cones configured in this fashion channel wind-borne pollen to the ovules for the best probability of pollination and reproduction.

This is an elegant example of evolution at work, for any small adjustment that resulted in an advantage in light gathering, optimal seed arrangement, or increased fertilization would put a plant at a competitive advantage, and would be selected for. Over millennia, plant cells have evolved a way to organize themselves for optimum performance, following precise mathematical models.

It’s something you can count on.

Additional resources:

- Explore another interesting aspect of the Fibonacci series, the ratio found by dividing one number by the one preceding it, which in all the higher pairs of numbers is around 1.618, known as Phi or the Golden Mean. This is compelling number that has many mathematical properties and appears not only in plants but in the proportions of the human body, throughout nature, and has been co-opted in great works of art and architecture.
- A quick look at the Golden Mean and the Golden Angle. The angle of divergence (angular distance between two successive leaves) in all of plants with Fibonacci spirals approaches an average of 137.5 degrees, or the Golden Angle.
- Click through models of how plants create new parts according to mathematical rules.

*Links to animated gifs. Let them load to show you the spirals.

That is really interesting. Would you consider cross-posting it on Transitions? It fits in really well with what I trying to do there.

Very interesting article. I really appreciate you explaining the way leaves spiral down the stems and which types of trees have which leaves.

I've had trouble identifying trees in the past. Maybe this will help. Thanks.

Great post. Here is a video explaining The Fibonacci Series"

http://www.textism.com/bucket/fib.html

Marvelous info. The fractions remind me of musical time signatures, which I like to think stem (pun intended) from the same natural logic.

I found you through Festival of the Trees at Via Negativa.

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