Tree Mathmatics

Okay, confession time: I’m not good at math.  I use it, it’s great, we went to the Moon with it, yada yada.  But I don’t think mathematically and must strive to understand it.  So I was amazed  to stumble across how trees use mathematics to arrange their leaves on a twig to optimize their collection of sunlight for the photosynthesis thing.  Leonardo da Vinci picked up on this long ago. Your impress your friends word for the day is phyllotaxis: the arrangement of leaves on a stem in relation to each other.


Trees arrange their leaves in three major ways: whorled, opposite, and alternate. Whorled is where several leaves encircle the stem at one point, which is uncommon.  Opposite is where the leaves are arranged in opposing pairs on each side of the twig.  Dogwood, all the maples, ash, and buckeye use this arrangement.  Usually each successive pair of opposing leaves forms at right angles to the original pair, thus minimizing shading each other from the sun.


The math really starts kicking in with the alternate leaf arrangement, which I used to define as leaves forming on the twig randomly and not in pairs.  I was totally wrong on the random part.   With alternate arrangement the leaves actually form in a spiral along the twig, and how many leaves form per complete spiral is predictable by species.  Imagine you have a tree twig in your hand; focus on one leaf and its location; from that leave start counting leaves  along the twig and how many turns around the twig it takes to return to a position matching the original leaf but further along the branch.  For example, if you’re looking at sycamore, you will count two leaves and one turn around the stem.  With cherry, you will count three leaves and two turns around.  Oaks and a bunch of other hardwoods have a 2/5 arrangement: 2 turns around the stem in 5 leaves.  There are other spiral leaf arrangement ratios as well: 1/3, 3/8, 5/13 and so on.  But there is a common thread in all of them in that they are all Fibonacci numbers, named after a mathematician that figured them out in the 13th century.  Okay, it’s getting a little deep so hang with me.   Fibonacci numbers include the sequence 2,3,5,8,13,21, etc.  A Fibonacci number is one that when you add two in the sequence together, they add up to the next highest number. So if you look, 2+3=5, 3+5=8 and so on.  This is interesting, but what’s critical about Fibonacci numbers is the ratio between them. If you divide a Fibonacci number by the next highest number it will be 0.618034 times as large as the number that follows.  The Greeks called this ratio the “golden mean” because the proportion 0.618034 to 1 is the mathematical basis for almost all spirals in nature: leaves spiraling up a twig, the arrangement of bracts on a pine cone, sunflower seeds on the flower head, even the spiral of a snail shell.


What it boils down to is that in nature, a spiral based on Fibonacci numbers is a very efficient design form, and in the case of trees, minimizes how much each leaf shades another, meaning better sunlight collection and more food production.   So next time you’re in the woods, keep in mind that nothing is random, that there is purpose in almost everything you observe.  I find that very humbling.

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