• 39°

OUTSIDE: Tree mathematics

By Steve Roark
Volunteer Interpreter, Cumberland Gap National Historical Park

Confession time: I am not good at math. I use it, it’s great, we went to the Moon with it, yada yada. But I
don’t think well mathematically and must strive to understand it. But I was amazed to discover that 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. The “impress your friends” word of
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, but they are not random. With alternate arrangement the
leaves 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 are 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 who
figured them out in the 13th century. I know it is getting a little deep so hang with me. Fibonacci
numbers include the sequence 1, 2, 3, 5, 8, 13, 21, etc. Note that when you add two numbers in the
sequence together, they add up to the next highest number. So if you look, 1+2=3, 2+3=5, 3+5=8, and so
on. Interesting yes, but what is 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 seed arrangement 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 it minimizes how much each leaf shades another, meaning better sunlight
collection and thus more food production. So next time you are in the woods, keep in mind that there is
purpose in almost everything you observe. That is both awesome and humbling.