Math in the News
Leonardo's Formula Explains Why Trees Don’t Splinter
November 17, 2011
A new study expands on Leonardo da Vinci’s observation that a tree usually grows so that the total thickness of the branches at a particular height is equal to the thickness of the trunk.
“Expressed mathematically, Leonardo's rule says that if a branch with diameter (D) splits into an arbitrary number (n) of secondary branches of diameters (d1, d2, et cetera), the sum of the secondary branches’ diameters squared equals the square of the original branch’s diameter,” wrote Kim Krieger in a recent article for Science.
Krieger writes that da Vinci’s rule holds true for almost all species of trees and that botanists have hypothesized that it is related to how a tree pumps water from its roots to its leaves.
Christophe Eloy, a visiting physicist at the University of California, San Diego, and a specialist in fluid dynamics, hypothesized that the rule had more to do with the force of wind blowing on a tree’s leaves and built a mathematical model to test his theory. Since most trees naturally grow in a fractal pattern, he modeled trees as cantilevered beams assembled to form a fractal network.
Because the leaves on a tree branch all grow at the same end of the branch, Eloy modeled the force of wind blowing on a tree’s leaves as a force pressing on the unanchored end of a cantilevered beam. When he plugged that wind-force equation into his model and assumed that the probability of a branch breaking due to wind stress is constant, he came up with Leonardo’s rule. He then tested it with a numerical computer simulation that comes at the problem from a different direction, calculating forces on branches and then using those forces to figure out how thick the branches must be to resist breakage . . . The numerical simulation accurately predicts the branch diameters and the 1.8-to-2.3 range of Leonardo’s exponent.
Eloy’s results will be published in Physical Review Letters.