Leonardo’s Formula Explains Why Trees Don’t Splinter
Leonardo’s rule holds true for almost all species of trees, and graphic artists routinely use it to create realistic computer-generated trees. The rule says that when a tree’s trunk splits into two branches, the total cross section of those secondary branches will equal the cross section of the trunk. If those two branches in turn each split into two branches, the area of the cross sections of the four additional branches together will equal the area of the cross section of the trunk. And so on.
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. Or, in formula terms: D2 = ∑di2, where i = 1, 2, … n. For real trees, the exponent in the equation that describes Leonardo’s hypothesis is not always equal to 2 but rather varies between 1.8 and 2.3 depending on the geometry of the specific species of tree. But the general equation is still pretty close and holds for almost all trees.