Leonardo da Vinciʹs Rule and Fractal Complexity in Dichotomous Trees

A coincidence involving Leonardo da Vinciʹs ratios of branch diameters (0·707) in bifurcating trees and the limit of branch length ratios where fractal branching of bifurcating trees becomes non-fractal (at 0·707) suggests that constancy of the diameter ratios forces a fractal elaboration of the branch tips. The cylinder equation r22 /r21 = b1 /2b2 suggests an equality that does not occur in fact (because the b2/b1 ratio is always smaller than da Vinciʹs ratio and further branching continues for fractal trees). The branching proliferates as a geometric progression. The resulting fractal complexity of the branch tips probably enhances the flow of fluids to and from leaves, creates a spacious bower, and lessens the crushing weight that would result from non-fractal branching. Tree growth upward is initially rapid but becomes regularly diminished by fractal elaboration. A hypothesis presented suggests the number of real branchings in some trees is limited by da Vinciʹs ratio. The old/(new + old) wood ratios converge to the original branch ratio meaning the dichotomous tree increases new wood by 2b2 × old wood at each branching. The linear dimension of the canopy (compared to the sum of the old and the end branches) also approaches the 0·707 constant at about five orders of branching, and if exceeding those orders there would seem to be adverse effects.