Experimental evidence for logarithmic fractal structure of botanical trees.

Abstract

The area-preserving rule for botanical trees by Leonardo da Vinci is discussed in terms of a very specific fractal structure, a logarithmic fractal. We use a method of the numerical Fourier analysis to distinguish the logarithmic fractal properties of the two-dimensional objects and apply it to study the branching system of real trees through its projection on the two-dimensional space, i.e., using their photographs. For different species of trees (birch and oak) we observe the Q^{-2} decay of the spectral intensity characterizing the branching structure that is associated with the logarithmic fractal structure in two-dimensional space. The experiments dealing with the side view of the tree should complement the area preserving Leonardo's rule with one applying to the product of diameter d and length l of the k branches: d_{i}l_{i}=kd_{i+1}l_{i+1}. If both rules are valid, then the branch's length of the next generation is sqrt[k] times shorter than previous one: l_{i}=sqrt[k]l_{i+1}. Moreover, the volume (mass) of all branches of the next generation is a factor of d_{i}/d_{i+1} smaller than previous one. We conclude that a tree as a three-dimensional object is not a logarithmic fractal, although its projection onto a two-dimensional plane is. Consequently, the life of a tree flows according to the laws of conservation of area in two-dimensional space, as if the tree were a two-dimensional object.