Experimental evidence for logarithmic fractal structure of botanical trees

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) = k d(i+1) l(i+1). If both rules are valid, then the branch's length of the next generation is root k times shorter than previous one: l(i) = root kl(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.

Automated summary

This paper discusses Leonardo da Vinci's area-preserving rule for botanical trees, focusing on its connection to a specific fractal structure called a logarithmic fractal. Using numerical Fourier analysis, the researchers analyze the logarithmic fractal properties of the branching system of real trees in two-dimensional space, based on photographs. They find that different species of trees exhibit a Q(-2) decay in the spectral intensity, indicating a logarithmic fractal structure in the two-dimensional space. Additionally, the experiments involving the side view of the trees suggest that a rule involving the product of branch diameter and length complements Leonardo's area-preserving rule. The findings suggest that while a tree as a three-dimensional object is not a logarithmic fractal, its projection onto a two-dimensional plane is.