TY - JOUR

T1 - Experimental evidence for logarithmic fractal structure of botanical trees

AU - Grigoriev, S. V.

AU - Shnyrkov, O. D.

AU - Pustovoit, P. M.

AU - Iashina, E. G.

AU - Pshenichnyi, K. A.

N1 - Publisher Copyright: © 2022 American Physical Society.

PY - 2022/4/29

Y1 - 2022/4/29

N2 - 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: dili=kdi+1li+1. If both rules are valid, then the branch's length of the next generation is k times shorter than previous one: li=kli+1. Moreover, the volume (mass) of all branches of the next generation is a factor of di/di+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.

AB - 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: dili=kdi+1li+1. If both rules are valid, then the branch's length of the next generation is k times shorter than previous one: li=kli+1. Moreover, the volume (mass) of all branches of the next generation is a factor of di/di+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.

UR - http://www.scopus.com/inward/record.url?scp=85129693398&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/18fdf1c4-2652-3f3f-a2c3-ce801c9081f9/

U2 - 10.1103/physreve.105.044412

DO - 10.1103/physreve.105.044412

M3 - Article

C2 - 35590611

AN - SCOPUS:85129693398

VL - 105

JO - Physical Review E

JF - Physical Review E

SN - 1539-3755

IS - 4

M1 - 044412

ER -