Leonardo's rule - why does the thickness of the branches obey a pattern?

2024 Author: Seth Attwood | [email protected]. Last modified: 2023-12-16 15:55

The graceful trunk of the tree is divided into branches, at first few and powerful, and those into thinner and thinner ones. This is so beautiful and so natural that hardly any of us paid attention to a simple pattern. The fact is that the total thickness of the branches at a certain height is always equal to the thickness of the trunk.

This fact was already noticed 500 years ago by Leonardo Da Vinci, who, as you know, was very observant. This relationship was called "Leonardo's Rule" and for a long time no one could understand why this is happening.

In 2011, the physicist Christoph Elloy of the University of California, proposed a curious explanation of his own.

The "Leonardo Rule" is true for almost all known tree species. The creators of computer games who create realistic three-dimensional models of trees are also aware of it. More precisely, this rule establishes that in the place where the trunk or branch is bifurcated, the sum of the sections of the bifurcated branches will be equal to the section of the original branch. When then this branch also bifurcates, the sum of the sections of its four branches will still be equal to the section of the original trunk. Etc.

This rule is written even more elegantly mathematically. If a trunk with diameter D is divided into an arbitrary number of branches n with diameters d1, d2, and so on, the sum of their squared diameters will be equal to the square of the trunk diameter. According to the formula: D2 = ∑di2, where i = 1, 2,… n. In real life, the degree is not always strictly equal to two and can vary within 1, 8-2, 3, depending on the peculiarities of the geometry of a particular tree, but in general, the dependence is strictly observed.

Before Elloy's work, the main version was considered the existence of a connection between Leonardo's rule and the nutrition of trees. To explain this phenomenon, botanists suggested that this ratio is optimal for the system of pipes through which water rises from the roots of the tree to the foliage. The idea looks quite reasonable, if only because the cross-sectional area, which determines the throughput of the pipe, directly depends on the square of the radius. However, the French physicist Christophe Eloy does not agree with this - in his opinion, such a pattern is connected not with water, but with air.

To substantiate his version, the scientist created a mathematical model that connects the foliage area of a tree with the wind force acting on a break. The tree in it was described as being fixed at only one point (the place of the conditional departure of the trunk under the ground), and representing a branching fractal structure (that is, one in which each smaller element is a more or less exact copy of the older one).

Adding wind pressure to this model, Elloy introduced a certain constant indicator of its limiting value, after which the branches begin to break. Based on this, he made calculations that would show the optimal thickness of the branching branches, such that the resistance to wind force would be the best. And what - he came to exactly the same relationship, with the ideal value of the same value lying between 1, 8 and 2, 3.

The simplicity and elegance of the idea and its proof have already been appreciated by experts. For example, Massachusetts engineer Pedro Reis comments: "The study places trees at the height of artificial structures specially designed to resist the wind - the best example of which is the Eiffel Tower." It remains to wait for what the botanists will say about this.

“In his work, Ella took a simple mechanical approach. He viewed the tree as a fractal (a figure with some degree of self-similarity), with each branch modeled as a beam with a free end. Under these assumptions (and also under the condition that the probability of a branch breaking under the influence of the wind is constant in time), it turned out that Leonardo's law minimizes the probability that tree branches will break under the pressure of the wind. Elloy's colleagues, on the whole, agreed with his calculations and even stated that the explanation was quite simple and obvious, but for some reason no one had thought of it before.