Modern biomechnanics first used to test da Vinci's "Rule of Trees"

Physicist Christopher Eloy published the article "Leonardo’s rule, self-similarity and wind-induced stresses in trees" in Physical Review Letters. Eloy, who is a professor in mechanical and aerospace engineering at University of California San Diego, was interested in studying wind behavior as it flows over objects. Eloy's investigation was inspired by a quote in da Vinci's journals which read “all the branches of a tree at every stage of its height when put together are equal in thickness to the trunk” (Ritcher 184-187). Da Vinci believed that the sum of surface areas of tree branches is equivalent to the total surface are of the larger branch or trunk they grew from. However, da Vinci's journals do not include experimental data to support this claim. While previous publications have quanitfied relationships between branch and trunk surface areas, Eloy's study is the first to employ modern biomechnaical models to attempt to explain why da Vinci's hypothesis largely holds true. 

Eloy concludes that da Vinci's proposed tree design allows for structural soundness in the face of wind. Eloy reached this conclusion by calculating the tensile and compressive shear stresses leaves and branches experience during various wind speeds. Comparing the results to other proposed models showed the favorability of da Vinci's "Rule of Trees" (Eloy).

Eloy's publications sparked other researchers to investigate the biomechanics of air flow through trees and provide reasoning as to why trees evolved to grow in accordance to da Vinci's hypothesis. Ryoko Minamino and Masaki Tateno from the University of Tokyo used field data to show that for the tree Fegus crenata, stress uniformity plays a larger role on branch formation than da Vinci's rule (Minamino and Masaki 10) . Ultimately, their work also showed that for several tree species, da Vinci's rule is applicable with a degree of deviation less than 1.2% (11).

Sources

Eloy, Christophe. “Leonardo’s Rule, Self-Similarity, and Wind-Induced Stresses in Trees.” Physical Review Letters, vol. 107, no. 25, Dec. 2011, doi:10.1103/physrevlett.107.258101.

Mandelbrot, Benoit B. “Fractal Geometry in Physics.” 1993, doi:10.21236/ada273271.

Minamino, Ryoko, and Masaki Tateno. “Tree Branching: Leonardo Da Vincis Rule versus Biomechanical Models.” PLoS ONE, vol. 9, no. 4, Aug. 2014, doi:10.1371/journal.pone.0093535.

Palca, Joe. “The Wisdom Of Trees (Leonardo Da Vinci Knew It).” NPR, NPR, 26 Dec. 2011, www.npr.org/2011/12/26/144127874/the-wisdom-of-trees-leonardo-da-vinci-k...

Richter, Irma A. "The Notebooks of Leonardo Da Vinci". Edward MacCurdy. Isis, vol. 35, no. 2, 1944, pp. 184–187., doi:10.1086/358695.

Image shows da Vinci's proposed hypothesis on branch formation (Minamino and Masaki 3).