Scaling in branch thickness and the fractal aesthetic of trees
Year: 2025
Type: article
Abstract: Leonardo da Vinci left guidelines for painting trees that have inspired landscape painters and tree physiologists alike, yet his prescriptions depend on a parameter, α, now known as the radius scaling exponent in self-similar branching. While da Vinci seems to imply α=2, contemporary vascular biology considers other exponents such as the case of α=3 known as Murray’s Law. Here we extend da Vinci’s theory of proportion to measure α in works of art, enabling comparison to modern tree physiology and fractal geometry. We explain how α determines proportions among branches and visual complexity, which in turn influence the fractal dimension D. We measure α in classic works of art drawn from 16th century Islamic architecture, Edo period Japanese painting and 20th century abstract art. We find α in the range 1.5 to 2.8 corresponding to the range of natural trees, as well as conformity and deviations from ideal branching that create stylistic effect or accommodate design and implementation constraints. Piet Mondrian’s cubist abstract Gray Tree furthermore foregoes explicit branching but conforms to the theoretically predicted distribution of branch thickness with α=2.8, suggesting that realistic scaling is as important as branching in conveying the form of a tree.
Source: PNAS Nexus
Authors: Jingyi Gao, Mitchell G Newberry
Language: English
Subfield: Visual Arts and Performing Arts
Field: Arts and Humanities
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $4,202
OpenAlex ID: W4407426916