Fractal scaling and the aesthetics of trees

Abstract

Trees in works of art have stirred emotions in viewers for millennia. Leonardo da Vinci described geometric proportions in trees to provide both guidelines for painting and insights into tree form and function. Da Vinci's Rule of trees further implies fractal branching with a particular scaling exponent $\alpha = 2$ governing both proportions between the diameters of adjoining boughs and the number of boughs of a given diameter. Contemporary biology increasingly supports an analogous rule with $\alpha = 3$ known as Murray's Law. Here we relate trees in art to a theory of proportion inspired by both da Vinci and modern tree physiology. We measure $\alpha$ in 16th century Islamic architecture, Edo period Japanese painting and 20th century European art, finding $\alpha$ in the range 1.5 to 2.5. We find that both conformity and deviations from ideal branching create stylistic effect and accommodate constraints on design and implementation. Finally, we analyze an abstract tree by Piet Mondrian which forgoes explicit branching but accurately captures the modern scaling exponent $\alpha = 3$, anticipating Murray's Law by 15 years. This perspective extends classical mathematical, biological and artistic ways to understand, recreate and appreciate the beauty of trees.