Scaling in branch thickness and the fractal aesthetic of trees

Leonardo da Vinci left guidelines for painting trees that have inspired landscape painters and tree physiologists alike, yet his prescriptions depend on a parameter, <i>α</i>, now known as the radius scaling exponent in self-similar branching. While da Vinci seems to imply <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn></mml:math> , contemporary vascular biology considers other exponents such as the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn></mml:math> known as Murray's Law. Here we extend da Vinci's theory of proportion to measure <i>α</i> in works of art, enabling comparison to modern tree physiology and fractal geometry. We explain how <i>α</i> determines proportions among branches and visual complexity, which in turn influence the fractal dimension <i>D</i>. We measure <i>α</i> in classic works of art drawn from 16th century Islamic architecture, Edo period Japanese painting and 20th century abstract art. We find <i>α</i> 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 <i>Gray Tree</i> furthermore foregoes explicit branching but conforms to the theoretically predicted distribution of branch thickness with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2.8</mml:mn></mml:math> , suggesting that realistic scaling is as important as branching in conveying the form of a tree.