Gao Jingyi, Newberry Mitchell G
Department of Computer Science, University of Wisconsin, Madison, WI 53706, USA.
Department of Biology, University of New Mexico, Albuquerque, NM 87131, USA.
PNAS Nexus. 2025 Feb 11;4(2):pgaf003. doi: 10.1093/pnasnexus/pgaf003. eCollection 2025 Feb.
Leonardo da Vinci left guidelines for painting trees that have inspired landscape painters and tree physiologists alike, yet his prescriptions depend on a parameter, , now known as the radius scaling exponent in self-similar branching. While da Vinci seems to imply , contemporary vascular biology considers other exponents such as the case of known as Murray's Law. Here we extend da Vinci's theory of proportion to measure in works of art, enabling comparison to modern tree physiology and fractal geometry. We explain how determines proportions among branches and visual complexity, which in turn influence the fractal dimension . We measure in classic works of art drawn from 16th century Islamic architecture, Edo period Japanese painting and 20th century abstract art. We find 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 furthermore foregoes explicit branching but conforms to the theoretically predicted distribution of branch thickness with , suggesting that realistic scaling is as important as branching in conveying the form of a tree.
列奥纳多·达·芬奇留下了关于绘制树木的准则，这些准则启发了风景画家和树木生理学家，但他的规定依赖于一个参数，现在这个参数在自相似分支中被称为半径缩放指数。虽然达·芬奇似乎暗示了 ，但当代血管生物学考虑其他指数，比如被称为默里定律的 情况。在这里，我们扩展了达·芬奇的比例理论，以测量艺术作品中的 ，从而能够与现代树木生理学和分形几何进行比较。我们解释了 如何决定树枝之间的比例和视觉复杂性，而这反过来又会影响分形维数 。我们测量了从16世纪伊斯兰建筑、江户时代日本绘画和20世纪抽象艺术中选取的经典艺术作品中的 。我们发现 的范围在1.5到2.8之间，与天然树木的范围相对应，同时还发现了与理想分支的一致性和偏差，这些偏差产生了风格效果或适应了设计与实施的限制。此外，皮特·蒙德里安的立体派抽象画虽然没有明确的分支，但符合理论上预测的分支厚度分布， 表明在传达树木形态方面，逼真的缩放与分支同样重要。
