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If you've ever spent time gazing up through a tree's foliage, you may well have noticed that branch thickness seems to obey a kind of conservation principle from root to leaf. History marks Leonardo da Vinci as having been the first to put this observation into writing. His notebooks record his reflection that the thicknesses of a tree's branches sum to that of its trunk, at every stage of its growth. More precisely, he claimed that the squared diameter of the trunk equates to the sum of the squared diameters of the ascendant branches.

Although the equality remains unverified, and the exact exponent the subject of some discussion, da Vinci's rule is widely regarded in the world of computer graphics as an effective means of generating realistic trees. Now, Christophe Eloy has gone beyond intuition and aesthetics to probe the physics underpinning the observation — suggesting that the rule follows naturally from a mechanism that helps the tree withstand wind-induced stresses (http://arxiv.org/abs/1105.2591; 2011).

The relationship between a tree's response to wind and the diameter of its trunk was noticed back in the nineteenth century, when Metzger proposed that trunk thickness might be tuned to maintain constant bending stress. Eloy has applied a similar idea to the task of understanding da Vinci's observation. Drawing on the underlying self-similarity of the tree skeleton, his study brings a continuous model for wind loading together with a discrete picture relating a tree's dimensions at each rank to its fractal dimension and the unknown exponent in da Vinci's equation. By assuming that wind acts on the tree's foliage, and that the probability of stress-induced fracture is uniform, Eloy has recovered da Vinci's elegant rule.

A three-dimensional numerical model that factors in the asymmetry and stochasticity of branching, and the variability in the angle of incoming wind, attests to the robustness of Eloy's theory, which accurately predicts the numerically calculated branch diameters and the exponent in da Vinci's relation. The results seem to suggest that da Vinci's insightful observation corresponds to an optimal geometry for resisting wind-induced fracture in self-similar trees.