Da Vinci’s fractal trees

Leonardo da Vinci observed that tree branches together are always as thick as the trunk beneath them. This is true, and there are some good ideas why.

Tree displaying fractal branching pattern
Philippe hirou, CC BY 4.0, via Wikimedia Commons

Leonardo da Vinci was a keen observer of nature – it’s part of what makes his drawings and paintings so good. Some of the copious notes he made were posthumously compiled and published in a book called A Treatise on Painting. And it is in this treatise that we find a singular and simple observation about the nature of nature:

All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk [below them].

Da Vinci

So, a tree is thickest at the base. Whenever the trunk branches off, each of those branches taken together are the same thickness as the trunk from which they branched. And this pattern continues with every split, all the way to the extremities of the tree. If you (somehow) gathered together every twig from all of the ultimate ends of the tree together, their combined thickness would be the same as the trunk itself.

Da Vinci compares this to a river in reverse – the width of the tributaries taken together is equal to the main stream. But it also bears similarities to a fractal. Trees branch off into smaller and smaller divisions, all the while keeping this simple division rule intact. And, in fact, we see tree-like structures occurring naturally elsewhere.

One beautiful example is the Lichtenberg figure. Take a big hunk of insulation like acrylic or plexiglass. Fire an electrical discharge into it. As the electricity branches through the block, the resulting pattern is startlingly tree-like:

Lichtenberg figure
Jan1959, CC BY-SA 4.0, via Wikimedia Commons

But why do trees follow this rule? Well, there are two schools of thought on that. One is that da Vinci’s comparison to a river is very apt, that this is the most efficient way for sap to travel up the tree. The other is that this tree structure is really really good at protecting the tree from wind damage. About a decade ago this second idea was tested experimentally in a virtual wind tunnel. Sure enough, trees following da Vinci’s branching rule survived simulated storms much better than those that didn’t.

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