Leonardo da Vinci got the trees wrong.
The multi-talented genius of the Renaissance wrote his “rule of trees” more than 500 years ago. He was describing the way he thought trees branch. Although it was a brilliant idea that helped him draw realistic landscapes, Leonardo’s rule breaks for many types of trees. Now a new branching rule, called “Leonardo-like”, works for virtually any leafy treethe researchers report in an article accepted April 13 in Physical Review E.
“Leonardo’s old rule describes the thickness of the branches, while the length of the branch was not taken into account,” says physicist Sergey Grigoriev of the Petersburg Institute of Nuclear Physics in Gatchina, Russia. “So the description using the above rule is not complete.”
Leonardo’s rule says that the thickness of a member before it branches off into smaller ones is the same as the combined thickness of the branches sprouting from it (Serial number: 06/01/11). But according to Grigoriev and his colleagues, it is the surface area that remains the same.
Using surface area as a guide, the new rule incorporates limb widths Y lengths, and predicts that long branches end up being thinner than short ones. Unlike Leonardo’s conjecture, the updated rule works for slender birches just as well as it does for stocky oaks, the team reports.
The connection between the surface area of the branches and the overall structure of the tree shows that it is the living outer layers that guide the structure of the tree, the researchers say. “The life of a tree flows according to the laws of conservation of area in two-dimensional space,” the authors write in their study, “as if the tree were a two-dimensional object.” In other words, it is as if only two dimensions, the width of each branch and the distance between the branches on a branch, determine the structure of any tree. As a result, when trees are rendered in two dimensions in a painting or on a screen, the new rule describes them particularly well.
The new Leonardo-like ruler is an improvement, says Katherine McCulloh, a botanist at the University of Wisconsin-Madison who was not involved in this study. But she has her doubts about the Russian group’s justification for it. In most trees, she says, the living part extends much deeper than the thin topsoil.
“It really depends on the species and even the age,” says McCulloh. “A giant old oak tree might have a centimeter of live wood… [but] certainly there are tropical tree species that have very deep sapwood and can have living wood in most of their cross sections.”
Still, the fact that Leonardo’s similar rule appears to hold for many trees intrigues McCulloh. “For me, it drives home the question of why they are [trees] retaining this geometry for its outer fabric, and how does that relate to the microscopic-level differences we see in wood,” he says. “It’s a really interesting question.”
To test their rule, Grigoriev and his colleagues took pictures of trees from a variety of species and analyzed the branches to confirm that the real-world patterns matched the predictions. The photos offer “a direct measure of a tree’s characteristics without actually touching it, which can be important when dealing with a living object,” says Grigoriev.
Although the team has yet to study evergreens, the rule holds for all deciduous trees the researchers have observed. “We have applied our methodology to maple, linden and apple,” says Grigoriev, as well as oak, birch and chestnut. “They show the same general structure and obey Leonardo’s rule.”
While it is possible to confirm the rule by measuring the branches by hand, it would be necessary to climb trees and check all the branches, a risky exercise for both trees and scientists. “Note,” the researchers write, “that not a single tree was damaged during these experiments.”