How Scientists Used 3D-Tech to Disprove da Vinci’s Rule of Trees

New ratio calculates wood’s carbon capture

Wed 27 Sep 23


While the ‘rule of trees’ has been a cornerstone in understanding forest ecology and management, recent studies reveal its limitations, mainly when applied to a tree’s internal vascular structures.

Leonardo da Vinci is synonymous with the Renaissance, remarkable for his contributions to art, engineering, and anatomy. But this prodigious polymath also dipped his toes into the realm of botany.

Among Leonardo’s 13,000 pages of notes and drawings – what we collectively refer to as his ‘Codex’ – lies an observation that’s come to be known as ‘Leonardo da Vinci’s rule of trees’. Simply put, this rule postulates that the total cross-sectional area of a tree’s branches is equal to the cross-sectional area of its trunk.

This sketch from one of Leonardo da Vinci’s notebooks illustrates his original ‘rule of trees’.
(Image Credit: British Library).

In even simpler terms, chop a tree down to a stump, and the collective cross-sectional areas of all its branches will equal the cross-sectional area of the trunk itself.

It is believed this insight helped the brilliant da Vinci to draw realistic landscapes.

Penned more than 500 years ago, the ‘rule of trees’ was almost immediately adopted by scientists seduced by such an elegant observation. To this day, da Vinci’s ‘rule of trees’ is still taken by many at face value as true. However, a new study has found that, while valid in some respects, the rule of trees “isn’t exactly correct” when applied to the internal vascular structures of trees.

Understanding the proportionality between a tree’s trunk and branches is crucial in predicting how forests respond to environmental stress, such as droughts or diseases. Moreover, the rule has direct applications in forest management. For instance, it allows for better estimating the biomass in a forest, which is essential for carbon accounting – a critical tool in combating climate change.

Studies conducted in the past few decades have found that da Vinci’s rule holds true for many tree species, although many species deviate from the rule.

Researchers using 3D scanning technology to assess tree architecture found that the total area of branches comes incredibly close to matching the trunk’s area, seemingly validating Leonardo’s centuries-old observation.

But other studies have since shown that da Vinci’s rule for how trees branch is close but not exactly right in describing the full complexity of branching.

In 2022, Russian scientists at the Petersburg Nuclear Physics Institute released a controversial study claiming that the surface area stays the same as tree limbs branch into smaller ones, not thickness.

Researchers from Bangor University in the UK and the Swedish University of Agricultural Sciences have also raised an eyebrow at Da Vinci’s rule. They argue that when you dig into the internal structure of trees, this celebrated principle starts to lose ground – a not-so-universal law, after all.

Verso A tree Leonardo da Vinci, c.1500. (Image Credit: Leonardo da Vinci, CC BY-SA 4.0, via Wikimedia Commons)

Many scientists have adopted da Vinci’s ‘rule of trees’ as part of the metabolic scaling theory, which explores how an organism’s metabolism, or the rate at which it uses energy, relates to its size. 

Scientists believed this rule applied to the appearance of trees and how water courses through them.

As water and nutrients flow through a tree – from its roots, up the trunk, and out to the tips of its leaves – this movement naturally faces a measure of hydraulic resistance.

When Bangor University researchers Ruben Valbuena and Stuart Sopp crunched the numbers, they came to a startling conclusion – at some point, the uniformity Leonardo da Vinci talked about just didn’t cut it.

Imagine a city’s plumbing system: if all pipes were the same size, water pressure would be lost at the furthest points. Similarly, trees must adjust the dimensions of their internal channels to ensure nutrients reach every leaf. This requires tapering off the volume, leading to a higher concentration of smaller channels at the edges.

It’s like a narrower and more turbulent river as it flows away from its source. “While da Vinci’s rule serves as a great ‘tip’ for artists, it doesn’t stand the test when zoomed in at the microscopic level,” said Ruben Valbuena.

The research team believes this new understanding will refine plant function theories. It might also explain why larger trees are more vulnerable to environmental stressors such as drought and climate change.

“One of our aims was to produce a ratio which could be used to estimate tree biomass and carbon in forests,” Stuart Sopp added. “This new ratio will assist in calculating global carbon capture by trees.”

• These findings appeared in the Proceedings of the National Academy of Sciences.

Editor’s note:  Leonardo da Vinci remains best known for two paintings – the Mona Lisa and the Last Supper. Largely self-educated, he filled dozens of secret notebooks with inventions, observations and theories about pursuits from aeronautics to human anatomy. His combination of intellect and imagination allowed him to create, at least on paper, such inventions as the bicycle, the helicopter and an airplane based on a bat’s physiology and flying ability.


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