Da Vinci Formula Explains Tree Growth

by Dr. Elizabeth Mitchell on November 19, 2011
Featured in News to Know

Tree trunk math . . . more complicated than meets the eye.

Centuries ago Leonardo da Vinci observed the taper of many tree trunks follows a mathematical division of diameters. Those divisions are used by graphic artists to make trees look realistic. But until now, no one has been sure why those divisions are important. Physicist Christophe Eloy, a specialist in fluid mechanics, has devised a computer model based on a new theory of “why,” and the answer goes beyond the simplistic explanations previously accepted.

Leonardo’s rule says that when a tree trunk divides, the cross-sectional area of the branches adds up to the cross-sectional area of the trunk. And when those branches divide, the sum of all those areas also equals that of the trunk. And so forth.

Mathematically, an approximation states that the sum of the squares of all the branch diameters is equal to the square of the trunk’s diameter. In reality, however, those numbers aren’t always squared. The real exponent can vary from 1.8 to 2.3. Thus the reality is already more complicated than the simple, even though the approximation is pretty close.


Why would this relationship exist? Botanists have assumed diameter needs to vary this way to supply sufficient water to the leaves. Eloy thought something more was involved. He noticed that branches really split the way fractals do, with each division forming the same angle. He also noticed wind-stress on a branch mainly affects the leaves growing near the end. Therefore, he applied the mathematics required to calculate the necessary branch diameter to successfully resist breakage. His wind-resistance model accurately predicts the diameter required and the actual exponent required to get precise answers.

Commenting on Eloy’s results, to be published in Physical Review Letters, UC Berkeley mathematician Marcus Roper commented, “Trees are very diverse organisms, and Christophe seems to have arrived at a simple and elegant physical principle that explains how branches taper in size as you go from the trunk, through the boughs, up to the twigs. It’s surprising and wonderful that no one thought of [the wind explanation] sooner.”

This study brings trees up to par with manmade structures that have been primarily designed taking into account wind-loading considerations, the Eiffel Tower being perhaps the most well-known example.

MIT engineer Pedro Reis commented, “This study brings trees up to par with manmade structures that have been primarily designed taking into account wind-loading considerations, the Eiffel Tower being perhaps the most well-known example.”

We must express a little amusement at Reis’s turn of phrase. God designed trees about 6,000 years ago. Leonardo da Vinci observed the mathematical relationship just a few centuries ago. And only now has anyone figured out a purpose inherent in such a mathematically complex design. Man’s architectural wonders mimic God’s designs. And perhaps with application of Eloy’s discovery, human engineers can better apply the Master Engineer’s principles to build more wind-resistant buildings as they seek to bring manmade structures “up to par” with God’s trees!

For more information on biomimicry see “Robotic Insects Mimic God’s Own Design” and “Ancient Protein a Potent Antimicrobial.” And don’t miss the January 2012 issue of Answers magazine. The theme will be “God Invented It First.” The issue will include articles about some fascinating examples of biomimetics.

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