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Two thousand years after the Indian mathematician and poet Acharya Pingala first described it, the Fibonacci sequence is still showing up in new places. The series—in which each number is the sum of the previous two—bears the nickname of the Italian mathematician who introduced it to Europe, Leonardo Bonacci (“Fibonacci” means “Son of Bonacci”). Recently, scientists have found that the sequence (0, 1, 1, 2, 3, 5, 8, 13 … ) maps beautifully onto some of the most complex systems in nature, offering a tantalizing clue about the mechanics of biological growth.
Michael Simon Toon was not looking for these famous numbers when he began designing synthetic trees for a solar energy project. Toon noticed that although other solar tree designs exist, no one had yet successfully replicated the structural stability and surface area efficiency that natural trees use to harvest light energy. The inspiration for Toon’s version came from another ancient insight: Leonardo da Vinci’s widely accepted area-preserving rule, which postulates that the sum of the thickness of all of a tree’s branches cannot exceed the thickness of the trunk.
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A Los Angeles, California-based artist, designer, and contractor, Toon had previously encountered the power of mathematical beauty in his modernist lighting fixtures and building designs, so the idea of basing a bio-inspired design on an ancient law of botany made perfect sense. “I’d seen tree-shaped solar projects before, but they looked nothing like real trees,” he tells Popular Mechanics. Most importantly, they were unrealistic: “You never see a symmetrical tree.”
To build his model, Toon planned to fit aluminum and PVC pipes of a few stock sizes (between 1 and 4 inches in diameter) into custom 3D-printed connectors with three openings. The connectors—or “crotches” in botanical parlance—would be responsible for making the tree conform to da Vinci’s rule by carefully balancing the relative sizes of the pipe holes.
To make a realistically asymmetrical tree shape, Toon constructed the three holes of each connector in three different sizes. “You have one single trunk coming out of the ground and it splits off into two smaller branches in a tree crotch. One branch is slightly smaller than the trunk itself, and the other is smaller than either the trunk or the other branch,” he explains. In other words, every branch connector in this biomechanical tree model connects three branches of different sizes, with the largest at the bottom and the smallest two at the top.
Toon thought the incorporation of a 550-year-old botanical insight would lend his model some of the beauty and structural resilience of a natural tree. What he didn’t expect was to discover a new instance of the Fibonacci sequence in his own design.
“All I did was make as many tree crotches as were required to complete the tree, and then I counted the number of crotches of each size that I needed,” Toon recalls. “And, lo-and-behold, it was the Fibonacci sequence.”
As the tree is built from the trunk up, the branch connectors decrease in size. In order to accommodate the tubelike branches, each crotch must share one to three hole sizes with another crotch. So, for example, the first connector has its largest (bottom) hole matched to the width of the tree’s trunk, while the middle and smallest size holes on top are each the same size as one of the bottom holes on the next-highest connector.
That’s when the sequence showed up. Toon found that, given these fixed relationships—da Vinci’s rule, the fixed unequal ratios of hole sizes in each connector, and the necessity of matching the bottom hole size of each connector to one of the top hole sizes of the next-lowest connector—the frequency of each size of connector he printed would follow the Fibonacci sequence. If you were to label each connector by size, there would be one of the largest connector (size A), one of size B, two of size C, three of size D, five of size E, eight of size F, and so on.
Fibonacci sequences are well-documented in the spirals of certain flower petals and have been reported in other natural branching systems, like rivers, roots, and bronchial tubes. Despite their fame and ubiquity, however, new examples of the numbers in nature don’t appear every day.
Structural regularities like this may emerge naturally in living forms due to the demands of biological survival. All trees have the same priorities—using minimal energy to transfer water and nutrients between the roots and the crown, without being knocked over—and the same available materials with which to achieve them, so optimizing resources for survival forces them into a limited range of possible architectures. The result is adherence to mathematical laws like da Vinci’s—and the emerging Fibonacci sequence Toon found in biosynthetic trees.
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The implications of this discovery are hard to predict. From a scientific perspective, observations like Toon’s can help guide research into the growth patterns of the original plants. A well-supported mathematical model of development can help scientists answer long-held questions about why plants grow the way they do. In theory, if you can prove that growth happens at a predictable rate or in a reliable geometric pattern, you can start testing theories relating to plant behavior against that pattern in order to validate them.
Since ancient times, physicists and architects, like da Vinci himself, have studied natural forms for inspiration. The past decade has seen an explosion of research into design approaches based on living forms—variously called biomechanical, bio-inspired, and biophilic design. Rather than mimicking existing organisms directly, these approaches use observed laws, like Fibonacci spirals and da Vinci’s rule above, to construct synthetic versions that retain some of the structural advantages of the living originals.
Designs based on these emergent natural laws also have the advantage of being aesthetically gorgeous. The golden ratio, in which the size relationship of a part to its whole is roughly 1:1.618, has been famous since the days of Euclid as a way of describing natural beauty in everything from Renaissance paintings to human faces.
Applying the Fibonacci sequence to branching systems certainly helps make Toon’s constructed trees and diagrams look breathtakingly realistic. The project, called the Tree of Water and Power, is currently in development—the first installation is due to open in December 2022—but Toon is encouraged by the emergence of the famous sequence.
“I didn’t do it on purpose,” he emphasized. “I just followed the rules of the tree.”
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