Designer Michael Simon Toon Finds the Fibonacci Sequence in Synthetic Tree Design


Two thousand years after the Indian mathematician and poet Acharya Pingala first described it, the Fibonacci sequence always appears in new places. The series, in which each number is the sum of the previous two, is named after the Italian mathematician who introduced it to Europe, Leonardo Bonacci (“Fibonacci” means “Son of Bonacci”). Recently, scientists discovered that the sequence (0, 1, 1, 2, 3, 5, 8, 13…) matches beautifully with some of nature’s most complex systems, offering a tantalizing clue into the mechanisms of growth. biological.

Michael Simon Toon wasn’t looking for those famous numbers when he started designing synthetic shafts for a solar energy project. Toon noticed that while other solar tree designs exist, no one has yet succeeded in replicating the structural stability and surface efficiency that natural trees use to harvest light energy. The inspiration for Toon’s version came from another old idea: the widely accepted theory of Leonardo da Vinci area preservation rulewhich postulates that the sum of the thickness of all the branches of a tree cannot exceed the thickness of the trunk.

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An artist, designer, and entrepreneur based in Los Angeles, California, Toon had previously encountered the power of mathematical beauty in his modernist lighting and building designs, so the idea of ​​basing a bio-inspired design on an ancient law of botany has become perfect. senses. ” I had seen tree shaped solar projects before, but they didn’t look anything like real trees,” he says. Popular mechanics. More 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 “crotchets” 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 create a realistic asymmetrical tree shape, Toon constructed the three holes in each connector in three different sizes. “You have a single trunk sticking out of the ground, and it splits into two smaller branches in a tree fork. One branch is slightly smaller than the trunk itself, and the other is smaller than the trunk or the other branch,” he explains. In other words, each branch connector in this biomechanical tree model connects three branches of different sizes, the largest at the bottom and the two smaller ones at the top.

Toon believed that incorporating a 550-year-old botanical idea would give 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 inseams as needed to complete the tree, then counted how many inseams of each size I needed,” recalls Toon. “And, oh surprise, it was the Fibonacci sequence.”

As the tree is built from the trunk, the branch connectors decrease in size. In order to accommodate the tube-like temples, each crotch must share one to three hole sizes with another crotch. So, for example, the first connector has its largest (bottom) hole matching the width of the tree trunk, while the middle and smallest holes at the top are each the same size as one of the holes of the next highest connector. .

That’s when the sequence appeared. Toon found that given these fixed relationships – da Vinci’s rule, the fixed unequal ratios of the hole sizes in each connector and the need to match the bottom hole size of each connector to one of the hole sizes higher of the next lowest connector – the frequency of each connector size it printed would follow the Fibonacci sequence. If you were to label each connector by size, there would be one of the largest connectors (size A), one size B, two size C, three size D, five size E, eight size F, etc. . to.

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The Fibonacci sequences are well documented in the spirals of some flower petals and have been reported in other natural branching systems, such as rivers, rootsand bronchi. Despite their fame and ubiquity, however, new examples of numbers in nature do not appear every day.

Structural regularities like this can arise 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 materials available to reach them, thus the optimization of resources for the survival forces them to limited range of possible architectures. The result is adherence to mathematical laws like those of Leonardo da Vinci and the emergent Fibonacci sequence found in biosynthetic trees.

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The implications of this finding are difficult to predict. From a scientific perspective, observations like Toon’s can help direct research into the growth patterns of the original plants. A well-supported mathematical model of development can help scientists answer long-standing questions about why plants grow the way they do. In theory, if you can prove that growth occurs at a predictable rate or according to a reliable geometric pattern, you can start testing the theories of plant behavior against this pattern 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 living form-based design approaches – variously referred to as biomechanical, bio-inspired and biophilic design. Rather than directly mimicking existing organisms, 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 emerging natural laws also have the advantage of being aesthetically stunning. The golden ratio, in which the size relationship of a part to its whole is approximately 1:1.618, has been famous since the time of Euclid as a way of describing natural beauty in everything from Renaissance paintings for human faces.

The application of the Fibonacci sequence to branching systems certainly helps to make the trees and diagrams constructed by Toon breathtakingly realistic. The project, called 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 said. “I just followed the rules of the tree.”

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