spaceSpace and Physics

Brand New Fibonacci Sequence Discovered By Accident In Attempt To Harvest Sunlight


Dr. Katie Spalding


Dr. Katie Spalding

Freelance Writer

Katie has a PhD in maths, specializing in the intersection of dynamical systems and number theory.

Freelance Writer


Looks like a tree, but it's totally synthetic. Image credit: synthetick/

It’s not every day you get to discover a brand-new Fibonacci sequence – but Simon Michael Toon, the designer behind an upcoming solar energy project based on artificial “trees”, has done just that.

The Fibonacci sequence is one of the prime examples (see what we did there?) of pure math cropping up in the real world. It’s a simple number pattern where each new term is found by adding together the two that preceded it:

The first terms are 0 and 1; each subsequent term is found by adding together the previous two. Image credit: Peter Hermes Furian/

Seems easy, right? But despite looking like the kind of exercise you’d give a first grader to practice mental math, the Fibonacci sequence is a surprisingly fundamental law in the natural world. In fact, Leonardo of Pisa – also known as Leonardo Bonacci, which is where “Fibonacci” comes from – famously rediscovered the sequence when he was working a problem concerning bunnies.

Now, plant life loves the Fibonacci sequence – and for a very good reason. Think about the leaves on a plant: the plant’s energy comes from the sun, so its goal as it grows is to maximize its leaves’ exposure to the sunlight. The obvious way to do that is to make sure new leaves grow a little way round the stem from the previous one – but how far round should it go?

The answer comes down to that old mathematical favorite: the Golden Ratio. The best way for our plant to arrange its leaves is by placing each new one about 61.8 percent of the way around the stem from the previous one – the reciprocal of the Golden Ratio. And the best way to approximate the Golden Ratio using whole numbers is – you guessed it – the Fibonacci sequence.

Each square's side length is the sum of the side length of the two squares before it. Taking the ratio of the pairs gets us closer to the golden ratio, or ?. In red, we see the golden spiral. Image credit: ducu59us/

So it’s perhaps not surprising that Toon discovered his own project was following this ancient pattern as well. But what is newsworthy is precisely where the sequence showed up: not in the leaves, or branch patterns, where the golden ratio was already well-documented – but in the very architecture of the tree itself.


“You have one single trunk coming out of the ground and it splits off into two smaller branches in a tree crotch,” Toon told Popular Mechanics.

“One branch is slightly smaller than the trunk itself, and the other is smaller than either the trunk or the other branch.”

So any tree crotch – that is, a splitting point where one branch (or trunk) becomes two branches – is connected to three branches, all of different sizes, with the thickest branch at the bottom and the smaller two at the top. For Toon, whose tree was made not by nature but out of stock size aluminum and PVC piping, it was just a matter of 3D printing the right number and size of crotches for his creation.

Using botanical laws from da Vinci and a little lateral thinking, Toon soon found that the structure of his tree was far less random than you might first assume.


“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,” he explained. “And, lo-and-behold, it was the Fibonacci sequence.”

Basically, as you get further and further into the tree, the branches and crotches decrease in size – the biggest crotch is size A, say, then the second largest is size B, and so on. What Toon found was that the numbers of each size needed to construct the tree followed the Fibonacci sequence: there was one size A, one size B, two size C, three size D, five size E, eight size F, and so on.

“I didn’t do it on purpose,” Toon said. “I just followed the rules of the tree.”

This isn’t the first time nature’s penchant for pure math has been harnessed for eco-friendly invention, but it certainly is a fun reminder that as smart as we like to think we are, there’s practically nothing out there that Mother Nature hasn’t already perfected over the last few billion years. And while Toon’s project may have set itself some pretty ambitious goals, it has at least already given us something special: a mathematical surprise, more than two millennia in the making.


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