Fibonacci, Da Vinci, and a Tree Walk into a Bar

written by Deborah S.

And Da Vinci said, “Hey, Tree, it looks like the cross-sections of your branches sum to the cross-section of the trunk!” Then Fibonacci said, “Hey, that sounds familiar…” The tree reacted, “Well, of course, it’s not like I want my branches breaking…Wait, did you really not know this?”

If you were wondering, yes, that hilarious “joke” is my own. It might not have a punchline, but it summarizes today’s topic!

Let’s introduce another mathematical idea related to the Fibonacci Sequence and trees. This observation and concept is credited to Leonardo Da Vinci. He proposed that if one takes the cross-sections of a tree’s branches at a specific height and sums them all, one would get the same area as the cross-section of the tree’s trunk. To further explain, let’s clarify the definition of a cross-section. If one takes a stick and saws it in half, the resulting circle you see when looking at a sawed-off portion is called the cross-section.

The cross-section of a log - photo by Crusier

Imagine a tree, perhaps a young one, with only five branches. Suppose one were to measure the circumference and radius of the five branches at the same height for each. From this info, one could calculate the cross-section’s area if we assume the cross-section is a perfect circle, and it is likely close enough for a decent computation.

(If you’re interested: circumference = c = 2πr, meaning c/2 = πr, and the area of a circle = A = πr2 = πr*r = c/2*r).  

For the sake of simplicity, let’s say the areas of the cross-sections of those five branches were 2, 2, 3, 3, and 4 units. Then, Leonardo’s Rule stipulates the trunk of that tree would be about 2+2+3+3+4 = 14 units in area.  

A sketch made by Leonardo da Vinci illustrating his cross-section theory

This notion doesn’t just work for the trunk but also branches with “daughter” constituents. One could view a branch as a “mini trunk” and any further branches are “daughters”. According to the rule, the cross-sections of those daughter branches’ would sum to the cross-section of the “mother” branch.

Leonardo's rule on branches - graphic from Wired/Science Now/Kim Krieger

Leonardo’s Rule seems to hold to scientific observation.  

However, scientists weren’t sure why. What was the purpose of branching in such a way? In 2011 – the same year that seventh-grader Aiden won an award for discovering tree branches spiral in ratios of Fibonacci numbers – an aerodynamics physicist named Christophe Eloy determined that growing and branching in this way is optimal for withstanding high winds!

Perhaps it should come as no surprise that organisms as ancient as trees would have evolved a highly efficient growing method. Even if we attribute these biological advancements to time’s trial and error, they still excite us when we make the realizations. I return to a musing from a previous article: is our math based on nature or do we find ways to place our math onto nature? Chickens and eggs.

You may have noticed we have now addressed Da Vinci and the Tree portions of my opener, but we have not yet brought Fibonacci into the mix.  

Whenever we think of branching in a tree, we usually think of it splitting twice, instead of three, four, or more. Sure, multiple splits happen, but they are not the most common occurrence. With Leonardo’s Rule, we notice that when two branches split, their cross-sections sum to the cross-section of the preceding branch. This process is essentially the concept of the Fibonacci sequence: summing the previous two values to create the next. While we traditionally start the Fibonacci Sequence with 1 and 1 to get 2, then 3, and so on, we could start with any two numbers. For example, we could begin with 2 and 5. This new genesis would create another Fibonacci-type sequence: 2, 5, 7, 12, 19, 31, 50, 81, 141… Mathematicians love to name everything: these orderings are called Lucas Sequences.

So, Leonardo’s Rule – in which a trunk or branch equals the next two routes – is a Fibonacci Sequence is reverse. Or, if we started at the top of a tree, we could find a Fibonacci or Lucas Sequence as we worked toward the main trunk! We saw in the first article an example using the Fibonacci Sequence in such a way to knit a tree:

A knitted Fibonacci tree - photo from Botanica Mathematica

Using 21 – a Fibonacci number – as the width of the trunk, the knitter split the tree into branches of 8 and 13, the Fib numbers which combined to create 21. Then the creator continually split those branches into the preceding Fib values until left with a width of 1 at the end/top of the knitted tree. A knitted tree using Leonardo’s Rule and Fibonacci numbers!  

Is there a better way to tie these concepts together?

But, perhaps the takeaway from this series theme is that trees are great at being self-architects. Collecting sunlight, flowing air, and withstanding winds, trees have developed efficient growth patterns to excel in these areas. I do not doubt that we will find more connections between nature and mathematical optimizations, and I’ll be delighted by every single one!

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