Dragon_treesImage by Solkoll,

If you look at a branch of a typical deciduous tree, you can see that it looks like a smaller tree. Likewise, that branch branches off into smaller branches that look like even smaller trees.

This characteristic of trees is an example of what mathematicians, biologists, and systems scientists call self-similarity. Self-similar systems repeat their basic geometry at smaller and smaller scales, creating multiple miniatures of themselves at different scales. In general, natural and mathematical systems in which self-similarity results in complex and detailed patterns are referred to as fractal systems.

Many natural phenomena are or can be fractal:



Photo of a 12-sided snowflake by Becky Ramotowski,

ocean waves,


Painting by Katsushika Hokusai,
/CC BY-NC-ND 3.0

and even broccoli.


Photo by Jon Sullivan,

Trees are loosely fractal. While the trunks don’t keep replicating, the branches do. As the Fractal Explorer observes:

If you don’t know anything about fractals a tree might seem as a very random object. No patterns, no rules. But if you know something about fractals and look closer you can see that basically a tree is a trunk with trees on it. That is a basic pattern that every tree follows.

Taxonomies are often described as taxonomic trees, or as having a tree-like structure. To carry the analogy further, we often refer to the progressively more specific and more numerous hierarchical subdivisions in a taxonomy as branches. The overall domain of a taxonomy, while sometimes referred to as its root, might also be viewed as its trunk.

So this begs the question: Are taxonomies fractal? As it turns out, several authors have written articles on the fractal nature of biological genus-and-species taxonomies. These articles discuss the branching characteristics of these taxonomies, the same branching characteristics that we see in taxonomies outside the realm of biological species categorization. They also discuss the mathematical tendencies of the proportions of the various branches, tendencies that could perhaps be a natural result of the degree to which things in a group need to be different before we find it appropriate to give them different names.

In recent years, interdisciplinary scientists such as Christophe Eloy have been studying the natural forces that make trees grow the way they do, and how their growth patterns might make them resilient in windstorms. Interestingly enough, these scientists have been inspired, in part, by an observation that another person with an interdisciplinary approach, Leonardo da Vinci, made 500 years ago.

As Joe Palca explains in “The Wisdom Of Trees (Leonardo Da Vinci Knew It)”:

Leonardo noticed that when trees branch, smaller branches have a precise, mathematical relationship to the branch from which they sprang. Many people have verified Leonardo’s rule, as it’s known, but no one had a good explanation for it. …

Leonardo’s rule is fairly simple, but stating it mathematically is a bit, well, complicated. Eloy did his best:

“When a mother branch branches in two daughter branches, the diameters are such that the surface areas of the two daughter branches, when they sum up, is equal to the area of the mother branch.”

Translation: The surface areas of the two daughter branches add up to the surface area of the mother branch.

Here’s another explanation, from Esther Inglis-Arkell’s article “Scientists Still Puzzled by a Fractal Discovered 500 Years Ago”, that might be more intuitive:

Strip the leaves off of the average tree, soak the whole thing in water until it gets mushy, bundle the branches up together, and you’ll get what looks like one long trunk. That’s what Leonardo Da Vinci said in the fifteen hundreds. If a tree trunk splits off into three main branches, each of the branches will be one third the size of the trunk. When each of those branches splits into three again, making nine branches on the second ‘tier’ of the tree, each of these second tier branches will be one ninth the side of the trunk. As the branches grow and split, they will always be a particular fraction of the size of the trunk, and adding together all the fractional bits of each ‘tier’ of branches will always add up to ‘one trunk.’ This isn’t the case in all trees, but the majority hold to this pattern.

Can we gain a new perspective on taxonomies from all this? I think the lesson might have to do with scope, specificity, and detail. According to da Vinci’s observation, tree branches uniformly become ever thinner until they taper off, yet their total bulk at most levels of the tree will be approximately the same. So, in a taxonomy that grows naturally, we might expect that the terms at any given depth might be at approximately the same level of specificity. At the same time, their individual scopes at any given depth will add up to a sum total that will ideally (I think) cover the same scope as the top level of terms. As with trees branches tapering off, though, this will be less true as the taxonomy branches naturally taper off and end at the most specific levels.

Inglis-Arkell sums up with some interesting observations about the beauty of branches:

This pattern of growth has a mathematical, as well as physical, beauty. Trees are natural fractals, patterns that repeat smaller and smaller copies of themselves. Each tree branch, from the trunk to the tips, is a copy of the one that came before it. Branches split off from the highest tip the same way they do from the trunk, and set of branches splits off at the same angle to each other. Physics, math, and biology come together to create the simplest and most efficient growth pattern. It just took Leonardo Da Vinci to first notice it, the big show-off.

 Barbara Gilles, Taxonomist
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