What Are Fractals?
What Are Fractals?
A Long Overdue Introduction To "Conceptual Fractals"
What do snowflakes, coastlines, and your productivity have in common?
They all exhibit fractal patterns.
The name of this publication — Fractal Productivity — was initially based on a mere inkling. “Fractal” came up as a term when asking my personal knowledge companion for a topic to kick off a Substack. Over two years later, I explored the first part of the name quite deeply; I’ve written a lot about what productivity is and what it isn’t. But to this day, I haven’t talked about fractals all that much.1
So, today, I want to add a long-overdue introduction about “fractals” and why this Substack still bears the working title it started with.
What Are Fractals?
Fractals are a concept from mathematics. They describe geometrical shapes with an extraordinary property: they exhibit interesting detail regardless of scale. What does that mean? For “normal” geometric shapes, such as a cube or a marble, the level of detail slowly begins to fade as we zoom in closer and closer. And eventually, we even arrive at a level where no detail is left. With a fractal, however, the level of interesting details remains roughly the same as we dial on the resolution.
A classic example here is that of the coastline. If one looks at a coastline from a bird’s perspective, it exhibits a certain ruggedness. If one then starts to zoom in, this ruggedness does not go away, but instead, finer and finer details are revealed, and, interestingly, the ruggedness prevails. This leads to the counterintuitive observation that coasts don’t have a well-defined length. The closer one looks, the longer the coast gets, as more and finer details are taken into account, increasing the overall length of the coast. This is called the Coastline Paradox.
For the longest time, mathematicians were somehow blinded by Euclidean geometry, which assumes all lines and surfaces are smooth and continuous. In a purely Euclidean world, zooming in means losing detail. It took mathematician Benoit Mandelbrot to observe that crinkliness, discontinuity, roughness, and, most importantly, “fractality” are, in fact, commonplace features of reality. And it was actually upon coming across the coastline paradox in an obscure scientific journal that Mandelbrot coined the term fractal. He based it on the Latin frāctus, which means broken, fractured, fragmented, or simply irregular.
Fractality usually suggests a bit more than having detail on every level. It suggests the notion of breaking something into smaller replicas of itself. That is, when we zoom in on a small region of a fractal, we end up looking at the same or a very similar shape we started with. Take a head of broccoli, for example. If you break it into pieces, you end up with small copies of the whole. The same features and patterns that “make” broccoli are repeated across different levels of magnification. So much so that if you were to take very close-up pictures of the whole and the parts, you’d struggle to tell them apart.
So, simply put, fractals are objects that look the same, or very similar, at all scales; no matter how deep you zoom in, you will find the same shapes and patterns. This property, the repetition of patterns across scales, is also called self-similarity. One of the properties of fractals is that they are, to varying degrees, self-similar. Since each level or scale of the fractal pattern has similar characteristics, and you can essentially scale up or down following the same principles regardless of where you start, this is also sometimes referred to as being “scale-free.”
The general ideas behind fractals (circular relationships, reflexive loops, and self-similarity) are older than Mandelbrot. The famous Russian Matryoshka dolls, for instance, which also represent a kind of “nested sameness,” were first produced in the late 19th century.
Mandelbrot, however, was the first to discover and particularly describe the ubiquitous appearance of fractals in nature. Since then, fractals have taken on a life of their own. They are now mentioned and used across many sciences, each with slightly different focuses.
It’s important to note, however, that fractals are not “everywhere.” Many of the examples you will encounter are mere “fractal-like”, and some people will confuse fractals with other concepts such as recursiveness and modularity. So, most things in nature are not fractal according to the mathematical definitions. If you are interested, Wikipedia has a list of mathematically validated examples. More interestingly, however, there is also a list of natural phenomena with fractal features.
What, however, has all of this to do with productivity?
Conceptual Fractals
With Fractal Productivity, I have used the term “fractal” in a more colloquial sense, one that is more closely related to our mental lives and psychology.
To illustrate, please take a second and imagine a cat. Even though I haven’t just specified anything in particular, you likely had no problem conjuring up in your mind one of our furred four-legged friends. We’ve all seen hundreds, if not thousands, of cats in our lives, so it’s easy to think about and talk about them collectively under a single template or stencil. Cats come in many types and shapes, and even cats from the same parents each look slightly different. Also, nowadays, we probably see more digital and animated cats than real ones. We can also quickly identify a cat sketched out with just a few lines, and even a cat that misses a leg or just has the shadow of a cat is enough to fall into that category. All these variations are conceptualized as one thing: cat. There is a many-to-one relationship. “Cat” is thus what we call a concept. There’s no single exact “cat” in reality. Rather, cat is a very broad term that encompasses all kinds of cats, whether real or not.
Concepts of real objects are a powerful way for us to communicate. But they become even more powerful when we use them to describe less tangible things or ideas such as “willpower,” “flow,” or “health.” These things have no direct object counterpart in reality; they just exist in our heads. And as humans, we rely heavily on such non-existent patterns to describe reality. That is, we conceptualize things and form mental models, analogies, and theories about them.
Fractals in nature usually serve a descriptive purpose. They help us describe general shapes of reality but don’t help us explain why things are how they are. However, we can sometimes use them to make predictions about the unknown. Discovering a fractal allows us to predict the tiny details we can’t see or the vast expanses we can’t observe (yet). Fractals can, in certain cases, show us how things “scale.”
Now, imagine we take multiple concepts from different levels of abstraction (like something really big, something medium-sized, and something really small) and find them to be self-similar or fractal-like. We could then say we’ve discovered a “conceptual fractal.” One example I have brought up many times is that of work units. Actions, tasks, and projects all have similar characteristics. They describe some “work to be done,” something to be achieved. They can all have “sub-parts”. They all give direction. Recognizing this, we can speculate whether they actually sit on a spectrum, repeating the same patterns and principles at different “scales.” Digging deeper, if we find that this holds true, we can then say that we have a fractal of “work units” at our hands. We could say that work units are a conceptual fractal.
Why Do Conceptual Fractals Matter?
If we discover a conceptual fractal, we can start to play around with it. We can zoom in or out beyond what is known. In the case of work units, we could explore things smaller than actions or bigger than projects. This latter exploration - zooming out beyond projects - led me to discover the personal program. The work unit fractal made me see something previously hidden in plain sight.
Realizing that work units are fractals opened my eyes. It made me see both the forest and the trees. This realization not only led me to discover a new concept but also gave me more clarity at every level. Understanding that actions, tasks, projects, and programs are not totally separate but very similar things—albeit on different magnitudes—makes it easier to “move between them” mentally. I call this scaling and zooming around a conceptual fractal “fractaling.”
By situating previously thought different and separate concepts within a single fractal relationship, we enhance the power of each concept involved. Without the conceptual fractal of work units, people talk about tasks and sub-tasks, projects and sub-projects, and even megaprojects. The terms “task” and “project” are not tied to any specific magnitude; they convey no information about “size” or even magnitude. Placing them on a fractal spectrum gives them more expressiveness and meaning and it makes the relationship between them also a lot clearer.
So, conceptual fractals can help us discover new concepts and provide power-ups for our language. But they can do even more. By defining concepts within a fractal relationship, we can transfer best practices from well-known concepts to lesser-known ones. This is exactly what I did in the very first series here on Fractal Productivity.
Summing Up
Today, I gave you a long-overdue introduction to fractals. Probably, the concept wasn’t all new to you. However, my research has not revealed any significant insights into how others have used the idea of a “conceptual fractal.” I might just be one of the first to use self-similar concepts in such a particular way. If you know of any such attempts, please do let me know.
Even though I was pretty euphoric about discovering the work unit fractal, I have since struggled to find other proper examples of conceptual fractals that revolve around personal productivity. I’ve already considered some candidates here on FP, such as the idea of a “work-rest” fractal, which posits that we must interleave work and rest at several scales in our lives. However, I debunked this due to several conceptual holes.
Some obvious and more naive concepts, such as “reviews” (as in weekly vs. monthly vs. quarterly reviews) and “goals” (e.g., annual vs. quarterly vs. weekly goals), can quickly be disqualified as they are essentially already a single concept. To discover new useful conceptual fractals, we need to investigate similar but seemingly different concepts.
However, these candidates may turn out to be part of something larger. For instance, one interesting contender I’m pondering is “focus,” which, on a small scale, may be related to the direction of our attention and, on a very large scale, refer to more of a sort of “essentialism” and in the middle, there may actually be situated some from of goals. I will need to dig deeper here, though.
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I’m curious, have you identified any conceptual fractals shaping your life?
Please let us know in the comments.
This was partly for practical reasons, such as time constraints, but also because I felt I had to build a solid knowledge repository on “productivity” first. There is so much shallow content out there that I felt needed revision, and there was so much confusion in terms of terminology.
However, since starting this blog, I have also learned a lot about fractals. I have watched documentaries on the fractals found in nature, read books on complexity and scaling, and learned about the history of this captivating and beautiful topic. In turn, I have continually contemplated what the true meaning of “Fractal Productivity” could be. I have yet to look more deeply into the mathematical and geometric side of things, though, and I won’t bother you here with that, as it gets very complicated pretty soon.