Fractal patterns are everywhere. From trees to lightning, from river deltas to coastlines, from the interior of your lungs to your fingers and hands, fractals are the mathematical patterns that define the natural world.
But why? Why does nature love these branching, self similar patterns? And what even is a fractal? Come with me as I explain the answers to all these questions, as well as why I find the fractal coasters I make so dang interesting.
Fractals pop up in so many unrelated places that it can seem deeply mysterious. After all, trees and lightning are about as unrelated as you can get, in terms of what they are; one is the culmination of 4 billions years of evolution, an unbroken optimization process stretching back to barren rocks, while the other is a split second phenomena, the result of physics trying to balance the charges built up by trillions of water molecules. And yet, there is an undeniable similarity in appearance between them
To understand this similarity, we first need to understand that nature, whether it be physics or biology, tends to run on simple rules that are repeated many times. We mostly understand why this is for biology; simple rules that can be repeated many times are much easier and more reliable to code into DNA.
For a tree to produce a relatively optimal sunlight-gathering shape, all it needs is the simple rule that after a section of trunk grows to a certain length, it should divide into some number of branches. When those branches get big enough, they will also divide into smaller branches, and so on. That rule, repeated across the course of a tree’s life, is enough to make the familiar tree structure.
Other examples of biology and growth running on simple rules include animal’s bilateral symmetry( you only need one data for one side of an animal, and then just mirror it for the other side) and the spiral structure of nautilus shells (just keep growing shell at a given angle, and you’ll end up with a spiral).
The story for why physics seems to run on simple mathematical rules is more complicated, and more speculative. Proposals range from our universe being simulated on an alien computer, to everything in the universe literally being made out of math, on some deep level; I’m not really qualified to weigh in on this topic, but the important thing is that physics does seem to run on simple rules.
One simple rule that Isaac Newton came up with to describe a falling apple turns out to describe why the vast majority of the matter in the universe is where it is. Another simple equation that was invented to describe how magnets pull on each other turns out to perfectly describe how light, electricity, and magnetism behave. Regardless of the reason, simple rules tend to have vast explanatory power
The simple rules that describe why lightning makes the shape that it does are a little more complicated than the one for trees, but it’s still quite understandable. Essentially, imagine a turtle on one edge of a chessboard. The turtle has the goal of reaching the other side, but it must follow the following rules; it can only move to a new square by rolling a 8 sided dice, where each side of the dice corresponds to one of the 8 chess squares directly adjacent to it. If the dice would send it to a square that it’s already stepped on, the dice will be-rerolled. Finally, if the turtle gets stuck somewhere, it can reposition itself to any square that it’s already stepped on.
If this game were to be played out, complete with dice rolls, the turtle would make slow, steady, progress forward, albeit at the cost of sometimes wandering backwards or sideways. In fact, as it played, it would make a branching pattern, reminiscent of a lightning bolt.
Above is one result of me playing this game. Due to the large cell size, and small grid, it doesn’t quite look right; however, as you repeated the same game with a smaller and smaller grid, you would get a result that looked more and more like a lightning bolt.
This weird game is what electricity in the air is doing, during a lightning strike. The charge wants to get to the ground to equalize positive and negative charges; however, air isn’t naturally conductive. Therefore, the lighting has to dump enough energy into, or ionize, each air molecule in order to turn it into a plasma, which is conductive. But which air molecule is ionized next is random; this is our dice roll from the previous game.
As the electricity jumps from molecule to molecule, forming a conductive network trailing behind it, that jagged lightning shape starts to emerge. The branching happens due to the fact that it’s not just that leading edge which can ionize the next molecule; any previous point that’s been ionized can also randomly ionize it’s neighbor, forming the start of a new potential branch. This pattern continues throughout the lightning strike, from small scale to large scale, simultaneously
So, to re-iterate: trees make their fractal shape due to following the simple rule of dividing after growing a certain length, while lightning gets its shape due to the rules of random ionization. There’s clear differences here, and yet, the fractal nature of both is dictated by clear, simple rules; in fact, they’re both fractal because they are created by those simple rules.
Fractalness, after all, is partially defined by being self-similar, of having the same shape whether you zoom in or out.
And the easiest way to achieve that self-similar property is to have a rule that produces the same result at a large or small scale. The branching of twigs is the same as the branching of the main trunk of a tree, and a lightning bolt is equally jagged whether you zoom in or out.
These simple rules are what creates these common, yet otherworldly, fractal structures, and they’re the thing I find most interesting about fractals. Any fractal shape is a reminder that our universe runs on surprisingly simple rules that still manage to result in the beautiful complexity that surrounds us.
Ultimately, all this is why I wanted to create my fractal coasters.
I wanted a physical, commonly used item that reminds me of the underlying patterns that connect and explain the natural world around us. And so, to do so, I harnessed the power of lightning itself. The exact same physics, the exact same rules, that create lightning create these coasters.
Now that we better understand the theory, join me for part 2 where I discuss the details about how exactly making fractally burned designs works,