When I was 14 I had a revelation that “Everything is Everything”. When I was 16, I saw my first video of a Mandelbrot set zooming in forever and I finally had the visual representation of that revelation.

I knew instantly when I saw this that it somehow held the secrets of life within its colorful kaleidoscopic shapes.

As I researched more and more, usually instead of studying, I learned that fractals are mathematical, iterative, self-similar, infinite, natural, chaotic, and efficient. If you don’t understand yet, don’t worry. I’ve got you covered.

In everything around us, there is a special energy at work. In all the universe, there are repeating patterns unfolding around us. From the way that atoms look like galaxies, to the way fiddlehead fern curls look like snail shells. The basic building blocks of this world are self-similar, repeating patterns. They are iterative, ever-changing, yet born of the same equations.

Mathematical

The same way we learned in math class that y=mx+b gets you this:

and (x−h)2+(y−k)2=r2 gets you this:

well, a fractal equation

gets you THIS:

It’s MATH, people! It’s not just swirly drawings on hippie posters or the way the world looks when you’re tripping. This is literally an equation being graphed, although our Ti-84 calculators from back in the day might not have been able to handle it.

And go look at that equation again. See the C in there? That’s the same C that’s in Einstein’s E = MC^2. Soo… It’s probably important.

Now, I won’t lie. My fascination and research took me much more clearly into the pretty pictures (don’t worry I’ll show you more soon!) than the intense math of it. So here’s my condolences for all of you who actually understand fractals mathematically. I swear I have several fractal mathematics books on my shelves which I have vigorously skimmed and then used as margherita coasters. That being said, here’s what has stuck out and made sense to me:

Iterative

Something that makes a fractal equation different than a “regular equation” is that it is iterative. (“regular” = Euclidian geometry, the field that we learned in high school for the SATs. Iterative is from the same roots as “iterate” as in, do a thing, then do it again, then do it again.

On the math side, our line equation y=mx+b happens once. You figure out what numbers M and B are, then you plug in any number for X (every number) and you solve for Y on the left side based on that set. The numbers balance out, you graph a line, everybody goes home. With a fractal equation, you plug in all the numbers, but then you use the number you solved for on the left to PLUG BACK IN to the letters on the right. Then you have a new answer. Then you plug that one in. And you keep doing this for, literally, ever.

What this means is that once an equation gets started, it will keep producing new points to graph. It will keep growing. (Just like life. Life is iterative. Life keeps growing and making new babies and they make newer babies and they just keep on going. More on that soon.)

Self Similar

Another way of saying this is that each equation has it own set of repetition rules. For example, if you start with an equilateral triangle, and then make the rule “place a smaller triangle in the middle third of each side of this shape” you get image two. And then if you take your new image, and plug it into the same rule again (so now each one of THOSE sides will get a triangle in the middle third of it), you get image 3. Do this once more and you’re at image 4.

Not only do fractal equations keep making new points to graph (and not only do those collections of points create distinct shapes) but all those shapes that keep getting made are very freaking similar to each other. This is called self-similar.

As in: since you’re using the same rule (make smaller triangles) with each new iteration, your new image will keep adding triangles, not something else. Now, if your rule was far more complicated and included adding (or subtracting) multiple shapes each time, you would see multiple shapes in your image, but they would still repeat and hold true to the order that the rule has created.

A more complicated example of self-similar iteration would be humans giving birth to other humans instead of, say, rocks or chickens.

Infinite and scalable

Notice how the triangles keep getting smaller and smaller? And if you really did keep repeating that rule forever, they would get smaller and smaller forever. And if you flipped the rule around and went in the other direction, they’d get bigger and bigger forever. So the patterns we have access to on our scale of sight, intelligence, etc. are likely repeating at scales too tiny and too large for us to comprehend yet. But it can help us predict that they’re there.

Natural

So far you’re like, “Who cares? These aren’t on the SATs bc they aren’t applicable, you’re just talking baby math to us.” And so far you’re right, but here’s where it gets interesting.

Remember how that little triangle equation begun to look suspiciously like a snowflake after only a couple iterations? (That one’s called Koch’s snowflake…) Anyways, that is not the only fractal equation that tends to begin looking suspiciously like something found in nature fairly quickly.

This equation quickly looks like a tree (There’s a whole series called Pythagorean Tree fractals if you’re interested)

This one begins to look like bubbles sticking together, or clusters of xylem and phloem inside plant stocks.

This one looks like the veins of a leaf close up. And the top corner has strands that resemble lightning strikes.

Here’s a few more examples.

At first it was just cool to think that some things in nature had these fractal things that looked alike, math that explained their shapes. But then I began realizing that everything in nature is a fractal, has an equation for itself, has a repeating pattern and a set of rules it follows as it iterates.

Every-thing from smoke spirals to species survival trends follows fractal patterns.

Check out these “fractally-generated-landscapes”

And since we’re a part of nature, whether we want to believe it or not, that means WE have fractal patterns.... In our veins (literally), in our brain wiring, in our experiences, emotions, and beliefs.

Humans are known for trying to create patterns when there are none, when it’s just random chaos, so I admit that there is a possibility of that happening when I say “every-thing”. But based on what comes next about fractals, I’m not so convinced chaos disproves order.

Chaotic

When a fractal equation is graphed (for example one that will look like a fern) it doesn’t plot points at the root and then systematically work its way up the stem, filling in each leaf shape in a linear, easily trackable way. Nope. It’s actually wildly unpredictable. As the equation is iterated; plugged in again and again to plot a new point each time, they show up all over the place. You would probably think the points are not even related to each other (the first image). But give it more time, more data points, and the shape begins to fill itself in until… Aha! something recognizable!

This process would essentially be the same as taking a 2 billion piece puzzle, dumping out the box, then placing random pieces inside the invisible outline until it’s complete. This is kind of called lateral processing (as opposed to linear processing) where a bunch of stuff is happening at once until the goal is complete, instead of one linear thing happening, then the next, then the next, until completion.

When we figured out how to make computers lateral processors, they got way smarter, and a lot closer to us…. (Read: fractal iteration is probably important in terms of AI development.)

Efficient

The last cool nuance I’ll say about fractal patterning is that it is simultaneously nature’s best way of conserving energy and best way of increasing energy. If all of life is trying to conserve and increase energy so they can iterate (that’s what the kids are calling it these days) then why would nature want to use anything less than the optimal procedures?

Interestingly, our brains often think that a linear path from point A to point B is the most effective. But that’s only true in undisturbed, independent systems. Aka, if I was drawing a line on a piece of paper and nothing else was in the way. But if I was, say, a mountain stream trying to get back to the ocean, a direct line from A to B would require WAY more energy. I’d have to bore through rocks and trees and ground, I’d have to push animals out of my way and who knows what else. I might have to defy gravity a little bit and float through the air sometimes to make it to the ocean in a perfectly straight line. And all those things would require a LOT of energy. And remember that C in the equation that’s related to Einstein’s energy equation? Well it just wouldn’t be having that. Instead, water, lightning, blood flow, etc. take the paths of least resistance to meander toward their destinations, which is actually more energy efficient, and all those paths happen to be self-similar.

On the other side of the coin, fractal shaping produces the most energy. Again, there’s a lot more math that goes on behind the scenes of all this, but consider this anecdote of why tree branches spread they way they do, which happens to be similar to our veins and other internal structures. A tree’s branches have the goal of getting it’s leaves the most sunshine per surface area possible, to optimize growth. A vein system wants to get fresh oxygenated blood to as many cells as possible with the least expenditure of energy to keep the body alive and strong. The shapes they take, splitting off into seemingly chaotic, smaller and smaller, self-similar branches with self-similar spacing between the leaves (or veins) is the most efficient system we know of for doing this. What might look like chaotic because it doesn’t fit easily into a “regular” geometry class is actually the smartest set of equations we know for optimizing growth. Oh shit.

SO WHAT?

Thanks for sticking with me through that math lesson. The people who don’t like math probably hated it, and the people who like math probably really hated it, so the fact that you made it this far means you’re just a good person. But yes, WHY would I take you down that journey during the middle of a self help blog? This is why:

YOU ARE FULL OF FRACTALS.

You have mathematical functions that are your natural programming to be a human.

Your are an iterative life form, always using new information to plug into your next experiences. You do a thing and get a result, which gives you a belief about the world. Then, you plug that belief back into your next experience and low and behold, it shapes the outcome of that experience so that the result you get is a more deeply-convinced belief. and on. forever.

Your beliefs and experiences and sensations are self-similar. You will experience the same type of energy, the same pattern, in many aspects of your life, because it’s all being fed by the same repetition rules.

These self-similar patterns will happen on the smallest scales - down to your cells, all the way up to the biggest scales of culture (and beyond) because the equation is infinite.

Until you change the equation...and that is the soul work we're here to do.

Like I said, since you are natural, and fractals are the structure of nature, you are fractals too.

Your most efficient way of acting, learning, changing, growing, and recharging will often seem chaotic until it’s completed. But just because you can’t see what the puzzle pieces in your life are pointing you to yet, doesn’t mean there isn’t a shape they are destined for.

Fractal God: How Math Met SPIRITUALITY and Had a Baby called Destiny