Fractals and a Broken World
To begin a few definitions and history. This is a fractal.
The first fractal ever made, by this guy.
His name is Benoit B. Mandelbrot. He was a mathematician who was born in Warsaw Poland, went to school in France and got his Graduate degree at Cal Tech before going to work at IBM. At IBM he used those first computers to create the first fractal, out of the Mandelbrot Set( a set is just a collection of things that fit in a given definition). What defines this set is not particularly important to this article.
So a Fractal, in English, is something that repeats itself on the micro and macro scales. To illustrate, draw a circle. Now focus on just one piece of that shape. It is probably just a line, or a curved line. Now focus in more onto a piece of that piece, it is still just a line. The point that I am trying to make is that you don’t keep seeing squares, or circles, or triangles when you focus in on a small piece of that shape. This shape belongs to the world of Euclidean Geometry ( this is normal geometry that you learn in high school, made by the Greek mathematician and philosopher Euclid around 300 BC ).
Now the shape that was shown at the beginning of the article belongs to Fractal Geometry (a shape that obeys the mathematical definition of a fractal and can be self repeating ). You can think of this as a geometry of roughness whereas Euclidean Geometry is a geometry of smoothness.
So let’s do the same exercise that we did before with the circle with something called the Koch Snowflake.
If you focus on one section of it, it looks exactly like the bigger picture,and if you “zoom” in far enough you see that this pattern emerge. Starting with a straight line, you take the middle of it and place a triangle there.
Now, a purely mathematical fractal goes on infinitely down and infinitely up. (Another article will explore what I mean by “purely mathematical”).
Since we live in a real world limited by laws that prevent going down to the smallest infinity, you will eventually just reach something non-fractal at a certain point. To be considered a fractal, it just needs to go three layers deep.
So why does this matter. Why do all of these definitions and largely theoretical things matter to anybody outside the mathematical and academic world.
This is why: Our world is broken. Its fractured, and jagged and mainly empty space.
The world is not by any means smooth. If you have ever had the privilege to climb a mountain, then you have noticed that mountains are not just cones, the way a child would draw them like so
When lightening strikes it is not just a straight line, its jagged and sharp and broken. So if you wanted to describe this world to someone why would you use something that only describes smooth shapes. Now there are mathematical methods to use smooth shapes to get an approximation of the world, but why use an approximation when you can have the exact answer. More importantly, what happens when the smooth shapes don’t work. This is the problem that was solved when Mandelbrot came up with this new geometry. There were patterns and problems in chemistry and physics that Euclidean Geometry couldn’t solve. We needed a new set of tools to solve these new problems.
But for those that are skeptic still, to the power and beauty of this new fractal geometry. Let me make it personal.
Your Brain. Nine pounds between your ears that simultaneously creates and perceives that world around you, and evolution had a particular problem to solve. Our brain is growing, new connections being made daily which means that it either needs to have a container that grows with it or it needs to grow inward. That is what all of those wrinkles and folds in the brain are, the brain is growing in on itself, allowing the surface area of it to grow, but it volume stays almost the same. But how is that possible?
This is because the brain follows a fractal pattern, known as a spiral fractal.
You can see all of the spiral shapes that are apart of this picture, and you can also image in a general spiral over the entire brain. This shape is known as a Logarithmic Spiral (This just means that the formula that makes this spiral comes from a function known as a logarithm).
These shapes and patterns, are what made me value mathematics beyond its many other benefits. Mathematics has a beauty to it, and brings beauty to the world around us. This spiral pattern is not just in our brain. It is in the Milky Way Galaxy, in the way flowers grow leaves and petals, in our vegetables, in our architecture, art, and bodies. Mathematics is what allows us to delve deeper into these patterns and try to understand them, which gives us a better understanding of the world around us.
From a philosophical perspective, if this was taken outside of the context of math. Our lives are broken, not in a depressing way. But more in a scattered, chaotic sort of way. There is always a little bit of “ I have no idea whats going on” no matter how together one has their life, but looking back you could see how all of these things culminate into the life you have now. Our life is self repeating, changing constantly but always turning out the same.
