One of the best-known features of Jupiter is the great red spot. It’s been there for hundreds of years, and it’s a big storm, a vast hurricane on Jupiter. And when the Voyager spacecraft got a look at that, it noticed you get complicated, turbulent vortices spinning off. What we see on Jupiter is turbulence. Big vortices, smaller vortices, tiny vortices, and if we could look closely — smaller and smaller and smaller ones.
That’s the example of chaos; more specifically, it’s the geometry of chaos. Any geometrical object with that structure is called a fractal. So fractals are the geometry of chaos. They’re unlike all the usual things you get in geometry.
If you take something like a circle and make it bigger and bigger and bigger, it just gets flattered. If you look closely enough at it, you will notice that pretty easily. Yes, there is nothing unusual for that example. If you start with a sphere and make it bigger and bigger, it just seems flattered and flattered like a plane. That’s why a lot of people thought the Earth was flat in ancient times.
Fractals are different. When you magnify them, you see more and more and more structure. And yet, that very intricate geometry can be created by straightforward mathematical rules. In this article, I will demonstrate an example of a fractal pattern created by simple mathematical rules.
To do this, we need a triangle marked with 1, 2, 3, a ruler, some pile of red dots, and a three-sided die. Our three-sided die is just an ordinary die, but we will put the same value on the opposite side.
So what we do is here, we start by putting a dot somewhere in the triangle randomly. In other words, we can put the dot anywhere in the triangle. After that, we roll the dice, and we get a number between 1 and 3. Then we will use the ruler to connect the corner and the first dot. We get a line. That line is crucial because we will put another dot in the middle of the line. After that, we will roll the dice and again and get a number. Then we will connect the corner and the last dot we put. Similarly, we will put another dot in the middle of our new line. We will keep applying this again and again and again.
As you can see, there are some simple mathematical rules;
- Roll the die.
- Start from wherever you are and go halfway towards the corner that the die has given you.
- And repeat these steps.
The question is, what shape would you expect to see? What happens if you do it thousands of times? If we use a computer and we’ll see what pattern builds up exactly. In the beginning, nothing will be interesting, but if we keep going, we will start to see a lovely, surprising mathematical pattern appearing.
It is called the Sierpinski gasket. All of our car mechanics know that a gasket is a thing you take out of your car, and it’s full of holes of different sizes. Sierpinski was a Polish mathematician who invented this shape for totally different reasons. It’s got big holes in the middle and then in pieces around, and there are smaller holes and smaller holes and smaller holes and smaller holes. Mathematically those holes go on forever.
That is typical of fractals. Simple rules create structure on all scales, even the finest. To learn more, you can read this book.