Universe, Game, Chaos

By Jonathan Lam on 08/03/17

Tagged: brain-dump

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You can see the relation between those words, right? Universe, game, chaos. It's all around us.

A simple example would be the second law of thermodynamics, that the universe always tends to move towards a state of higher entropy (chaos). Humans are major players in a game that attempts to create order in all of that chaos, and history has shown that the darkness and chaos can be ordered.

But what prompted me to post this is what is known as the "Chaos Game." According to Wikipedia, it "refer[s] to a method of creating a fractal, using a polygon and an initial point selected at random inside it." It's not what you would think of when you hear the name; what comes to mind to me is an arbitrary strategy board game that quickly degrades to anarchy and hilarious laughter. But it's not. It's actually something beautiful.

I came across the method when I was trying to create a Sierpinski fractal, of which one of the noted methods on Wikipedia was the Chaos Game. Here's the result. Scaled to fit a screen, its shape is recognizable, but when zoomed in you can see the colors. It's colorful static, color-coded to the chronology of the points. The randomization of the colors shows the randomization of the filling-in of the fractal.

What's so fascinating is that, for most polygons, if you choose a random spot, move it a specified ratio of the distance to a random vertex, plot the point, and repeat the moving and plotting, there's a high chance of ending up with a fractal. Sometimes a small "restriction," such as preventing repetition of chosen vertices, can dramatically change the fractal. For the Sierpinski fractal, it's as simple as a equilateral triangle and a half-distance ratio.

It's so simple. So elegant. And that's one of three things* I like about math.

* Another is being able to logically explain, prove, or understand a topic in many different ways, such as exponent properties and trigonometric proofs. Another interesting concept is when such unoptimized, unmathematical methods (as most of my coding solutions to difficult math problems are) work, mathematically. See the Monte Carlo method for a great example of this, which depends on probabilistic means (dumb luck) and brute-force even more than the Chaos Game.