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Chaos Games

The Chaos Games are really interesting. They are an example of strange attractors in random dynamical systems. Instead of deterministic rules being used to create fractals like seeds and Iterative Function Systems, global rules applied to random systems are also used to generate fractals.

Sierpinski Triangle

This one you can actually do in real life with some paper, a pen, and a 6 sided die, if you have some serious patience. Alternatively, its really easy for programming savvy people to whip up a program that does this, ive linked a webpage below for you to play with an interactive version.

Here is an interactive webpage which you may play the chaos game yourself, made in geogebra by Ben Sparks

Here is a video explaining the Chaos Game by Numberphile

Here is a short video explaining the specific mechanics of why the sierpinski triangle appears when doing this

Here is a video showcasing other examples of the Chaos Games with different shapes and rulesets

To start, draw a triangle, label the three verticies

     A
     /\
    /  \
   /    \
  /      \
B----------C

Now, grab a regular 6 sided dice. The rules of the game are this:

1. Start by picking any random point inside the triangle
2. roll the die, if it lands on 1 or 2 draw a point MIDWAY between your randomly chosen point and vertex A. If it lands on 3 or 4 do the same thing but with B, and 5 or 6 is C.
3. Once your midway point is plotted, roll the die again and repeat step 2, drawing a midway point between the point you just plotted and a random vertice.
4. do this a couple hundred times until a shape starts to emerge from the seeming randomness

At first, it will seem completely random where the points are plotted, but eventually with enough points a shape starts to appear. Yup, you guessed it, that shape is the sierpinski triangle! The edges of the sieripnski triangle act as whats known as a "strange attractor" because they quite literally attract points to specific areas.

There is no easy way to show this in ascii, so i recommend looking at the videos linked above or trying it out yourself through the geogebra website. Again, you could do it in real life with a paper, pen and a die but it takes hundreds of points to get a clear enough image to really see it, which means hundreds of dice throws.