I greatly recommend watching it and subscribing the channel if you haven't yet done so - it's full of all sorts of mathematical goodness for the mathematical geek in you.
I made an interactive JavaScript implementation of the game, which you can check out HERE (see fig. 1).
The code for the demo is available on GitHub: https://github.com/dagothar/chaos-game. Let me know if you spot any bugs!
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| Fig. 1. Chaos game demonstration. |
So, what is the Chaos Game?
It's a method for creating fractals. You start with a blank page, a set of vertices arranged in a polygon and a randomly selected starting point. Each turn you select one of the vertices of the polygon and travel towards it covering a certain portion of the distance (for example 1/2 - it is also the same each turn). You draw a dot each time you stop to pick another vertex to travel to. The procedure, simple enough as it sounds, yields some surprisingly complex results.
With a starting set of vertices arranged in a triangle, and covering half the distance to the randomly selected vertex each time, a Sierpiński triangle emerges (see fig. 2).
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| Fig. 2. Sierpiński triangle is made on a triangular grid with the step parameter set to 0.5. |
More complicated setups yield deservedly rewarding fruits (see fig. 3).
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| Fig. 3. Pentagonal fractal with step=0.67. |
A very curious thing happens when a portion of the game board is barred against the intrusion of the randomly wandering point. In that case, the shape of the restricted area is repeated fractalised all over the board in form of the areas untouched by the point iterations (see fig. 4). The figure shows the blog icon repeated within a square setup of control points. Is it possible to do the same inside a triangle or a pentagon? Does the step size parameter has to be changed?
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| Fig. 4. DeltaStep logo fractalised in a square after 100,000 iterations. |
How to use the demo?
Simply click Start. The game will be played automatically between the vertices of the polygon specified by the red dots. You can Pause the game at any time.
Clear removes the dots, but not the vertices. You can also Download the currently generated image yo your computer.
The Step size slider controls the distance which is covered towards the next randomly selected vertex each iteration (0.0 to 1.0).
You can control the speed at which the new dots are generated. By default, the callback is invoked every 1 ms, but that speed is mostly likely limited by your hardware. You can slow the game down up to 1 dot/sec by using the slider.
You can also customize the colors and the dot size.
Controlling the control points. You can move the control points by hovering over them, clicking and dragging them around. To remove the control point, simply drag it outside the board.
You can add new control points by clicking the Add new point button on the left side of the interface. The new point is placed in the middle of the board, from where you can drag it to wherever you like.
You can make any regular polygon setups by using the Make n-gon button. The number on the right of the button controls the number of the regular polygon vertices.
Defining the restricted area. You can paint on the board with your mouse. Simply click and drag around outside the control points. The painted area will be blocked for the game iterations, thus allowing you to create the fractals of custom patterns. (Note: do not block too much of the board - the script may get stuck trying to find a new possible jump target).
There are several unanswered questions the game leaves us with. What happens when the control points are placed on the vertices of a concave polygon? What if the polygon is distorted? What does the step size actually do? How do you achieve the fractalisation effect with more or less points than four?
The chaos game invites.
Literature
[1] Numberphile, Chaos game video, available: https://www.youtube.com/watch?v=kbKtFN71Lfs
[2] Wikipedia, Chaos game, available online: https://en.wikipedia.org/wiki/Chaos_game
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