Structure from Chaos: Fractals Generated on the Circle of Fifths via the Chaos Game

Running the chaos game on the circle of fifths, producing music to accompany the beautiful fractals that result.

We start using a triangle on the notes C, E, and A♭ (forming an augmented chord), and play the chaos game to generate the Sierpiński triangle.

Then we use a hexagon on the notes C, D, E, G♭, A♭, and B♭ (a whole-tone scale). For optimal packing, the ratio used to divide the lines in the chaos game is 0.667 for a hexagon.

Next we use all 12 notes (the chromatic scale) to form a dodecagon fractal. The dodecagon is optimally packed with a ratio of 0.789 to divide the lines.

Finally, we use a square (i.e. a diamond) on the notes C, E♭, G♭, and A (forming a diminished 7th chord). Playing the normal chaos game on a square, however, doesn't yield a fractal. It only produces uniform noise within the square. When a simple restriction is added: not allowing any corner to be repeated twice in a row, a beautiful fractal results.

0:00 Sierpiński Triangle
3:54 Hexagon Chaos Game
7:05 Dodecagon Chaos Game
10:27 Square Chaos Game
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Interested in learning more about fractals, algorithms, and how to program? Here are some useful and/or classic textbooks that I recommend (these are affiliate links, if you buy one, I get a small commission):

▶ "The Fractal Geometry of Nature" by Benoit B. Mandelbrot: https://amzn.to/4apa7pT
▶ "Fractals Everywhere" by Michael F. Barnsley: https://amzn.to/4apgh9r
▶ “Algorithms” (4th Edition) by Robert Sedgewick & Kevin Wayne: https://amzn.to/3uo25xR
▶ “Effective Java” (3rd Edition) by Joshua Bloch: https://amzn.to/3HOnYJL
▶ “Design Patterns: Elements of Reusable Object-Oriented Software” by Erich Gamma, Richard Helm, Ralph Johnson, & John Vlissides: https://amzn.to/49fpr7R
▶ “Discrete Algorithmic Mathematics” by Stephen B. Maurer & Anthony Ralston: https://amzn.to/4bmsOvG

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