🎲 Chaos Game
The chaos game (Michael Barnsley, 1988): start anywhere, repeatedly pick a random target vertex and jump a fraction of the way toward it. After thousands of iterations a perfect fractal attractor emerges — despite the randomness. The fraction and rules (e.g. forbidding the same vertex twice) change which fractal appears. This is an instance of an Iterated Function System (IFS). 🇺🇦 Українська
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Why Does Order Emerge from Chaos?
Each "jump halfway" operation is a contraction mapping. By the Banach fixed-point theorem, repeated contractions must converge to a unique fixed set — the attractor of the IFS. The Sierpiński triangle with r = 1/2 has fractal dimension log(3)/log(2) ≈ 1.585. Changing r shifts the dimension and changes the attractor — at r = 2/3 you get a filled triangle. The Barnsley fern uses four different affine transforms with different probabilities: 85% of the time the fern grows a leaflet, 7% rotates to the right sub-frond, etc.