billiard-fractals

Billiard Fractals

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This project explores how simple rules - like a billiard ball bouncing in a rectangular grid - can produce complex, structured patterns when translated into symbolic sequences.

By reducing 2D trajectories to 1D symbolic sequences:

$Q_k=\left\lfloor k \sqrt{x} \right\rfloor \bmod 2$

we uncover recursive, quasi-fractal structures emerging purely from irrational steps and modular thresholds. These binary sequences, when rendered spatially, exhibit self-similarity, despite being entirely deterministic.

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Further extending this idea with nonlinear functions:

$Q_k=\left\lfloor k^2 \sqrt{x} \right\rfloor \bmod 2$

we observe patterns resembling interference textures or symbolic holography - generated not by waves, but by curved discretization.

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The work presented here is a translation and adaptation of my articles on Habr (Part 1, Part 2, Part 3, Part 4)

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License

MIT License. See LICENSE for details.

Contact

Serhii Herasymov

sergeygerasimofff@gmail.com

https://github.com/xcontcom

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