Skip to content

Exploring fractals generated from symbolic billiards, irrational discretization, and curved surface slicing. A bridge between number theory and emergent geometry.

License

Notifications You must be signed in to change notification settings

xcontcom/billiard-fractals

Folders and files

NameName
Last commit message
Last commit date

Latest commit

 
 
 
 
 
 
 
 
 

Repository files navigation

Billiard Fractals

Preview

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.

Preview

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.

Preview

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

About

Exploring fractals generated from symbolic billiards, irrational discretization, and curved surface slicing. A bridge between number theory and emergent geometry.

Topics

Resources

License

Stars

Watchers

Forks

Releases

No releases published

Packages

No packages published