Mathematical Billiards (2024)
6 hours ago
- Inner billiards studies trajectories of a point ball bouncing inside a convex shape following the law of reflection, with questions about periodic trajectories.
- Outer billiards is an abstract game where a point outside a shape is reflected through tangent points, leading to orbits that can be periodic or chaotic.
- Singularities in outer billiards occur when the map is undefined, such as along extensions of polygon sides, and can produce tiling-like or fractal singularity sets.
- In hyperbolic geometry, outer billiards on regular polygons can yield an infinite family of shapes with all trajectories periodic, unlike the Euclidean case.
- Outer length billiards is a newer variant where the distance traveled depends on the shape, and its dynamics show a mix of simplicity and complexity.
- Computer experiments are used to build intuition and discover patterns, but are not rigorous proofs; they complement theoretical work.
- The author developed software using Angular Typescript and Three.JS for visualization, and recommends ShaderToy for quick graphical experiments.