# Mathematical Billiards: Bouncing Balls and Hidden Patterns

> How computer experiments reveal periodic orbits in billiards games.
- Title: Mathematical Billiards: Bouncing Balls and Hidden Patterns
- Summary: How computer experiments reveal periodic orbits in billiards games. Computer simulations show that for specific triangle shapes in hyperbolic geometry…
- Keywords: mathematical billiards, periodic orbits, hyperbolic geometry, singularity sets, triangles, computer experiments, dynamical systems, technology, startups, Mathematical, Billiards, Bouncing
- Source: Uni Heidelberg — https://structures.uni-heidelberg.de/blog/posts/2024_01_costa/index.php
- Published: 2024-01-10T00:00:00+00:00
- Read time: 1 min
- Topics: mathematical billiards, periodic orbits, hyperbolic geometry, singularity sets, triangles, computer experiments, dynamical systems, technology, startups
## What's that structure?
I enjoy pocketing balls with complex shots in pool but miss often due to friction and collisions.

> I miss a lot of shots
## Inner Billiards
Mathematicians study ball paths bouncing inside convex shapes like ellipses.

> The resulting sequence of chords is called a trajectory
## Outer Billiards
A game where balls move outside the table following new mathematical rules.

> The point x is sent to the point y by what mathematicians call the outer billiards map
## Orbit Behavior
Orbits can be periodic or chaotic depending on the table shape.

> For an ellipse, every orbit lies on an ellipse sharing its foci
## Singularities
Flat sides cause map undefined points where orbits get stuck.

> The map has a singularity at x when reflecting across a flat side
## Periodic Islands
Some regions contain points that return to their starting position after orbits.

> Every starting point in that region returns to its starting point
## Hyperbolic Triangles
For specific triangle side lengths in hyperbolic geometry, orbits form tilings.

> For any integer k ≥ 7, a triangle with side length ℓ(k) creates a tiling pattern
## Outer Length Billiards
A variation where balls move using tangent circles and new points.

> Construct the unique circle tangent to the table at p and to the two lines
## Computer Experiments
Software helps discover patterns in billiard orbits that are hard to see manually.

> I discovered this by playing with the software until the pattern revealed itself
## Why Mathematicians Care
Billiards models physical systems and reveals hidden complexity in familiar spaces.

> I know Euclidean geometry like the back of my hand but there is so much still to learn
## Software Tools
Web-based tools use Angular Typescript and Three.JS for interactive billiards simulations.

> The code is written for the web using Angular Typescript with Three.JS
## Computer Experiments ≠ Proofs
Visualizations suggest patterns but rigorous proofs require deeper mathematical work.

> Once I have a little evidence, I can turn to the chalkboard and work out the proof
## Hyperbolic Tiling
For specific triangles, orbits tile the plane with predictable patterns.

> The singularity set looks exactly like a tiling of the plane by triangles and k-sided polygons
## Mathematical Motivation
Billiards connects to physics and reveals hidden complexity in familiar spaces.

> The standard billiards models a ball bouncing in a table or a photon in a mirrored room
## Future Research
Computers help find patterns but rigorous proofs remain challenging.

> I try to implement the behavior and play with the system to get a feel for what happens
## Conclusion
Computer experiments reveal periodic orbits in specific billiard games.

> For any integer k ≥ 7, a triangle with side length ℓ(k) creates a tiling pattern
## Key takeaway

Computer simulations show that for specific triangle shapes in hyperbolic geometry, billiard orbits form predictable tilings of the plane.