# Reflection on Billiards and Math Feb 2024

Reflection on Billiards and Math: Variations on a Theme

Quanta article

Unfolding the Mysteries of Polygonal Billiards

This recent article is an excellent survey of the modern-day work on the subject of reflections in polygonal billiards tables.

Wolfram post

Illumination Problem – MathWorld

Good starting point to many good links.

Numberphile Video

The Illumination Problem – Numberphile

This recent article is an excellent survey of the modern-day work on the subject of reflections in polygonal billiards tables.

Steve Mould Video

Penrose Unilluminable Room Is Impossible To Light

Very good graphical and physical presentations of the Penrose room.

The Action Lab Video

I Made a Real-Life Unilluminable Room

An insightful physical experiment with both light reflection and water waves.

Berglund video

Tutorial: How to make billiard simulations

Great presentation by Nils Berglund.

Mathematical Puzzles, Revised Edition – Peter Winkler
Author of the wonderful ACM Puzzled series

From the book Mathematical Puzzles

Laser Gun (a related problem)

You find yourself standing in a large rectangular room with mirrored walls. At
another point in the room is your enemy, brandishing a laser gun. You and
she are fixed points in the room; your only defense is that you may summon
bodyguards (also points) and position them in the room to absorb the laser rays
for you. How many bodyguards do you need to block all possible shots by the
enemy?

The Assassin Puzzle (a related problem)

These articles give lay readers, like me, insight into a problem of physical reality and the development of abstractions in solving the problem.

I first saw the problem in reading Geometric Transformations by I. M. Yaglom (translated by Allen Shields). Yaglom solved problems similar to the billiard problem using mostly traditional geometric transformations such as translation, rotation, half-rotation, reflection, and glide-reflection. These transformations in the hands of Yaglom were shown to be mathematically powerful.

I am, at this later date, still intrigued by the billiards problem. Above, I offer several links regarding the 2D form of the billiard problem. I am still looking for similar discussion/solutions regarding a 3D version of the problem.

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