# Polygonal billiards and uniform distribution

According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional.

My question is motivated by a videogame I've been playing lately, which can be seen at http://www.youtube.com/watch?v=LLLmfwxNJYU.

Essentially, the "physics" of the game involves several billiards in a polygon, and the player has to slash off pieces of the polygon while avoiding the billiards. Also, the removed piece has to be void of billiards.

I've been assuming that the distribution of billiards is in the long run uniform, in some hand waving sense. That's assuming that initial distributions and velocities are random. Is that true, or can the polygon be shaped in various ways to make that distribution non-uniform? In other words, are certain regions of a polygon are more likely to be void than other regions?

(I believe that a term like "ergodic" applies to this, but I'm not confident using it).

• A nice general survey of the mathematics of billiards problems can be found in the book: Geometry of Billiards, Serge Tabachnikov, American Mathematical Society, 2005. Oct 4, 2010 at 19:54
• A preprint of the book Geometry and Billiards is currently online at personal.psu.edu/sot2/books/billiardsgeometry.pdf. Lots of interesting related papers at personal.psu.edu/sot2/prints/preprints.html Feb 22, 2021 at 22:54

## 3 Answers

A periodic billiard path is one which returns to a point with the same direction it had before at that point. A dense billiard path is one that covers the whole region. (A periodic billiard path is not dense and thus cannot be uniformly distributed.)

Many polygons have periodic billiard paths. (For example, every acute triangle and every right triangle has a periodic billiard path. But it is not yet known whether every obtuse triangle has a periodic billiard path.) So it is possible for a polygon to have a path that is not dense and thus not uniformly distributed.

It is possible for a billiard path to be dense and yet not be uniformly distributed. This is true for the triangle with angles $0.4\pi,0.3\pi,0.3\pi$. (See, for example, theorem 1.3 of http://homepages.math.uic.edu/~demarco/billiards.pdf.)

You might also want to visit https://mathoverflow.net/questions/53641/dense-orbits-in-billiards.

I agree with Joel's answer, but it doesn't address the question of random directions. See the first sentence of

http://arxiv.org/pdf/math/0701658.pdf

which was subsequently published. For rational billiards (ie, all angles a rational multiple of $\pi$), the dynamics is ergodic in almost all directions, and in particular it is uniformly distributed in space.

Much less is known about irrational polygons; it is possible this is an open question.

The very difficult questions that has emerged over the past 20 years in billiard theory is the following one : is there a non-periodic orbit that is not everywhere dense in the billiard. a few years ago, Galperin thought that he had found one but an other mathematician, Tokarsky, proved that Galperin's orbit was actually a periodic one. So this is still an open question for specialists but it seems from computer simulations and theoretical results (see Masur, Boshernitzan, Tabachnikov...) that a non periodic orbit will be everywhere dense in the polygon.