# Billiard scale

**Billiard scales** are motivated by considering a point particle (a "billiard ball") bouncing off walls in a closed cubic room. Given a scale signature *a*_{1}X_{1} ... *a*_{r}X_{r} (i.e. stipulating that our scale has *r* distinct, not-necessarily-linearly-independent step sizes X_{1}, ..., X_{r}, and the number of X_{i} steps in the scale is *a*_{i} > 0), we imagine our billiard ball in an *r*-dimensional cubic room (with side length 1). We first fire off the billiard ball in the direction (*a*_{1}, ..., *a*_{r}) given by the scale signature. For integer *a*_{i}, the particle's trajectory will be periodic, and with probability one, the particle will only collide with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.

## Formal definition

Formally, let

*w*be a scale word with signature*a*_{1}X_{1}...*a*_{r}X_{r}(i.e.*w*is a scale word with*a*_{i}-many X_{i}steps);*L*be a line of the form*L*(*t*) = (*a*_{1}, ...,*a*_{r})*t*+**v**_{0}, where**v**_{0}is a constant vector in**R**^{r}. We say that*L*is*in generic position*if*L*intersects the hyperplane*x*_{1}= 0 at a point (0, α_{1}, α_{2}, ... α_{r-1}) where α_{i}and α_{j}/α_{i}for*i*≠*j*are irrational.

We call *w* a **rank-***r* **billiard scale** if any line in generic position of the form (*a*_{1}, ..., *a*_{r})*t* + **v**_{0} has intersections with coordinate level planes *x*_{i} = *k* ∈ **Z** that spell out the scale as you move in the positive *t* direction along that line. (This definition is equivalent to the definition given in terms of a billiard ball in a cubic room.)

## Properties

Proofs to be added

- Mosses are rank-2 billiard scales
- Fokker blocks are billiard scales. (?)
- A billiard scale becomes a billiard scale of lower rank when one removes all instances of some subset of its step sizes
- That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.

- There are only finitely many billiard scales with a given signature up to rotation
- Finiteness is obvious; how does the number of billiard scales with a given signature depend on
*r*or on the signature?

- Finiteness is obvious; how does the number of billiard scales with a given signature depend on