Hypercubic billiard word

Formally, let

• w be a scale word with signature a₁X₁, ..., a_rX_r (i.e. w is a scale word with a_i-many X_i steps);
• n = a₁ + ... + a_r be the length of w;
• L be a line of the form L(t) = (a₁, ..., a_r)t + v₀, where v₀ is a constant vector in R^r. We say that L is in generic position if L intersects the hyperplane x₁ = 0 at a point (0, α₁, α₂, ... α_r-1) where α_i and α_j/α_i for i ≠ j are irrational.

We say that w is a billiard scale if any line in generic position of the form (a₁, ..., a_r)t + v₀ 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.