Hypercubic billiard word

Billiard scales are one possible generalization of MOS scales to higher arities. Considered as infinite words, a standard term for them is cutting words. They can be visualized by considering a point particle (a "billiard ball") bouncing off walls in a closed cubic room. The ratio between the numbers of the step sizes (associated with the direction of the billiard ball) may be rational (resulting in a periodic scale) or irrational (resulting in an aperiodic scale). In the binary case with irrational slope, billiard scales correspond to Sturmian words, which are aperiodic "MOS" (i.e. strict variety 2) scales.

Mathematical overview

In the rational case, 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) and let a = (a₁, ..., a_r), which we call the velocity. We call w a rank-r billiard scale if there exists a vector b ∈ R^r such that the line at + b 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: We first fire off the billiard ball in the direction a = (a₁, ..., a_r) given by the scale signature. For integer a_i, the particle's trajectory will be periodic, and for almost any starting point, 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 of the given signature, though for arity higher than 2, this can yield rotationally inequivalent scales depending on the starting point.

Identifying opposite sides of the cubic room, thereby producing an r-torus, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity a, and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the lines parallel to a that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — subdivides the r-torus into finitely many regions each of which gives rise to a billiard scale.

Properties

Proofs to be added

• A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
• Not all billiard scales are Fokker blocks; blackdye can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
• A billiard scale becomes a billiard scale of lower rank when one removes all instances of some subset of its step sizes. However, the converse is false.
  • That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
• Not all billiard words of arity higher than 2 are balanced. (In binary scales, the term "(1-)balanced" becomes one characterization of the MOS property.) Ternary billiard scales that are 1-balanced are MV3.

Determining whether a scale word is a billiard scale

The following discussion documents a naive algorithm for answering whether a circular word s of arity r with signature vector a = ∑_i a_i e_i ∈ ℝ^r corresponding to the signature a₁x₁...a_rx_r is a billiard word:

Consider the r-dimensional prism P = ∏^r_i=1 [0, a_i]. Since the pattern in which the billiard line L = L(t) = at + b hits the integer coordinate hyperplanes (i.e. the sets x_i = n for n ∈ ℤ) is periodic with period 1 in t, we may first regard P as an r-torus and L : ℝ → P as a periodic function with period 1. Because s is a billiard word, L cannot meet any point q ∈ ℝ^r where two coordinates are integers. Thus for two distinct integers i < j in {1, ..., r}, any choice of two integers m_i ∈ {0, ..., a_i} and n_j ∈ {0, ..., b_j} corresponds to the affine hyperplane (which we call a constraint hyperplane)

[math]\displaystyle{ H(m_i, n_j) = \operatorname{span}(\mathbf{a}, \mathbf{e}_1, ..., \hat{\mathbf{e}}_i, ..., \hat{\mathbf{e}}_j, ..., \mathbf{e}_r) + (m_i \mathbf{e}_i + n_j \mathbf{e}_j), }[/math]

where the circumflexes indicate that the ith and jth basis vectors are to be omitted. In particular, L and H(m_i, n_j) are disjoint for any i < j, any m_i ∈ {0, ..., a_i − 1}, and any n_j ∈ {0, ..., b_j − 1}.

Now, using the identifications e_i = 0 for i in {1, ..., r} on P results in a smaller r-torus C whose fundamental domain in ℝ^r is the unit cube C̄ = ∏^r_i=1 [0, 1].

The path L descends to L : ℝ → C which is still periodic with period 1. The constraint hyperplanes also descend to C. Now unwrap C to C̄, and regard L as a subset of C̄ that is partitioned into disjoint line segments that travel from one facet (i.e. an (r − 1)-dimensional face) of C̄ to another. The reader is warned that to find (the images of) all of the constraint hyperplanes in C̄, any constraint hyperplane that does not meet C̄ should be shifted by integer increments in coordinates so that the shifted hyperplane does meet C̄. The constraint hyperplanes partition C̄ into finitely many regions (as they do for P), and any valid billiard path L in C̄ must meet len(s)-many of these regions before returning to its starting point.

Now we use the projection π, a linear map on ℝ^r whose kernel is generated by a, to project C̄ to an (r − 1)-dimensional convex polytope π(C̄). The constraint hyperplanes now become (r − 2)-dimensional hyperplanes that partition π(C̄) into finitely many convex regions. The components of L now become points in π(C̄), and each region in the partition has at most one point of π(L). When L hits an integer coordinate hyperplane x_i = (some integer), the corresponding point in π(L) now shifts by −π(e_i). Since L hits len(s) coordinate hyperplanes before returning to its starting region, if we choose any point in π(L) and shift it len(s) times, each corresponding to the coordinate of the hyperplane hit by L. To find all billiard scales with signature a, we simply iterate the procedure described in the previous sentence over all regions in the partition we obtained in π(C̄); we may choose the centroid of the region (which is a convex polytope) as the starting point of π(L).

Questions

1. A circular word s is d-balanced if for any k ≥ 1 and for any pair of length-k subwords w and w' of s,

[math]\displaystyle{ \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d, }[/math]

where |u|_xi is the number of occurrences of the letter x_i in the word u. Is it the case that for all r ≥ 1, all r-ary billiard circular words are (r − 1)-balanced? (The answer is known to be "yes" for r = 1, 2.)

2. Must ternary billiard scales have maximum variety at most 4, the MV4 ones being exactly the ones of balance 2? Does billiardness impose a deterministic relationship between arity, maximum variety and balance?

See also

• List of ternary billiard words of length at most 31