Hypercubic billiard word: Difference between revisions

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We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ''d'' such that the line '''a'''''t'' + '''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''' = (''a''1, ..., ''a''''d'') 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.

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 velocity '''a''' = (''a''1, ..., ''a''''d'') 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 a ''d''-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 ''d''-torus into finitely many regions each of which gives rise to a billiard scale.

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 We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ<sup>''d''</sup> such that the line '''a'''''t'' + '''b''' has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''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''' = (''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>) given by the scale signature. For integer ''a''<sub>''i''</sub>, 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.
+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 velocity '''a''' = (''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>) given by the scale signature. For integer ''a''<sub>''i''</sub>, 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 a ''d''-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 &mdash; namely those that have a point that has multiple integer coordinates &mdash; subdivides the ''d''-torus into finitely many regions each of which gives rise to a billiard scale.