Hypercubic billiard word: Difference between revisions
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== Mathematical overview == | == Mathematical overview == | ||
In the | In the periodic case, let ''w'' be a word representing a periodic scale with signature ''a''<sub>1</sub>'''X'''<sub>1</sub> ... ''a''<sub>''d''</sub>'''X'''<sub>''d''</sub> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many '''X'''<sub>''i''</sub> steps) and let '''a''' = (''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>), which we call the ''velocity''. | ||
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. | 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. | ||