Hypercubic billiard word: Difference between revisions

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'''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''1X1 ... ''a''''r''X''r'' (i.e. stipulating that our scale has ''r'' distinct~~, not-necessarily-linearly-independent~~ step sizes X1, ..., 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 of the given signature, though for [[arity]] higher than 2, this can yield rotationally inequivalent scales depending on the starting point.

'''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''1X1 ... ''a''''r''X''r'' (i.e. stipulating that our scale has ''r'' distinct step sizes X1, ..., 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 of the given signature, though for [[arity]] higher than 2, this can yield rotationally inequivalent scales depending on the starting point.

== Formal definition ==

Formally, let ''w'' be a scale word with signature ''a''1X1 ... ''a''''r''X''r'' (i.e. ''w'' is a scale word with ''a''''i''-many X''i'' steps) and let '''a''' = (''a''1, ..., ''a''''r'').

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-'''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''<sub>1</sub>X<sub>1</sub> ... ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. stipulating that our scale has ''r'' distinct, not-necessarily-linearly-independent step sizes X<sub>1</sub>, ..., X<sub>''r''</sub>, and the number of X<sub>''i''</sub> steps in the scale is ''a''<sub>''i''</sub> > 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''<sub>1</sub>, ..., ''a''<sub>''r''</sub>) given by the scale signature. For integer ''a''<sub>''i''</sub>, 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 of the given signature, though for [[arity]] higher than 2, this can yield rotationally inequivalent scales depending on the starting point.
+'''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''<sub>1</sub>X<sub>1</sub> ... ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. stipulating that our scale has ''r'' distinct step sizes X<sub>1</sub>, ..., X<sub>''r''</sub>, and the number of X<sub>''i''</sub> steps in the scale is ''a''<sub>''i''</sub> > 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''<sub>1</sub>, ..., ''a''<sub>''r''</sub>) given by the scale signature. For integer ''a''<sub>''i''</sub>, 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 of the given signature, though for [[arity]] higher than 2, this can yield rotationally inequivalent scales depending on the starting point.
 == Formal definition ==
 Formally, let ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub> ... ''a''<sub>''r''</sub>X<sub>''r''</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>''r''</sub>).