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. ~~The~~ particle's trajectory will be periodic, and with probability one, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.

'''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, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.

== Formal definition ==

Formally, let

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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. The particle's trajectory will be periodic, and with probability one, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.
+'''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, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.
 == Formal definition ==
 Formally, let