Hypercubic billiard word: Difference between revisions

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'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity|arities]]. Considered as infinite words, a standard term for them is '''cutting words'''<ref>Vuillon, L. (2003). Balanced words. Bulletin of the Belgian Mathematical Society-Simon Stevin, 10(5), 787-805.</ref> ~~with rational slope~~. 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 or irrational.

'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity|arities]]. Considered as infinite words, a standard term for them is '''cutting words'''<ref>Vuillon, L. (2003). Balanced words. Bulletin of the Belgian Mathematical Society-Simon Stevin, 10(5), 787-805.</ref>. 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 or irrational.

== Mathematical overview ==

In the rational case, 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''), which we call the ''velocity''.

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-'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity|arities]]. Considered as infinite words, a standard term for them is '''cutting words'''<ref>Vuillon, L. (2003). Balanced words. Bulletin of the Belgian Mathematical Society-Simon Stevin, 10(5), 787-805.</ref> with rational slope. 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 or irrational.
+'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity|arities]]. Considered as infinite words, a standard term for them is '''cutting words'''<ref>Vuillon, L. (2003). Balanced words. Bulletin of the Belgian Mathematical Society-Simon Stevin, 10(5), 787-805.</ref>. 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 or irrational.
 == Mathematical overview ==
 In the rational case, 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>), which we call the ''velocity''.