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, standard terms for them include '''cutting sequences''', '''cutting words'''<ref name="vuillon">Vuillon, L. (2003) or '''hypercubic billiard words'''. 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 (resulting in a periodic scale) or irrational (resulting in an aperiodic scale).

'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity|arities]]. Considered as infinite words, standard terms for them include '''cutting sequences''', '''cutting words'''<ref name="vuillon">Vuillon, L. (2003)</ref>, or '''hypercubic billiard words'''. 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 (resulting in a periodic scale) or irrational (resulting in an aperiodic scale).

In the binary case with irrational slope, billiard scales correspond to '''Sturmian words''', which are aperiodic "MOS" (i.e. [[strict variety]] 2) scales.

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	'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity\|arities]]. Considered as infinite words, standard terms for them include '''cutting sequences''', '''cutting words'''<ref name="vuillon">Vuillon, L. (2003) or '''hypercubic billiard words'''. 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 (resulting in a periodic scale) or irrational (resulting in an aperiodic scale).		'''Billiard scales''' are one possible generalization of [[MOS]] scales to higher [[arity\|arities]]. Considered as infinite words, standard terms for them include '''cutting sequences''', '''cutting words'''<ref name="vuillon">Vuillon, L. (2003)</ref>, or '''hypercubic billiard words'''. 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 (resulting in a periodic scale) or irrational (resulting in an aperiodic scale).
	In the binary case with irrational slope, billiard scales correspond to '''Sturmian words''', which are aperiodic "MOS" (i.e. [[strict variety]] 2) scales.		In the binary case with irrational slope, billiard scales correspond to '''Sturmian words''', which are aperiodic "MOS" (i.e. [[strict variety]] 2) scales.