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 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''' = (''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''' = (''a''1, ..., ''a''''r'') given by the scale signature. For integer ''a''''i'', the particle's trajectory will be periodic, and for almost any '''b''', 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.

Identifying opposite sides of the cubic room, thereby producing an ''r''-torus, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity '''a''', and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the set of lines that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — subdivides the ''r''-torus into finitely many regions each of which gives rise to a billiard scale.

Revision as of 11:19, 1 November 2023 view source Inthar (talk \| contribs) autopatrolled, emailconfirmed 15,674 edits No edit summary Tags: Mobile edit Mobile web edit ← Older edit		Revision as of 11:33, 1 November 2023 view source Inthar (talk \| contribs) autopatrolled, emailconfirmed 15,674 edits No edit summary Tags: Mobile edit Mobile web edit Newer edit →
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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 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''' = (''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''' = (''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 for almost any '''b''', 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.

	Identifying opposite sides of the cubic room, thereby producing an ''r''-torus, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity '''a''', and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the set of lines that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — subdivides the ''r''-torus into finitely many regions each of which gives rise to a billiard scale.		Identifying opposite sides of the cubic room, thereby producing an ''r''-torus, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity '''a''', and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the set of lines that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — subdivides the ''r''-torus into finitely many regions each of which gives rise to a billiard scale.