Hypercubic billiard word - Revision history

Inthar: /* Properties */

2026-01-01T20:22:51Z

Properties

← Older revision		Revision as of 20:22, 1 January 2026
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	== Properties ==		== Properties ==
	Proofs to be added		Proofs to be added
	* A (circular) scale word is a rank-2 billiard scale if it is a MOS scale.		* A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
	* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position <math>\left(1, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{3}}\right)</math>, but it is not a Fokker block.		* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position <math>\left(1, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{3}}\right)</math>, but it is not a Fokker block.
	* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.		* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.

Inthar: /* Determining whether a scale word is a billiard scale */

2025-12-29T03:27:08Z

Determining whether a scale word is a billiard scale

← Older revision		Revision as of 03:27, 29 December 2025
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	The following discussion documents a naive algorithm for answering whether a circular word ''s'' over ''d'' letters with step signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub> is a billiard word with velocity <math>\mathbf{a} = \sum_{i}a_{i}\mathbf{e}_{i} \in \mathbb{Z}^{d}</math>:		The following discussion documents a naive algorithm for answering whether a circular word ''s'' over ''d'' letters with step signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub> is a billiard word with velocity <math>\mathbf{a} = \sum_{i}a_{i}\mathbf{e}_{i} \in \mathbb{Z}^{d}</math>:

	Consider the ''d''-dimensional prism <math>P = \prod^{d}_{i = 1}\left[0, a_{i}\right]</math>. Since the pattern in which the billiard line {{nowrap\|''L'' {{=}} ''L''(''t'')}} {{nowrap\|{{=}} '''a'''''t'' + ''b''}} hits integer coordinate hyperplanes (i.e. sets {{nowrap\|''x''<sub>''i''</sub> {{=}} ''n''}} for {{nowrap\|''n'' ∈ ℤ}}) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and {{nowrap\|''L'' : ℝ → ''P''}} as a periodic function with period 1. Because ''s'' is a billiard word, ''L cannot'' meet any point {{nowrap\|'''q''' {{=}} (''q''<sub>1</sub>, ..., ''q''<sub>''d''</sub>)}} ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap\|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap\|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap\|''m''<sub>''i''</sub> ∈ {{(}}0, ..., ''a''<sub>''i''</sub>{{)}}}} and {{nowrap\|''n''<sub>''j''</sub> ∈ {{(}}0, ..., ''b''<sub>''j''</sub>{{)}}}} corresponds to the affine hyperplane (which we call a ''constraint hyperplane'')		Consider the ''d''-dimensional prism <math>P = \prod^{d}_{i = 1}\left[0, a_{i}\right].</math> Since the pattern in which the billiard line {{nowrap\|''L'' {{=}} ''L''(''t'')}} {{nowrap\|{{=}} '''a'''''t'' + ''b''}} hits integer coordinate hyperplanes (i.e. sets {{nowrap\|''x''<sub>''i''</sub> {{=}} ''n''}} for {{nowrap\|''n'' ∈ ℤ}}) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and {{nowrap\|''L'' : ℝ → ''P''}} as a periodic function with period 1. Because ''s'' is a billiard word, ''L cannot'' meet any point {{nowrap\|'''q''' {{=}} (''q''<sub>1</sub>, ..., ''q''<sub>''d''</sub>)}} ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap\|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap\|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap\|''m''<sub>''i''</sub> ∈ {{(}}0, ..., ''a''<sub>''i''</sub>{{)}}}} and {{nowrap\|''n''<sub>''j''</sub> ∈ {{(}}0, ..., ''b''<sub>''j''</sub>{{)}}}} corresponds to the affine hyperplane (which we call a ''constraint hyperplane'')

	<math>H(m_i, n_j) = \operatorname{span}(\mathbf{a}, \mathbf{e}_1, ..., \hat{\mathbf{e}}_i, ..., \hat{\mathbf{e}}_j, ..., \mathbf{e}_r) + (m_i \mathbf{e}_i + n_j \mathbf{e}_j),</math>		<math>H(m_i, n_j) = \operatorname{span}(\mathbf{a}, \mathbf{e}_1, ..., \hat{\mathbf{e}}_i, ..., \hat{\mathbf{e}}_j, ..., \mathbf{e}_r) + (m_i \mathbf{e}_i + n_j \mathbf{e}_j),</math>

Inthar: /* Determining whether a scale word is a billiard scale */

2025-02-23T18:54:06Z

Determining whether a scale word is a billiard scale

← Older revision		Revision as of 18:54, 23 February 2025
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	== Determining whether a scale word is a billiard scale ==		== Determining whether a scale word is a billiard scale ==
	The following discussion documents a naive algorithm for answering whether a circular word ''s'' over ''d'' letters with step signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub> is a billiard word with velocity <math>\mathbf{a} = \sum_{i}a_{i}e_{i} \in \mathbb{Z}^{d}</math>:		The following discussion documents a naive algorithm for answering whether a circular word ''s'' over ''d'' letters with step signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub> is a billiard word with velocity <math>\mathbf{a} = \sum_{i}a_{i}\mathbf{e}_{i} \in \mathbb{Z}^{d}</math>:

	Consider the ''d''-dimensional prism <math>P = \prod^{d}_{i = 1}\left[0, a_{i}\right]</math>. Since the pattern in which the billiard line {{nowrap\|''L'' {{=}} ''L''(''t'')}} {{nowrap\|{{=}} '''a'''''t'' + ''b''}} hits integer coordinate hyperplanes (i.e. sets {{nowrap\|''x''<sub>''i''</sub> {{=}} ''n''}} for {{nowrap\|''n'' ∈ ℤ}}) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and {{nowrap\|''L'' : ℝ → ''P''}} as a periodic function with period 1. Because ''s'' is a billiard word, ''L cannot'' meet any point {{nowrap\|'''q''' {{=}} (''q''<sub>1</sub>, ..., ''q''<sub>''d''</sub>)}} ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap\|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap\|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap\|''m''<sub>''i''</sub> ∈ {{(}}0, ..., ''a''<sub>''i''</sub>{{)}}}} and {{nowrap\|''n''<sub>''j''</sub> ∈ {{(}}0, ..., ''b''<sub>''j''</sub>{{)}}}} corresponds to the affine hyperplane (which we call a ''constraint hyperplane'')		Consider the ''d''-dimensional prism <math>P = \prod^{d}_{i = 1}\left[0, a_{i}\right]</math>. Since the pattern in which the billiard line {{nowrap\|''L'' {{=}} ''L''(''t'')}} {{nowrap\|{{=}} '''a'''''t'' + ''b''}} hits integer coordinate hyperplanes (i.e. sets {{nowrap\|''x''<sub>''i''</sub> {{=}} ''n''}} for {{nowrap\|''n'' ∈ ℤ}}) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and {{nowrap\|''L'' : ℝ → ''P''}} as a periodic function with period 1. Because ''s'' is a billiard word, ''L cannot'' meet any point {{nowrap\|'''q''' {{=}} (''q''<sub>1</sub>, ..., ''q''<sub>''d''</sub>)}} ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap\|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap\|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap\|''m''<sub>''i''</sub> ∈ {{(}}0, ..., ''a''<sub>''i''</sub>{{)}}}} and {{nowrap\|''n''<sub>''j''</sub> ∈ {{(}}0, ..., ''b''<sub>''j''</sub>{{)}}}} corresponds to the affine hyperplane (which we call a ''constraint hyperplane'')

ArrowHead294: /* Open questions */ Lol, "cleverer"

2024-12-27T15:52:46Z

Open questions: Lol, "cleverer"

← Older revision		Revision as of 15:52, 27 December 2024
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	== Open questions ==		== Open questions ==
	* Does the generating function for ternary deletion-MOS necklaces admit an elegant expression? What about billiard scales over a given number of letters?		* Does the generating function for ternary deletion-MOS necklaces admit an elegant expression? What about billiard scales over a given number of letters?
	* Is there a ~~cleverer~~ algorithm for checking whether a scale on more than 2 letters is a billiard scale?		* Is there a more clever algorithm for checking whether a scale on more than 2 letters is a billiard scale?

	== See also ==		== See also ==

ArrowHead294 at 15:51, 27 December 2024

2024-12-27T15:51:41Z

Show changes

BudjarnLambeth: /* Properties */ typo

2024-02-18T06:13:20Z

Properties: typo

← Older revision		Revision as of 06:13, 18 February 2024
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	== Properties ==		== Properties ==
	Proofs to be added		Proofs to be added
	* A (circular) scale word is a rank-2 billiard scale ~~iff~~ it is a MOS scale.		* A (circular) scale word is a rank-2 billiard scale if it is a MOS scale.
	* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.		* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
	* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.		* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.

Inthar: /* Properties */

2024-02-14T19:47:38Z

Properties

← Older revision		Revision as of 19:47, 14 February 2024
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	Proofs to be added		Proofs to be added
	* A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.		* A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
	* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.		* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
	* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.		* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.
	** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.		** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.

Inthar: /* Mathematical overview */

2024-02-14T19:46:14Z

Mathematical overview

← Older revision		Revision as of 19:46, 14 February 2024
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	== Mathematical overview ==		== Mathematical overview ==
	In the ~~rational~~ case, let ''w'' be a scale ~~word~~ with signature ''a''<sub>1</sub>'''X'''<sub>1</sub> ... ''a''<sub>''d''</sub>'''X'''<sub>''d''</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>''d''</sub>), which we call the ''velocity''.		In the periodic case, let ''w'' be a word representing a periodic scale with signature ''a''<sub>1</sub>'''X'''<sub>1</sub> ... ''a''<sub>''d''</sub>'''X'''<sub>''d''</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>''d''</sub>), which we call the ''velocity''.
	We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ<sup>''d''</sup> such that the line '''a'''''t'' + '''b''' has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ ℤ that spell out the scale as you move in the positive ''t'' direction along that line.		We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ<sup>''d''</sup> such that the line '''a'''''t'' + '''b''' has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ ℤ that spell out the scale as you move in the positive ''t'' direction along that line.

Inthar at 19:45, 14 February 2024

2024-02-14T19:45:01Z

← Older revision		Revision as of 19:45, 14 February 2024
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	== Mathematical overview ==		== Mathematical overview ==
	In the rational case, let ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub> ... ''a''<sub>''d''</sub>X<sub>''d''</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>''d''</sub>), which we call the ''velocity''.		In the rational case, let ''w'' be a scale word with signature ''a''<sub>1</sub>'''X'''<sub>1</sub> ... ''a''<sub>''d''</sub>'''X'''<sub>''d''</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>''d''</sub>), which we call the ''velocity''.
	We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ<sup>''d''</sup> such that the line '''a'''''t'' + '''b''' has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ ℤ that spell out the scale as you move in the positive ''t'' direction along that line.		We call ''w'' a '''rank-'''''d'' '''billiard scale''' if there exists a vector '''b''' ∈ ℝ<sup>''d''</sup> such that the line '''a'''''t'' + '''b''' has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ ℤ that spell out the scale as you move in the positive ''t'' direction along that line.

Inthar: /* Determining whether a scale word is a billiard scale */

2024-02-14T19:44:04Z

Determining whether a scale word is a billiard scale

← Older revision		Revision as of 19:44, 14 February 2024
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	Now, using the identifications '''e'''<sub>''i''</sub> = 0 for ''i'' in {1, ..., ''d''} on ''P'' results in a smaller ''d''-torus ''C'' whose fundamental domain in ℝ<sup>''d''</sup> is the unit cube ''C̄'' = ∏<sup>''d''</sup><sub>''i''=1</sub> [0, 1]. The path ''L'' descends to ''L'' : ℝ → ''C'' which is still periodic with period 1. The constraint hyperplanes also descend to ''C''. Now unwrap ''C'' into ''C̄'', and regard ''L'' as a subset of ''C̄'' that is partitioned into disjoint line segments that travel from one facet (i.e. a (''d'' − 1)-dimensional face) of ''C̄'' to another. The reader is warned that to find (the images of) all of the constraint hyperplanes in ''C̄'', any constraint hyperplane that does not meet ''C̄'' should be shifted by integer increments in coordinates so that the shifted hyperplane does meet ''C̄''. The constraint hyperplanes partition ''C̄'' into finitely many regions (as they do for ''P''), and any valid billiard path ''L'' in ''C̄'' must meet len(''s'')-many of these regions before returning to its starting point.		Now, using the identifications '''e'''<sub>''i''</sub> = 0 for ''i'' in {1, ..., ''d''} on ''P'' results in a smaller ''d''-torus ''C'' whose fundamental domain in ℝ<sup>''d''</sup> is the unit cube ''C̄'' = ∏<sup>''d''</sup><sub>''i''=1</sub> [0, 1]. The path ''L'' descends to ''L'' : ℝ → ''C'' which is still periodic with period 1. The constraint hyperplanes also descend to ''C''. Now unwrap ''C'' into ''C̄'', and regard ''L'' as a subset of ''C̄'' that is partitioned into disjoint line segments that travel from one facet (i.e. a (''d'' − 1)-dimensional face) of ''C̄'' to another. The reader is warned that to find (the images of) all of the constraint hyperplanes in ''C̄'', any constraint hyperplane that does not meet ''C̄'' should be shifted by integer increments in coordinates so that the shifted hyperplane does meet ''C̄''. The constraint hyperplanes partition ''C̄'' into finitely many regions (as they do for ''P''), and any valid billiard path ''L'' in ''C̄'' must meet len(''s'')-many of these regions before returning to its starting point.

	Now we use the projection π, a linear map on ℝ<sup>''d''</sup> whose kernel is generated by '''a''', to project ''C̄'' to a (''d'' − 1)-dimensional convex polytope π(''C̄''). The constraint hyperplanes now become (''d'' − 2)-dimensional hyperplanes that partition π(''C̄'') into finitely many convex regions. The components of ''L'' now become points in π(''C̄''), and each region in the partition has at most one point of π(''L''). When ''L'' hits an integer coordinate hyperplane ''x''<sub>''i''</sub> = (some integer), the corresponding point in π(''L'') now shifts by −π('''e'''<sub>''i''</sub>), since the corresponding point in ''C̄'' must undergo a shift by −'''e'''<sub>''i''</sub> upon ''L'' hitting the coordinate hyperplane. Since ''L'' hits len(''s'') coordinate hyperplanes before returning to its starting region, we choose some region ~~and any~~ point in it and advance the point len(''s'') times, each point in the orbit corresponding to the coordinate of the hyperplane hit by ''L''. To find all billiard scales with signature '''a''', we simply iterate the procedure described in the previous sentence over all regions (all of which are convex polytopes) in the partition we obtained in π(''C̄''); we may choose the centroid of the region as the starting point of π(''L'').		Now we use the projection π, a linear map on ℝ<sup>''d''</sup> whose kernel is generated by '''a''', to project ''C̄'' to a (''d'' − 1)-dimensional convex polytope π(''C̄''). The constraint hyperplanes now become (''d'' − 2)-dimensional hyperplanes that partition π(''C̄'') into finitely many convex regions. The components of ''L'' now become points in π(''C̄''), and each region in the partition has at most one point of π(''L''). When ''L'' hits an integer coordinate hyperplane ''x''<sub>''i''</sub> = (some integer), the corresponding point in π(''L'') now shifts by −π('''e'''<sub>''i''</sub>), since the corresponding point in ''C̄'' must undergo a shift by −'''e'''<sub>''i''</sub> upon ''L'' hitting the coordinate hyperplane. Since ''L'' hits len(''s'') coordinate hyperplanes before returning to its starting region, we choose some region with a point of π(''L'') in it and advance the point len(''s'') times, each point in the orbit corresponding to the coordinate of the hyperplane hit by ''L''. To find all billiard scales with signature '''a''', we simply iterate the procedure described in the previous sentence over all regions (all of which are convex polytopes) in the partition we obtained in π(''C̄''); we may choose the centroid of the region as the starting point of π(''L'').

	The preceding method is redundant in that for chiral scales, one need not generate both chiralities manually using this method. This fact is realized via the symmetry of the coordinate planes under reflection about the orthogonal complement of '''a''', which reverses the orientation of the billiard trajectory.		The preceding method is redundant in that for chiral scales, one need not generate both chiralities manually using this method. This fact is realized via the symmetry of the coordinate planes under reflection about the orthogonal complement of '''a''', which reverses the orientation of the billiard trajectory.