Hypercubic billiard word: Difference between revisions

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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''1''x''1...''a''''d''''x''''d'' 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''''i'' {{=}} ''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''1, ..., ''q''''d'')}} ∈ ℝ''d'' where two coordinates, ''q''''i'' and ''q''''j'', {{nowrap|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap|''m''''i'' ∈ {{(}}0, ..., ''a''''i''{{)}}}} and {{nowrap|''n''''j'' ∈ {{(}}0, ..., ''b''''j''{{)}}}} 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''''i'' {{=}} ''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''1, ..., ''q''''d'')}} ∈ ℝ''d'' where two coordinates, ''q''''i'' and ''q''''j'', {{nowrap|''i'' < ''j''}}, are integers. Thus for two distinct integers {{nowrap|''i'' < ''j''}} in {1, ..., ''d''}, any choice of two integers {{nowrap|''m''''i'' ∈ {{(}}0, ..., ''a''''i''{{)}}}} and {{nowrap|''n''''j'' ∈ {{(}}0, ..., ''b''''j''{{)}}}} 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>

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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>:
-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>)}} ∈&nbsp;ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap|''i'' &lt; ''j''}}, are integers. Thus for two distinct integers {{nowrap|''i'' &lt; ''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>)}} ∈&nbsp;ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, {{nowrap|''i'' &lt; ''j''}}, are integers. Thus for two distinct integers {{nowrap|''i'' &lt; ''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>