Hypercubic billiard word: Difference between revisions

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The following discussion documents a naive algorithm for answering whether a circular word ''s'' of arity ''d'' with velocity '''a''' = ∑''i'' ''a''''i'' '''e'''''i'' ∈ ℤ''d'' (representing the signature ''a''1''x''1...''a''''d''''x''''d'') is a billiard word:

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

Consider the ''d''-dimensional prism ''P'' = ∏''d''''i''=1 [0, ''a''''i'']. Since the pattern in which the billiard line ''L'' = ''L''(''t'') = '''a'''''t'' + ''b'' hits integer coordinate hyperplanes (i.e. sets ''x''''i'' = ''n'' for ''n'' ∈ ℤ) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and ''L'' : ℝ → ''P'' as a periodic function with period 1. Because ''s'' is a billiard word, ''L'' ''cannot'' meet any point '''q''' ∈ ℝ''d'' where two coordinates, ''q''''i'' and ''q''''j'', ''i'' < ''j'', are integers. Thus for two distinct integers ''i'' < ''j'' in {1, ..., ''d''}, any choice of two integers ''m''''i'' ∈ {0, ..., ''a''''i''} and ''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'' of arity ''d'' with velocity '''a''' = ∑<sub>''i''</sub> ''a''<sub>''i''</sub> '''e'''<sub>''i''</sub> ∈ ℤ<sup>''d''</sup> (representing the signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub>) is a billiard word:
-Consider the ''d''-dimensional prism ''P'' = ∏<sup>''d''</sup><sub>''i''=1</sub> [0, ''a''<sub>''i''</sub>]. Since the pattern in which the billiard line ''L'' = ''L''(''t'') = '''a'''''t'' + ''b'' hits the integer coordinate hyperplanes (i.e. the sets ''x''<sub>''i''</sub> = ''n'' for ''n'' ∈ ℤ) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and ''L'' : ℝ → ''P'' as a periodic function with period 1.  Because ''s'' is a billiard word, ''L'' ''cannot'' meet any point '''q''' ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, ''i'' < ''j'', are integers. Thus for two distinct integers ''i'' < ''j'' in {1, ..., ''d''}, any choice of two integers ''m''<sub>''i''</sub> ∈ {0, ..., ''a''<sub>''i''</sub>} and ''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 ''P'' = ∏<sup>''d''</sup><sub>''i''=1</sub> [0, ''a''<sub>''i''</sub>]. Since the pattern in which the billiard line ''L'' = ''L''(''t'') = '''a'''''t'' + ''b'' hits integer coordinate hyperplanes (i.e. sets ''x''<sub>''i''</sub> = ''n'' for ''n'' ∈ ℤ) is periodic with period 1 in ''t'', we may first regard ''P'' as a ''d''-torus and ''L'' : ℝ → ''P'' as a periodic function with period 1.  Because ''s'' is a billiard word, ''L'' ''cannot'' meet any point '''q''' ∈ ℝ<sup>''d''</sup> where two coordinates, ''q''<sub>''i''</sub> and ''q''<sub>''j''</sub>, ''i'' < ''j'', are integers. Thus for two distinct integers ''i'' < ''j'' in {1, ..., ''d''}, any choice of two integers ''m''<sub>''i''</sub> ∈ {0, ..., ''a''<sub>''i''</sub>} and ''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>