Hypercubic billiard word: Difference between revisions

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Now, using the identifications '''e'''''i'' = 0 for ''i'' in {1, ..., ''d''} on ''P'' results in a smaller ''d''-torus ''C'' whose fundamental domain in ℝ''d'' is the unit cube ''C̄'' = ∏''d''''i''=1 [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''d'' 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''''i'' = (some integer), the corresponding point in π(''L'') now shifts by −π('''e'''''i''), since the corresponding point in ''C̄'' must undergo a shift by −'''e'''''i'' 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 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''d'' 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''''i'' = (some integer), the corresponding point in π(''L'') now shifts by −π('''e'''''i''), since the corresponding point in ''C̄'' must undergo a shift by −'''e'''''i'' 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'').

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.

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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'' &minus; 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'' &minus; 1)-dimensional convex polytope π(''C̄''). The constraint hyperplanes now become (''d'' &minus; 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 &minus;π('''e'''<sub>''i''</sub>), since the corresponding point in ''C̄'' must undergo a shift by &minus;'''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 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'' &minus; 1)-dimensional convex polytope π(''C̄''). The constraint hyperplanes now become (''d'' &minus; 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 &minus;π('''e'''<sub>''i''</sub>), since the corresponding point in ''C̄'' must undergo a shift by &minus;'''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'').
 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.