Hypercubic billiard word: Difference between revisions
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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, if we choose any point in π(''L'') and shift it 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'' − 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, if we choose any point in π(''L'') and shift it 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''). | ||
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 | 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'''. | ||
== Open questions == | == Open questions == | ||