Hypercubic billiard word: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
'''Billiard scales''' are motivated by considering a point particle (a "billiard ball") that moves in an ''r''-dimensional cubic room and room's (''r'' − 1)-dimensional walls. Given a scale [[signature]] ''a''<sub>1</sub>X<sub>1</sub>, ..., ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. stipulating that our scale has ''r'' distinct, not-necessarily-linearly-independent step sizes X<sub>1</sub>, ..., X<sub>''r''</sub>, and ''a''<sub>''i''</sub>-many X<sub>''i''</sub> steps), we imagine firing the billiard ball off in the corresponding direction (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>) given by the scale signature. The particle's trajectory will be periodic, and with probability one, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale. | |||
== Formal definition == | == Formal definition == | ||
Formally, let | Formally, let | ||
| Line 5: | Line 6: | ||
* ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''<sub>1</sub> = 0 at a point (0, α<sub>1</sub>, α<sub>2</sub>, ... α<sub>''r''-1</sub>) where α<sub>''i''</sub> and α<sub>''j''</sub>/α<sub>''i''</sub> for ''i'' ≠ ''j'' are irrational. | * ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''<sub>1</sub> = 0 at a point (0, α<sub>1</sub>, α<sub>2</sub>, ... α<sub>''r''-1</sub>) where α<sub>''i''</sub> and α<sub>''j''</sub>/α<sub>''i''</sub> for ''i'' ≠ ''j'' are irrational. | ||
We | We call ''w'' a '''rank-'''''r'' '''billiard scale''' if any line in generic position of the form (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + ''v''<sub>0</sub> 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. | ||
== Properties == | == Properties == | ||
Revision as of 04:02, 2 June 2022
Billiard scales are motivated by considering a point particle (a "billiard ball") that moves in an r-dimensional cubic room and room's (r − 1)-dimensional walls. Given a scale signature a1X1, ..., arXr (i.e. stipulating that our scale has r distinct, not-necessarily-linearly-independent step sizes X1, ..., Xr, and ai-many Xi steps), we imagine firing the billiard ball off in the corresponding direction (a1, ..., ar) given by the scale signature. The particle's trajectory will be periodic, and with probability one, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.
Formal definition
Formally, let
- w be a scale word with signature a1X1, ..., arXr (i.e. w is a scale word with ai-many Xi steps);
- n = a1 + ... + ar be the length of w;
- L be a line of the form L(t) = (a1, ..., ar)t + v0, where v0 is a constant vector in Rr. We say that L is in generic position if L intersects the hyperplane x1 = 0 at a point (0, α1, α2, ... αr-1) where αi and αj/αi for i ≠ j are irrational.
We call w a rank-r billiard scale if any line in generic position of the form (a1, ..., ar)t + v0 has intersections with coordinate level planes xi = k ∈ Z that spell out the scale as you move in the positive t direction along that line.
Properties
Proofs to be added
- Mosses are rank-2 billiard scales
- A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes