Hypercubic billiard word: Difference between revisions
No edit summary |
|||
| Line 7: | Line 7: | ||
We say that ''w'' is 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. | We say that ''w'' is 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 == | ||
Proofs to be added | |||
* [[Mos]]ses are rank-2 billiard scales | * [[Mos]]ses 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 | * A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes | ||