Hypercubic billiard word: Difference between revisions
m →Properties: typo |
|||
| Line 7: | Line 7: | ||
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. (This definition is equivalent to the definition given in terms of a billiard ball in a cubic room.) | 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. (This definition is equivalent to the definition given in terms of a billiard ball in a cubic room.) | ||
== Properties == | == Properties ==<nowiki> | ||
Proofs to be added | Proofs to be added | ||
* [[Mos]]ses are rank-2 billiard scales | * [[Mos]]ses are rank-2 billiard scales | ||
| Line 15: | Line 15: | ||
* There are only finitely many billiard scales with a given signature up to rotation | * There are only finitely many billiard scales with a given signature up to rotation | ||
** Finiteness is obvious; how does the number of billiard scales with a given signature depend on ''r'' or on the signature? | ** Finiteness is obvious; how does the number of billiard scales with a given signature depend on ''r'' or on the signature? | ||
* Billiard scales are balanced, which implies that an n-ary billiard scale is MV(n choose floor(n/2)). (Bulgakova et al., 2023) | |||
[[Category:Scale]] | [[Category:Scale]]</nowiki> | ||