Hypercubic billiard word: Difference between revisions

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Proofs to be added

* A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.

* ~~Not~~ all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.

* All [[distributionally even]] scales on any finite number of letters are billiard scales.<ref name="sano"/>, but the converse does not hold. This is because not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.

* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. However, the converse is false.

** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.

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* If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4.

* A ''d''-ary billiard scale satisfies the two generalizations of balance listed on the article [[balanced word]].

* All [[distributionally even]] scales on any finite number of letters are billiard scales.<ref name="sano"/>

== Determining whether a scale word is a billiard scale ==

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 Proofs to be added
 * A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
-* Not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
+* All [[distributionally even]] scales on any finite number of letters are billiard scales.<ref name="sano"/>, but the converse does not hold. This is because not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
 * A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. However, the converse is false.
 ** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
@@ Line 26: / Line 26: @@
 * If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4.
 * A ''d''-ary billiard scale satisfies the two generalizations of balance listed on the article [[balanced word]].
-* All [[distributionally even]] scales on any finite number of letters are billiard scales.<ref name="sano"/>
 == Determining whether a scale word is a billiard scale ==