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Rank-3 scale theorems: Difference between revisions

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* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink]
* A scale imprint is that of a Fokker block if and only if it is the [[product word|product]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink]
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* Any convex object on the lattice can be converted into a hexagon.
* Any convex object on the lattice can be converted into a hexagon.
* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
 
* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].
== Unproven Conjectures ==
== Conjectures ==
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.
== MV3 proofs ==
Under construction
=== Definitions and theorems ===
Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used).
==== Definition: Unconditionally MV3 ====
''S'' is ''unconditionally MV3'' or ''intrinsically MV3'' if ''S'' is MV3 for all possible choices of step ratio x:y:z.
==== Definition: EMOS ====
''S'' is ''elimination-MOS'' (EMOS) if the result of removing (all instances of) any one of the step sizes is a MOS.
==== Definition: PMOS ====
''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
==== Definition: GO ====
''S'' satisfies the ''generator-offset property'' (GO) if it satisfies the following equivalent properties:
# ''S'' can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form  either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.
These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.
For theorems relating to the GO property, see [[generator-offset property]].
==== Definitions: Billiard scale ====
Let
* w be a scale word with signature a<sub>1</sub>X<sub>1</sub>, ..., a<sub>r</sub>X<sub>r</sub>;
* n = a<sub>1</sub> + ... + a<sub>r</sub> be the length of w;
* 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>i</sub>/α<sub>j</sub> for i ≠ j are irrational.
We say that ''w'' is a ''billiard scale'' if any appropriate line in generic position, (a<sub>1</sub>, ..., a<sub>r</sub>)t + v<sub>0</sub>, has intersections with coordinate level planes x<sup>i</sup> = k that spell out the scale as you move in the positive t direction.


[[Category:Fokker block]]
[[Category:Fokker block]]
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[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Scale]]
[[Category:Scale]]
[[Category:Theory]]
[[Category:Pages with open problems]]
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