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]].
==== Definition: 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> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many X<sub>''i''</sub> steps);
* ''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>j</sub>/α<sub>i</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'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.


[[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]]

Latest revision as of 00:05, 28 June 2025

Theorems

  • Every triple Fokker block is max variety 3.
  • 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 trihexagonal tiling on the lattice.
  • A scale imprint is that of a Fokker block if and only if it is the product of two DE scale imprints with the same number of notes. See 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
  • 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.
  • 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.

Conjectures

  • Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.
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