Rank-3 scale theorems

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  • 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 word 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.

Unproven Conjectures

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