Rank-3 scale theorems

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 word of two DE scale imprints with the same number of notes. See http://www.springerlink.com/content/c23748337406x463/
• 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.

Unproven Conjectures

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