Rank-3 scale theorems: Difference between revisions

=Theorems=

<ul><li>Every triple [[Fokker_blocks|Fokker block]] is max variety 3.</li><li>Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)</li><li>Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.</li><li>A scale imprint is that of a Fokker block if and only if it is the [[Product_word|product word]] of two DE scale imprints with the same number of notes. See <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;">[http://www.springerlink.com/content/c23748337406x463/ http://www.springerlink.com/content/c23748337406x463/]</span></li><li>If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &gt; m &gt; n &gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s</li><li>Any convex object on the lattice can be converted into a hexagon.</li><li>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.</li></ul>

=Unproven Conjectures=

<ul><li>Every rank-3 Fokker block has mean-variety &lt; 4, meaning that some interval class will come in less than 4 sizes.</li></ul>      [[Category:fokker_block]]

[[Category:math]]

[[Category:rank_3]]

[[Category:scales]]

[[Category:theory]]