Rank-3 scale theorems
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2011-11-17 03:09:16 UTC.
- The original revision id was 276418896.
- The revision comment was: linked product word
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=Theorems= * Every triple Fokker block is max variety 3. * Every max variety 3 block is a triple 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 <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;">[[@http://www.springerlink.com/content/c23748337406x463/]]</span> * 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 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.
Original HTML content:
<html><head><title>Rank-3 scale theorems</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Theorems"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theorems</h1> <ul><li>Every triple Fokker block is max variety 3.</li><li>Every max variety 3 block is a triple Fokker block.</li><li>Triple Fokker blocks form a trihexagonal tiling on the lattice.</li><li>A scale imprint is that of a Fokker block if and only if it is the <a class="wiki_link" href="/product%20word">product word</a> of two DE scale imprints with the same number of notes. See <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank">http://www.springerlink.com/content/c23748337406x463/</a></span></li><li>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</li><li>Any convex object on the lattice can be converted into a hexagon.</li><li>Any 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><br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Unproven Conjectures"></a><!-- ws:end:WikiTextHeadingRule:2 -->Unproven Conjectures</h1> <ul><li>Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.</li></ul></body></html>