Rank-3 scale theorems: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 275237972 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 275491866 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-11- | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-15 00:17:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>275491866</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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* Every Fokker block is the product word of two MOS scales. See <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;">[[@http://www.springerlink.com/content/c23748337406x463/]]</span> | * Every Fokker block is the product word of two MOS scales. 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 | * 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= | =Unproven Conjectures= | ||
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<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Rank-3 scale theorems</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theorems"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theorems</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Rank-3 scale theorems</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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>Every Fokker block is the product word of two MOS scales. 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 &gt; m &gt; n &gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s</li></ul><br /> | <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>Every Fokker block is the product word of two MOS scales. 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 &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 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:&lt;h1&gt; --><h1 id="toc1"><a name="Unproven Conjectures"></a><!-- ws:end:WikiTextHeadingRule:2 -->Unproven Conjectures</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><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 &lt; 4, meaning that some interval class will come in less than 4 sizes.</li></ul></body></html></pre></div> | <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></body></html></pre></div> | ||