Rank-3 scale theorems: Difference between revisions
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Wikispaces>keenanpepper **Imported revision 276832258 - Original comment: ** |
Wikispaces>guest **Imported revision 406883482 - 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: | : This revision was by author [[User:guest|guest]] and made on <tt>2013-02-14 01:55:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>406883482</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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<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;white-space: pre-wrap ! important" class="old-revision-html">=Theorems= | <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;white-space: pre-wrap ! important" class="old-revision-html">=Theorems= | ||
* Every triple [[Fokker blocks|Fokker block]] is max variety 3. | * Every triple [[Fokker blocks|Fokker block]] is max variety 3. | ||
* Every max variety 3 block is a triple Fokker block. | * 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. | * 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> | * 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> | ||
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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 <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> 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 &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><br /> | <ul><li>Every triple <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> 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 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 &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><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> | ||
Revision as of 01:55, 14 February 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2013-02-14 01:55:38 UTC.
- The original revision id was 406883482.
- The revision comment was:
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 blocks|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 <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 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.
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 <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> 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 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 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><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>