Rank-3 scale theorems: Difference between revisions

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Wikispaces>xenwolf
**Imported revision 276418896 - Original comment: linked product word**
Wikispaces>xenwolf
**Imported revision 276419386 - Original comment: is there a page alout fokker blocks in this wiki?**
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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:xenwolf|xenwolf]] and made on <tt>2011-11-17 03:09:16 UTC</tt>.<br>
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-11-17 03:12:55 UTC</tt>.<br>
: The original revision id was <tt>276418896</tt>.<br>
: The original revision id was <tt>276419386</tt>.<br>
: The revision comment was: <tt>linked [[product word]]</tt><br>
: The revision comment was: <tt>is there a page alout fokker blocks in this wiki?</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>
<h4>Original Wikitext content:</h4>
<h4>Original Wikitext 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;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 block is max variety 3.
* Every triple [[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.
* Triple Fokker blocks form a trihexagonal tiling on the lattice.
* Triple Fokker blocks form a trihexagonal tiling on the lattice.
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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Rank-3 scale theorems&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theorems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theorems&lt;/h1&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Rank-3 scale theorems&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theorems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theorems&lt;/h1&gt;
  &lt;ul&gt;&lt;li&gt;Every triple Fokker block is max variety 3.&lt;/li&gt;&lt;li&gt;Every max variety 3 block is a triple Fokker block.&lt;/li&gt;&lt;li&gt;Triple Fokker blocks form a trihexagonal tiling on the lattice.&lt;/li&gt;&lt;li&gt;A scale imprint is that of a Fokker block if and only if it is the &lt;a class="wiki_link" href="/product%20word"&gt;product word&lt;/a&gt; of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank"&gt;http://www.springerlink.com/content/c23748337406x463/&lt;/a&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &amp;gt; m &amp;gt; n &amp;gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s&lt;/li&gt;&lt;li&gt;Any convex object on the lattice can be converted into a hexagon.&lt;/li&gt;&lt;li&gt;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.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
  &lt;ul&gt;&lt;li&gt;Every triple &lt;a class="wiki_link" href="/Fokker%20block"&gt;Fokker block&lt;/a&gt; is max variety 3.&lt;/li&gt;&lt;li&gt;Every max variety 3 block is a triple Fokker block.&lt;/li&gt;&lt;li&gt;Triple Fokker blocks form a trihexagonal tiling on the lattice.&lt;/li&gt;&lt;li&gt;A scale imprint is that of a Fokker block if and only if it is the &lt;a class="wiki_link" href="/product%20word"&gt;product word&lt;/a&gt; of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank"&gt;http://www.springerlink.com/content/c23748337406x463/&lt;/a&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &amp;gt; m &amp;gt; n &amp;gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s&lt;/li&gt;&lt;li&gt;Any convex object on the lattice can be converted into a hexagon.&lt;/li&gt;&lt;li&gt;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.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Unproven Conjectures"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Unproven Conjectures&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Unproven Conjectures"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Unproven Conjectures&lt;/h1&gt;
  &lt;ul&gt;&lt;li&gt;Every rank-3 Fokker block has mean-variety &amp;lt; 4, meaning that some interval class will come in less than 4 sizes.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
  &lt;ul&gt;&lt;li&gt;Every rank-3 Fokker block has mean-variety &amp;lt; 4, meaning that some interval class will come in less than 4 sizes.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 03:12, 17 November 2011

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 03:12:55 UTC.
The original revision id was 276419386.
The revision comment was: is there a page alout fokker blocks in this wiki?

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:&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%20block">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 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>
 <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>
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