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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | == Theorems == |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-11-17 03:12:55 UTC</tt>.<br>
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| : The original revision id was <tt>276419386</tt>.<br>
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| : The revision comment was: <tt>is there a page alout fokker blocks in this wiki?</tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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=
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| * 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. (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 [http://en.wikipedia.org/wiki/Trihexagonal_tiling 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|product]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink] |
| * 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 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. | | * 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. |
| | * An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]]. |
| | == Conjectures == |
| | * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. |
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| =Unproven Conjectures=
| | [[Category:Fokker block]] |
| * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.</pre></div>
| | [[Category:Math]] |
| <h4>Original HTML content:</h4>
| | [[Category:Rank 3]] |
| <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>
| | [[Category:Scale]] |
| <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 />
| | [[Category:Pages with open problems]] |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Unproven Conjectures"></a><!-- ws:end:WikiTextHeadingRule:2 -->Unproven Conjectures</h1>
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| <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>
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