Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 360215369 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 360216571 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2012-08-27 13: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-27 13:22:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>360216571</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> | ||
| Line 100: | Line 100: | ||
The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | ||
Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An | Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1. | ||
[[image:dekany.png]] | [[image:dekany.png]] | ||
| Line 108: | Line 108: | ||
The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group. | The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group. | ||
[[image:myna11.png]] | |||
==The marveldene== | ==The marveldene== | ||
| Line 236: | Line 238: | ||
The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | ||
<br /> | <br /> | ||
Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An | Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:63:&lt;img src=&quot;/file/view/dekany.png/359810971/dekany.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/dekany.png/359810971/dekany.png" alt="dekany.png" title="dekany.png" /><!-- ws:end:WikiTextLocalImageRule:63 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Examples-Myna[11]"></a><!-- ws:end:WikiTextHeadingRule:28 -->Myna[11]</h2> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Examples-Myna[11]"></a><!-- ws:end:WikiTextHeadingRule:28 -->Myna[11]</h2> | ||
| Line 244: | Line 246: | ||
<br /> | <br /> | ||
The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextLocalImageRule:64:&lt;img src=&quot;/file/view/myna11.png/360216505/myna11.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/myna11.png/360216505/myna11.png" alt="myna11.png" title="myna11.png" /><!-- ws:end:WikiTextLocalImageRule:64 --><br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Examples-The marveldene"></a><!-- ws:end:WikiTextHeadingRule:30 -->The marveldene</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Examples-The marveldene"></a><!-- ws:end:WikiTextHeadingRule:30 -->The marveldene</h2> | ||