Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 564238161 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 564274971 - 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:genewardsmith|genewardsmith]] and made on <tt>2015-10-28 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-10-28 19:45:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>564274971</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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An [[https://en.wikipedia.org/wiki/Isometry|isometry]] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the [[https://en.wikipedia.org/wiki/Isometry_group|isometry group]]. | An [[https://en.wikipedia.org/wiki/Isometry|isometry]] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the [[https://en.wikipedia.org/wiki/Isometry_group|isometry group]]. | ||
In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where "i" denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S.</pre></div> | In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where "i" denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group. | ||
An interesting example of this is given by the [[https://en.wikipedia.org/wiki/Hexany|hexany]], 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]], from which we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. </pre></div> | |||
<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>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | <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>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
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An <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry" rel="nofollow">isometry</a> between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry_group" rel="nofollow">isometry group</a>.<br /> | An <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry" rel="nofollow">isometry</a> between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry_group" rel="nofollow">isometry group</a>.<br /> | ||
<br /> | <br /> | ||
In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where &quot;i&quot; denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S.</body></html></pre></div> | In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where &quot;i&quot; denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group.<br /> | ||
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An interesting example of this is given by the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Hexany" rel="nofollow">hexany</a>, 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]], from which we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany.</body></html></pre></div> | |||