Structure metric: Difference between revisions

Line 31:

== Isometry ==

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 {{w|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 {{w|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~~ = ~~D⋅S, 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.

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 {{nowrap|'''SD''' {{=}} '''DS'''}}. 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]]. If we set f(1) = 1, f(~~15/14~~) = 9/8~~, f(5/4~~) = 6/5~~, f(10/7~~) = 5/4~~, f(3/2~~) = 9/5 and f(~~12/7~~) = 15/8~~, ~~then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-~~2 ~~are the same as for the hexany; this scale, the [[hexagon|hexagon]], is isometric to the hexany~~. Also, by mapping the hexany to itself 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. It should be noted, however, that precise JI tuning is not required--both [[27edo|27edo]] and [[31edo|31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.

An interesting example of this is given by the {{w|hexany}}, {{nowrap|{{dash|1, 15/14, 5/4, 10/7, 3/2, 12/7, 2}}}}. This has distance matrix"

Even though the [[Graph-~~theoretic_properties_of_scales~~|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[~~star|~~star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[~~nova|~~Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera Euler(15^n) have the group of the square for both groups, Euler(105^n) gives the group of the cube, and the 5-limit diamond the group of the hexagon.

<math>

\left[ \begin{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

\end{matrix} \right]

</math>

If we set {{nowrap|''f''(1) {{=}} 1|''f''(15/14) {{=}} 9/8|''f''(5/4) {{=}} 6/5|''f''(10/7) {{=}} 5/4|''f''(3/2) {{=}} 9/5}}, and {{nowrap|''f''(12/7) {{=}} 15/8}}, then the distances we get from the new scale {{nowrap|{{dash|1, 9/8, 6/5, 5/4, 9/5, 15/8, 2}}}} are the same as for the hexany; this scale, the [[hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself 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, {{nowrap|(''x'' − 21)(''x'' + 3)2(''x'' + 5)3}}, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required—both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.

Even though the [[Graph-theoretic properties of scales|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is {{nowrap|Cps([2, 3, 5, 7, 11], 2)}}, the {{nowrap|2{{!}}5}} dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera {{nowrap|Euler(15''n'')}} have the group of the square for both groups, {{nowrap|Euler(105''n'')}} gives the group of the cube, and the 5-limit diamond the group of the hexagon.

== Invariants ==

@@ Line 31: / Line 31: @@
 == Isometry ==
-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 {{w|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''),&nbsp;''f''(''b'')) in ''Y'' equals '''d'''(''a'',&nbsp;''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 {{w|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 = D⋅S, 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.
+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'',&nbsp;''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 {{nowrap|'''SD''' {{=}} '''DS'''}}. 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]]. If we set f(1) = 1, f(15/14) = 9/8,  f(5/4) = 6/5, f(10/7) = 5/4, f(3/2) = 9/5 and f(12/7) = 15/8, then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-2 are the same as for the hexany; this scale, the [[hexagon|hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself 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. It should be noted, however, that precise JI tuning is not required--both [[27edo|27edo]] and [[31edo|31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.
+An interesting example of this is given by the {{w|hexany}}, {{nowrap|{{dash|1, 15/14, 5/4, 10/7, 3/2, 12/7, 2}}}}. This has distance matrix"
-Even though the [[Graph-theoretic_properties_of_scales|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star|star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[nova|Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera Euler(15^n) have the group of the square for both groups, Euler(105^n) gives the group of the cube, and the 5-limit diamond the group of the hexagon.
+<math>
+\left[ \begin{matrix}
+& 4 & 4 & 4 & 4 & 5 \\
+& 0 & 4 & 4 & 5 & 4 \\
+& 4 & 0 & 5 & 4 & 4 \\
+& 4 & 5 & 0 & 4 & 4 \\
+& 5 & 4 & 4 & 0 & 4 \\
+& 4 & 4 & 4 & 4 & 0
+\end{matrix} \right]
+</math>
+If we set {{nowrap|''f''(1) {{=}} 1|''f''(15/14) {{=}} 9/8|''f''(5/4) {{=}} 6/5|''f''(10/7) {{=}} 5/4|''f''(3/2) {{=}} 9/5}}, and {{nowrap|''f''(12/7) {{=}} 15/8}}, then the distances we get from the new scale {{nowrap|{{dash|1, 9/8, 6/5, 5/4, 9/5, 15/8, 2}}}} are the same as for the hexany; this scale, the [[hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself 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, {{nowrap|(''x'' − 21)(''x'' + 3)<sup>2</sup>(''x'' + 5)<sup>3</sup>}}, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required—both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.
+Even though the [[Graph-theoretic properties of scales|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is {{nowrap|Cps([2, 3, 5, 7, 11], 2)}}, the {{nowrap|2{{!}}5}} dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera {{nowrap|Euler(15<sup>''n'')}} have the group of the square for both groups, {{nowrap|Euler(105<sup>''n''</sup>)}} gives the group of the cube, and the 5-limit diamond the group of the hexagon.
 == Invariants ==