Editing Platonic solid (section)

=== Symmetry groups ===
In mathematics, the concept of [[symmetry]] is studied with the notion of a [[group (mathematics)|mathematical group]]. Every polyhedron has an associated [[symmetry group]], which is the set of all transformations ([[Euclidean isometry|Euclidean isometries]]) which leave the polyhedron invariant. The [[order (group theory)|order]] of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the ''full symmetry group'', which includes [[reflection (mathematics)|reflections]], and the ''proper symmetry group'', which includes only [[rotation (mathematics)|rotations]].

The symmetry groups of the Platonic solids are a special class of [[point groups in three dimensions|three-dimensional point groups]] known as [[polyhedral group]]s. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the [[Group action (mathematics)|action]] of the symmetry group, as are the edges and faces. One says the action of the symmetry group is [[transitive action|transitive]] on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is ''regular'' if and only if it is [[vertex-uniform]], [[edge-uniform]], and [[face-uniform]].

There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are:

* the [[tetrahedral group]] ''T'',
* the [[octahedral group]] ''O'' (which is also the symmetry group of the cube), and
* the [[icosahedral group]] ''I'' (which is also the symmetry group of the dodecahedron).

The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are ''centrally symmetric,'' meaning they are preserved under [[reflection through the origin]].

The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parentheses (likewise for the number of symmetries). [[Wythoff's construction|Wythoff's kaleidoscope construction]] is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.

{| class="wikitable" style="text-align:center"
|-
!rowspan=2|Polyhedron
!rowspan=2|[[Schläfli symbol|Schläfli<br/>symbol]]
!rowspan=2|[[Wythoff symbol|Wythoff<br/>symbol]]
!rowspan=2|[[Dual polyhedron|Dual<br/>polyhedron]]
!colspan=5|[[Symmetry group]] (reflection, rotation)
|-
![[Polyhedral group|Polyhedral]]
![[Schönflies notation|Schön.]]
![[Coxeter notation|Cox.]]
![[Orbifold notation|Orb.]]
![[group order|Order]]
|-
| [[tetrahedron]]
| {3, 3} || 3 {{pipe}} 2 3 || tetrahedron
| style="text-align:right;" | [[tetrahedral symmetry|Tetrahedral]] [[File:Disdyakis 6 spherical.png|50px]]
| ''T''<sub>d</sub><BR/>''T''
| [3,3]<BR/>[3,3]<sup>+</sup>
| *332<BR/>332
| 24<BR/>12
|-
| [[cube]]
| {4, 3} || 3 {{pipe}} 2 4 || octahedron
| rowspan=2 style="text-align:right;" | [[octahedral symmetry|Octahedral]] [[File:Disdyakis 12 spherical.png|50px]]
| rowspan=2 | ''O''<sub>h</sub><BR/>''O''
| rowspan=2 | [4,3]<BR/>[4,3]<sup>+</sup>
| rowspan=2 | *432<BR/>432
| rowspan=2 | 48<BR/>24
|-
| [[octahedron]]
| {3, 4} || 4 {{pipe}} 2 3 || cube
|-
| [[dodecahedron]]
| {5, 3} || 3 {{pipe}} 2 5 || icosahedron
| rowspan=2 style="text-align:right;" | [[icosahedral symmetry|Icosahedral]] [[File:Disdyakis 30 spherical.png|50px]]
| rowspan=2 | ''I''<sub>h</sub><BR/>''I''
| rowspan=2 | [5,3]<BR/>[5,3]<sup>+</sup>
| rowspan=2 | *532<BR/>532
| rowspan=2 | 120<BR/>60
|-
| [[icosahedron]]
| {3, 5} || 5 {{pipe}} 2 3 || dodecahedron
|}