Editing Tetrahedron (section)

=== Symmetry ===
[[Image:Tetraeder animation with cube.gif|thumb|The cube and tetrahedron]]
The vertices of a [[cube]] can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. The [[Symmetry in mathematics|symmetries]] of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid not mapped to itself by [[point inversion]].

[[Image:Symmetries of the tetrahedron.svg|thumb|upright=2|The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron]]
The regular tetrahedron has 24 isometries, forming the [[symmetry group]] known as [[full tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{d} </math>. This symmetry group is [[Isomorphism|isomorphic]] to the [[symmetric group]] <math> S_4 </math>. They can be categorized as follows:
* It has rotational tetrahedral symmetry <math> \mathrm{T} </math>. This symmetry is isomorphic to [[alternating group]] <math> A_4 </math>&mdash;the identity and 11 proper rotations&mdash;with the following [[conjugacy class]]es (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the [[Quaternions and spatial rotation|unit quaternion representation]]):
** identity (identity; 1)
** 2 conjugacy classes corresponding to positive and negative rotations about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together( {{nowrap|4 (1 2 3)}}, etc., and  {{nowrap|4 (1 3 2)}}, etc.; {{sfrac|1 ± ''i'' ± ''j'' ± ''k''|2}}).
** rotation by an angle of 180° such that an edge maps to the opposite edge: {{nowrap|3 ((1 2)(3 4)}}, etc.; {{nowrap|''i'', ''j'', ''k''}})
* reflections in a plane perpendicular to an edge: 6
* reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion ('''x''' is mapped to −'''x'''): the rotations correspond to those of the cube about face-to-face axes