Editing Rotational symmetry (section)

===Multiple symmetry axes through the same point===
For [[discrete symmetry]] with multiple symmetry axes through the same point, there are the following possibilities:
*In addition to an {{mvar|n}}-fold axis, {{mvar|n}} perpendicular 2-fold axes: the [[dihedral group]]s {{mvar|D<sub>n</sub>}} of order&nbsp;{{math|2''n''}} ({{math|''n'' ≥ 2}}). This is the rotation group of a regular [[prism (geometry)|prism]], or regular [[bipyramid]]. Although the same notation is used, the geometric and abstract {{mvar|D<sub>n</sub>}} should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see [[Dihedral group#Dihedral symmetry groups in 3D|dihedral symmetry groups in 3D]].
*4×3-fold and 3×2-fold axes: the rotation group {{mvar|T}} of order&nbsp;12 of a regular [[tetrahedron]]. The group is [[isomorphic]] to [[alternating group]] {{math|''A''<sub>4</sub>}}. 
*3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group&nbsp;''O'' of order&nbsp;24 of a [[cube]] and a regular [[octahedron]]. The group is isomorphic to [[symmetric group]] {{math|''S''<sub>4</sub>}}. 
*6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group&nbsp;{{mvar|I}} of order&nbsp;60 of a [[dodecahedron]] and an [[icosahedron]]. The group is isomorphic to alternating group&nbsp;{{math|''A''<sub>5</sub>}}. The group contains 10 versions of {{math|''D''<sub>3</sub>}} and 6 versions of {{math|''D''<sub>5</sub>}} (rotational symmetries like prisms and antiprisms).

In the case of the [[Platonic solid]]s, the 2-fold axes are through the midpoints of opposite edges, and the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.