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A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron. The group of orientation-preserving symmetries is ''S''4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron. ==Details== Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system. O, 432, or ()+ of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetry ''T'', but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the midpoints of the edges of the cube. ''Td'' and ''O'' are isomorphic as abstract groups: they both correspond to ''S''4, the symmetric group on 4 objects. ''Td'' is the union of ''T'' and the set obtained by combining each element of ''O'' \ ''T'' with inversion. ''O'' is the rotation group of the cube and the regular octahedron. Oh, *432, (), or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as ''O'', but with mirror planes, comprising both the mirror planes of ''Td'' and ''Th''. This group is isomorphic to ''S''4.''C''4, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral group for ''n'' = 3. See also the isometries of the cube. With the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ ''x'' ≤ ''y'' ≤ ''z''. An object with this symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by ''z'' = 1, and the octahedron by ''x'' + ''y'' + ''z'' = 1 (or the corresponding inequalities, to get the solid instead of the surface). ''ax'' + ''by'' + ''cz'' = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron. Faces are 8-by-8 combined to larger faces for ''a'' = ''b'' = ''0'' (cube) and 6-by-6 for ''a'' = ''b'' = ''c'' (octahedron). The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: D2h, and Td. D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations. |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Octahedral symmetry」の詳細全文を読む スポンサード リンク
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