geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the polyhedral group is any of the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s of the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

Groups

There are three polyhedral groups: *The

tetrahedral group 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

of order 12, rotational symmetry group of the

regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

. It is isomorphic to ''A''₄. ** The

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...

es of ''T'' are: ***identity ***4 × rotation by 120°, order 3, cw ***4 × rotation by 120°, order 3, ccw ***3 × rotation by 180°, order 2 *The octahedral group of order 24, rotational symmetry group of the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

and the

regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

. It is isomorphic to ''S''₄. **The conjugacy classes of ''O'' are: ***identity ***6 × rotation by ±90° around vertices, order 4 ***8 × rotation by ±120° around triangle centers, order 3 ***3 × rotation by 180° around vertices, order 2 ***6 × rotation by 180° around midpoints of edges, order 2 *The icosahedral group of order 60, rotational symmetry group of the

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, ...

and the

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

. It is isomorphic to ''A''₅. **The conjugacy classes of ''I'' are: ***identity ***12 × rotation by ±72°, order 5 ***12 × rotation by ±144°, order 5 ***20 × rotation by ±120°, order 3 ***15 × rotation by 180°, order 2 These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, ,3can be seen as the union of 6 tetrahedral symmetry ,3mirrors, and 3 mirrors of

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

Dih₂, ,2 Pyritohedral symmetry is another doubling of tetrahedral symmetry. The conjugacy classes of full tetrahedral symmetry, ''T_d≅ S₄'', are: *identity *8 × rotation by 120° *3 × rotation by 180° *6 × reflection in a plane through two rotation axes *6 × rotoreflection by 90° The conjugacy classes of pyritohedral symmetry, ''T_h'', include those of ''T'', with the two classes of 4 combined, and each with inversion: *identity *8 × rotation by 120° *3 × rotation by 180° *inversion *8 × rotoreflection by 60° *3 × reflection in a plane The conjugacy classes of the full octahedral group, ''O_h≅ S₄ × C₂'', are: *inversion *6 × rotoreflection by 90° *8 × rotoreflection by 60° *3 × reflection in a plane perpendicular to a 4-fold axis *6 × reflection in a plane perpendicular to a 2-fold axis The conjugacy classes of full icosahedral symmetry, ''I_h≅ A₅ × C₂'', include also each with inversion: *inversion *12 × rotoreflection by 108°, order 10 *12 × rotoreflection by 36°, order 10 *20 × rotoreflection by 60°, order 6 *15 × reflection, order 2

Chiral polyhedral groups

Full polyhedral groups

References

* Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (''The Polyhedral Groups.'' §3.5, pp. 46–47)

External links

* {{mathworld , urlname = PolyhedralGroup, title =PolyhedralGroup Polyhedra

Groups

Chiral polyhedral groups

Full polyhedral groups

See also

References

External links