Tetrahedral symmetry

A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation.

The group of all symmetries is isomorphic to the group S₄ of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A₄ of S₄.

Details

Chiral and full (or achiral) tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system.

Chiral tetrahedral symmetry

In the tetrakis cube one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.

T or 332 or 23, of order 12 - chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D₂ or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A₄, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).

The conjugacy classes of T are:

identity
4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321)
4 × rotation by 120° anti-clockwise (ditto)
3 × rotation by 180°

The rotations by 180°, together with the identity, form a normal subgroup of type Dih₂, with quotient group of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

A₄ is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A_{4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C₆ or D₃, but neither applies.}

Subgroups

T
D₂
C₃ and C₂
E

Achiral tetrahedral symmetry

T_d oder *332 oder ${\bar {4}}3m$ , of order 24 - achiral oder full tetrahedral symmetry. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S₄ ( ${\bar {4}}$ ) axes. T_d and O are isomorphic as abstract groups: they both correspond to S₄, the symmetric group on 4 objects. T_d is the union of T and the set obtained by combining each element of O \ T with inversion. See also the isometries of the regular tetrahedron.

The conjugacy classes of T_d are:

identity
8 × rotation by 120°
3 × rotation by 180°
6 × reflection in a plane through two rotation axes
6 × rotoreflection by 90°

Subgroups

T_d
T
D_2d
D₃ and D₂
C_3v and C_2v
C₃ and C₂
S₄ and S₂=C_i
E and C_s

Pyritohedral symmetry

T_h oder 3*2 oder $2/m\ {\bar {3}}$ , of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S₆ ( ${\bar {3}}$ ) axes, and there is inversion symmetry. T_h is isomorphic to T × Z₂: every element of T_h is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D_2h (that of a cuboid), of type Dih₂ × Z₂ = Z₂ × Z₂ × Z₂ . It is the direct product of the normal subgroup of T (see above) with C_i. The quotient group is the same as above: of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.

The conjugacy classes of T_h include those of T, with the two classes of 4 combined, and each with inversion: