Tetrahedral symmetry

Tetrahedral Symmetry

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Tetrahedral symmetry

A regular tetrahedron

Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
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.

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A regular tetrahedron

Tetrahedron

Details

Chiral and full (or achiral) tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere....
(or equivalently, symmetries on the sphere

List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
). They are among the crystallographic point groups

Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
of the cubic crystal system.

Chiral tetrahedral symmetry

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

Dihedral symmetry in three dimensions

This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
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

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
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 class

Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
es 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

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
of type Dih₂, with quotient group

Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
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

Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G....
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.

Achiral tetrahedral symmetry
T_d or *332 or , of order 24 - achiral or 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 axes. T_d and O are isomorphic as abstract groups: they both correspond to S₄, the symmetric groupSymmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
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 tetrahedronTetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
.

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°

Pyritohedral symmetry
T_h or 3*2 or , 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 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 cuboidCuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
), of type Dih₂ × Z₂ = Z₂ × Z₂ × Z₂ . It is the direct product of the normal subgroup of T (see above) with C_iInversion (geometry)

In geometry, inversive geometry is the study of a type of transformation of the Plane , called inversions. These transformations preserve angles and function generalized circles into generalized circles, where a generalized circle means either a circle or a line ....
. The quotient groupQuotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
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 pyritohedronPyritohedron

In geometry, a pyritohedron is an irregular dodecahedron. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 corners....
, which is 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 symmetryIcosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
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:
identity
8 × rotation by 120°
3 × rotation by 180°
inversion
8 × rotoreflection by 60°
3 × reflection in a plane

Solids with chiral tetrahedral symmetry

The Icosahedron colored as a snub tetrahedron has chiral symmetry.

Solids with full tetrahedral symmetry

Platonic solidPlatonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
:

Archimedean solidArchimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
:

(semi-regular: vertex-uniform)

Catalan solidCatalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
:

(semi-regular dual: face-uniform)

Name picture Dual Archimedean solid Faces Edges Vertices Face polygon
triakis tetrahedronTriakis tetrahedron

A triakis tetrahedron is an Archimedean solid solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with Tetrahedron added to each face....

(Video) truncated tetrahedronTruncated tetrahedron

The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....
12 18 8 isosceles triangle

e also

octahedral symmetryOctahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
icosahedral symmetryIcosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
binary tetrahedral groupBinary tetrahedral group

In mathematics, the binary tetrahedral group is an group extension of the tetrahedral group T of order 12 by a cyclic group of order 2.It is the binary polyhedral group corresponding to the tetrahedral group, and as such can be defined as the preimage of the tetrahedral group under the 2:1 covering homomorphism...}