- Tetrahedral symmetry
tetrahedronhas 12 rotational (or orientation-preserving) symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation.
The group of symmetries that includes reflections is isomorphic to "S"4, or the group 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 "A"4.
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
"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 "D"2 or 222, with in addition four 3-fold axes, centered "between" the three orthogonal directions. This group is
isomorphicto "A"4, the alternating groupon 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).
conjugacy classes of "T" are:
*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 subgroupof type Dih2, with quotient groupof 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.
"A"4 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.
Achiral tetrahedral symmetry
"Td" 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 "S"4 () axes. "Td" and "O" are isomorphic as abstract groups: they both correspond to "S"4, the
symmetric groupon 4 objects. "Td" 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 "Td" are:
*8 × rotation by 120°
*3 × rotation by 180°
*6 × reflection in a plane through two rotation axes
*6 × rotoreflection by 90°
"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" × "Z"2: 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 "D2h" (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 "Ci". The quotient groupis 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 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 symmetrygroup (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.
The conjugacy classes of "Th" include those of "T", with the two classes of 4 combined, and each with inversion:
*8 × rotation by 120°
*3 × rotation by 180°
*8 × rotoreflection by 60°
*3 × reflection in a plane
Solids with chiral tetrahedral symmetry
The Icosahedron colored as a snub tetrahedron has chiral symmetry.
olids with full tetrahedral symmetry
Platonic solid: Archimedean solid:
(semi-regular dual: face-uniform)
Name picture Dual Archimedean solid Faces Edges Vertices Face polygon triakis tetrahedron ) truncated tetrahedron 12 18 8 isosceles triangle
binary tetrahedral group
Wikimedia Foundation. 2010.
Look at other dictionaries:
Tetrahedral molecular geometry — In a Tetrahedral molecular geometry a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are cos−1(−1/3) ≈ 109.5° when all four substituents are the same, as in CH4. This … Wikipedia
Symmetry group — Not to be confused with Symmetric group. This article is about the abstract algebraic structures. For other meanings, see Symmetry group (disambiguation). A tetrahedron can be placed in 12 distinct positions by rotation alone. These are… … Wikipedia
Octahedral symmetry — The cube is the most common shape with octahedral symmetry 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… … Wikipedia
Icosahedral symmetry — A Soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. A regular icosahedron has 60 rotational (or orientation preserving) symmetries, and a symmetry order of 120 including transformations that… … Wikipedia
Molecular symmetry — in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule s chemical… … Wikipedia
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 the finite ones.There are four fundamental symmetry classes which have triangular fundamental domains:… … Wikipedia
Disphenoid tetrahedral honeycomb — Type convex uniform honeycomb dual Cell type disphenoid tetrahedron … Wikipedia
Tetrahedron — For the academic journal, see Tetrahedron (journal). Regular Tetrahedron (Click here for rotating model) Type Platonic solid Elements F = 4, E = 6 V = 4 (χ = 2) Faces by s … Wikipedia
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. It is a subgroup of the orthogonal group O(3), the group of all isometries… … Wikipedia
Uniform polyhedron — A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high… … Wikipedia