HOME

TheInfoList



OR:

150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A 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 ...
has 12 rotational (or orientation-preserving) symmetries, and a
symmetry order The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, that is, it is the order of its symmetry group. The object can be a molecule, crystal latti ...
of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric 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 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 In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal u ...
of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points.


Chiral tetrahedral symmetry

''T'', 332, ,3sup>+, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral
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, g ...
''D''2 or 222, with in addition four 3-fold axes, centered ''between'' the three orthogonal directions. This group is isomorphic to ''A''4, the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
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 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 wo ...
es of T are: *identity *4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321) *4 × rotation by 120° counterclockwise (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 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 of chiral tetrahedral symmetry


Achiral tetrahedral symmetry

Td, *332, ,3or 3m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3)
triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangl ...
. 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. Td and O are isomorphic as abstract groups: they both correspond to S4, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on 4 objects. Td is the union of T and the set obtained by combining each element of with inversion. See also the isometries of the regular tetrahedron. 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 wo ...
es of Td are: *identity *8 × rotation by 120° (C3) *3 × rotation by 180° (C2) *6 × reflection in a plane through two rotation axes (Cs) *6 × rotoreflection by 90° (S4)


Subgroups of achiral tetrahedral symmetry


Pyritohedral symmetry

Th, 3*2, ,3+or m, 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 a central inversion symmetry. Th is isomorphic to : every element of Th 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 . 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 In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentago ...
, 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 In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
group (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: *identity *8 × rotation by 120° (C3) *3 × rotation by 180° (C2) *inversion (S2) *8 × rotoreflection by 60° (S6) *3 × reflection in a plane (Cs)


Subgroups of pyritohedral symmetry


Solids with chiral tetrahedral symmetry

The Icosahedron colored as a snub tetrahedron has chiral symmetry.


Solids with full tetrahedral symmetry


See also

*
Octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
*
Icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
*
Binary tetrahedral group In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...
*


References

* Peter R. Cromwell, ''Polyhedra'' (1997), p. 295 * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, * ''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

* Norman Johnson (mathematician), N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups


External links

* {{DEFAULTSORT:Tetrahedral Symmetry Finite groups Rotational symmetry