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150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular
tetrahedron In , a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular , is a composed of four , six straight , and four . The tetrahedron is the simplest of all the ordinary and the only one that has fewer than 5 faces. The t ...

tetrahedron
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, i.e. the Order (group theory), order of its symmetry group. The object can be a molecule, crys ...
of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
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 Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "cold drop, frozen drop", ...
of the
cubic crystal system In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' " ...
. Seen in
stereographic projection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

stereographic projection
the edges of the
tetrakis hexahedron In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
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 (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest examples ...
''D''2 or 222, with in addition four 3-fold axes, centered ''between'' the three orthogonal directions. This group is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to ''A''4, the
alternating group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
on 4 elements; in fact it is the group of
even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ord ...
s 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of type Dih2, with
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a ...

cuboid
), of type . It is the direct product of the normal subgroup of T (see above) with C''i''. The
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

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 A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron A regular dodecahedron or pentagonal dodec ...
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 In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed ...
*
Icosahedral symmetry A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron A regular dodecahedron or pentagonal dodec ...
*
Binary tetrahedral group In mathematics, the binary tetrahedral group, denoted 2T or 2,3,3, is a certain nonabelian group of order (group theory), order 24. It is an group extension, extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order ...
*


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