Tetrahedral symmetry

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Point groups in three dimensions
Sphere symmetry group cs.png
Involutional symmetry
Cs, (*)
[ ] = CDel node c2.png
Sphere symmetry group c3v.png
Cyclic symmetry
Cnv, (*nn)
[n] = CDel node c1.pngCDel n.pngCDel node c1.png
Sphere symmetry group d3h.png
Dihedral symmetry
Dnh, (*n22)
[n,2] = CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c1.png
Polyhedral group, [n,3], (*n32)
Sphere symmetry group td.png
Tetrahedral symmetry
Td, (*332)
[3,3] = CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group oh.png
Octahedral symmetry
Oh, (*432)
[4,3] = CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group ih.png
Icosahedral symmetry
Ih, (*532)
[5,3] = CDel node c2.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c2.png
A regular tetrahedron, an example of a solid with full tetrahedral symmetry

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

The group of all symmetries is isomorphic to the group S4, symmetric group, as a 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[edit]

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.

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.

Orthogonal Stereographic projections
4-fold 3-fold 2-fold
Chiral tetrahedral symmetry, T, (332), [3,3]+ = [1+,4,3+], CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
Sphere symmetry group t.png Tetrakis hexahedron stereographic D4 gyrations.png Tetrakis hexahedron stereographic D3 gyrations.png Tetrakis hexahedron stereographic D2 gyrations.png
Pyritohedral symmetry, Th, (3*2), [4,3+], CDel node c2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
Sphere symmetry group th.png Disdyakis dodecahedron stereographic D4 pyritohedral.png Disdyakis dodecahedron stereographic D3 pyritohedral.png Disdyakis dodecahedron stereographic D2 pyritohedral.png
Achiral tetrahedral symmetry, Td, (*332), [3,3] = [1+4,3], CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png = CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group td.png Tetrakis hexahedron stereographic D4.png Tetrakis hexahedron stereographic D3.png Tetrakis hexahedron stereographic D2.png


Chiral tetrahedral symmetry[edit]

Sphere symmetry group t.png
The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face
Tetrahedral group 2.svg
A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions.
Tetrakishexahedron.jpg
In the tetrakis hexahedron 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, 332, [3,3]+, 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° 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 G = A4 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[edit]

Chiral tetrahedral symmetry subgroups
Schoe. Coxeter Orb. H-M Structure Cycle graph|Cyc Order Index
T [3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel node h2.pngCDel split1.pngCDel branch h2h2.pngCDel label2.png 332 23 A4 GroupDiagramMiniA4.svg 12 1
D2 [2,2]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png = CDel node h2.pngCDel split1-22.pngCDel branch h2h2.pngCDel label2.png 222 222 Dih2 GroupDiagramMiniD4.svg 4 3
C3 [3]+ CDel node h2.pngCDel 3.pngCDel node h2.png 33 3 Z3 GroupDiagramMiniC3.svg 3 4
C2 [2]+ CDel node h2.pngCDel 2x.pngCDel node h2.png 22 2 Z2 GroupDiagramMiniC2.svg 2 6
C1 [ ]+ CDel node h2.png 11 1 Z1 GroupDiagramMiniC1.svg 1 12

Achiral tetrahedral symmetry[edit]

The full tetrahedral group Td with fundamental domain

Td, *332, [3,3] or 43m, of order 24 - achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. 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 (4) axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 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:

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

Subgroups of achiral tetrahedral symmetry[edit]

Achiral tetrahedral subgroups
Schoe. Coxeter Orb. H-M Structure Cyc Order Index
Td [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png *332 43m S4 Symmetric group 4; cycle graph.svg 24 1
C3v [3] CDel node.pngCDel 3.pngCDel node.png *33 3m Dih3=S3 GroupDiagramMiniD6.svg 6 4
C2v [2] CDel node.pngCDel 2.pngCDel node.png *22 mm2 Dih2 GroupDiagramMiniD4.svg 4 6
Cs [ ] CDel node.png * 2 or m Dih1 GroupDiagramMiniC2.svg 2 12
D2d [2+,4] CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node.png 2*2 42m Dih4 GroupDiagramMiniD8.svg 8 3
S4 [2+,4+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 4.pngCDel node h2.png 4 Z4 GroupDiagramMiniC4.svg 4 6
T [3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 332 23 A4 GroupDiagramMiniA4.svg 12 2
D2 [2,2]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png 222 222 Dih2 GroupDiagramMiniD4.svg 4 6
C3 [3]+ CDel node h2.pngCDel 3.pngCDel node h2.png 33 3 Z3=A3 GroupDiagramMiniC3.svg 3 8
C2 [2]+ CDel node h2.pngCDel 2x.pngCDel node h2.png 22 2 Z2 GroupDiagramMiniC2.svg 2 12
C1 [ ]+ CDel node h2.png 11 1 Z1 GroupDiagramMiniC1.svg 1 24

Pyritohedral symmetry[edit]

The pyritohedral group Th with fundamental domain
The seams of a volleyball have pyritohedral symmetry

Th, 3*2, [4,3+] or m3, 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 (3) axes, and there is inversion symmetry. Th is isomorphic to T × Z2: 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 Dih2 × Z2 = Z2 × Z2 × Z2 . It is the direct product of the normal subgroup of T (see above) with Ci. 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 Th 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

Subgroups of pyritohedral symmetry[edit]

Pyritohedral subgroups
Schoe. Coxeter Orb. H-M Structure Cyc Order Index
Th [3+,4] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.png 3*2 m3 A4×2 GroupDiagramMiniA4xC2.png 24 1
D2h [2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png *222 mmm Dih2×Dih1 GroupDiagramMiniC2x3.svg 8 3
C2v [2] CDel node.pngCDel 2.pngCDel node.png *22 mm2 Dih2 GroupDiagramMiniD4.svg 4 6
Cs [ ] CDel node.png * 2 or m Dih1 GroupDiagramMiniC2.svg 2 12
C2h [2+,2] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png 2* 2/m Dih1×Z2 GroupDiagramMiniD4.svg 4 6
S2 [2+,2+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png × 1 2 or Z2 GroupDiagramMiniC2.svg 2 12
T [3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 332 23 A4 GroupDiagramMiniA4.svg 12 2
D3 [2,3]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 3.pngCDel node h2.png 322 3 Dih3 GroupDiagramMiniD6.svg 6 4
D2 [2,2]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png 222 222 Dih4 GroupDiagramMiniD4.svg 4 6
C3 [3]+ CDel node h2.pngCDel 3.pngCDel node h2.png 33 3 Z3 GroupDiagramMiniC3.svg 3 8
C2 [2]+ CDel node h2.pngCDel 2x.pngCDel node h2.png 22 2 Z2 GroupDiagramMiniC2.svg 2 12
C1 [ ]+ CDel node h2.png 11 1 Z1 GroupDiagramMiniC1.svg 1 24

Solids with chiral tetrahedral symmetry[edit]

Snub tetrahedron.png The Icosahedron colored as a snub tetrahedron has chiral symmetry.

Solids with full tetrahedral symmetry[edit]

Class Name Picture Faces Edges Vertices
Platonic solid tetrahedron Tetrahedron 4 6 4
Archimedean solid truncated tetrahedron Truncated tetrahedron 8 18 12
Catalan solid triakis tetrahedron Triakis tetrahedron 12 18 8
Near-miss Johnson solid Truncated triakis tetrahedron Truncated triakis tetrahedron.png 16 42 28
Tetrated dodecahedron Tetrated Dodecahedron.gif 28 54 28
Uniform star polyhedron Tetrahemihexahedron Tetrahemihexahedron.png 7 12 6

See also[edit]

References[edit]

  • Peter R. Cromwell, Polyhedra (1997), p. 295
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
  • 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, ISBN 978-0-471-01003-6 [1]
  • N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups

External links[edit]