List of spherical symmetry groups

Selected point groups in three dimensions

Involutional symmetry
Cs, (*)
[ ] =
Cyclic symmetry
Cnv, (*nn)
[n] =
Dihedral symmetry
Dnh, (*n22)
[n,2] =
Polyhedral group, [n,3], (*n32)

Tetrahedral symmetry
Td, (*332)
[3,3] =
Octahedral symmetry
Oh, (*432)
[4,3] =
Icosahedral symmetry
Ih, (*532)
[5,3] =

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.

This article lists the groups by Schoenflies notation, Coxeter notation,[1] orbifold notation,[2] and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.[3]

Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.[4]

Involutional symmetry

There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci).

Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
1 1 11 C1 C1 ][
[ ]+		 1 Z1
2 2 22 D1
= C2 		D2
= C2 		[2]+ 2 Z2
1 22 × Ci
= S2 		CC2 [2+,2+] 2 Z2
2
= m 		1 * Cs
= C1v
= C1h 		±C1
= CD2 		[ ] 2 Z2

Cyclic symmetry

There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry)

Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
4 42 2× S4 CC4 [2+,4+] 4 Z4
2/m 22 2* C2h
= D1d 		±C2
= ±D2 		[2,2+]
[2+,2]
4 Z4
Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
2
3
4
5
6
n 		2
3
4
5
6
n 		22
33
44
55
66
nn 		C2
C3
C4
C5
C6
Cn 		C2
C3
C4
C5
C6
Cn 		[2]+
[3]+
[4]+
[5]+
[6]+
[n]+




2
3
4
5
6
n 		Z2
Z3
Z4
Z5
Z6
Zn
2mm
3m
4mm
5m
6mm
nm (n is odd)
nmm (n is even) 		2
3
4
5
6
n 		*22
*33
*44
*55
*66
*nn 		C2v
C3v
C4v
C5v
C6v
Cnv 		CD4
CD6
CD8
CD10
CD12
CD2n 		[2]
[3]
[4]
[5]
[6]
[n]




4
6
8
10
12
2n 		D4
D6
D8
D10
D12
D2n
3
8
5
12
- 		62
82
10.2
12.2
2n.2 		3×
4×
5×
6×
n× 		S6
S8
S10
S12
S2n 		±C3
CC8
±C5
CC12
CC2n / ±Cn 		[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]


6
8
10
12
2n 		Z6
Z8
Z10
Z12
Z2n
3/m=6
4/m
5/m=10
6/m
n/m 		32
42
52
62
n2 		3*
4*
5*
6*
n* 		C3h
C4h
C5h
C6h
Cnh 		CC6
±C4
CC10
±C6
±Cn / CC2n 		[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]



6
8
10
12
2n 		Z6
Z2×Z4
Z10
Z2×Z6
Z2×Zn
≅Z2n (odd n)

Dihedral symmetry

There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).

Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
222 2.2 222 D2 D4 [2,2]+
 4 D4
42m 42 2*2 D2d DD8 [2+,4]
 8 D4
mmm 22 *222 D2h ±D4 [2,2]
 8 Z2×D4
Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
32
422
52
622 		3.2
4.2
5.2
6.2
n.2 		223
224
225
226
22n 		D3
D4
D5
D6
Dn 		D6
D8
D10
D12
D2n 		[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+



6
8
10
12
2n 		D6
D8
D10
D12
D2n
3m
82m
5m
12.2m
 62
82
10.2
12.2
n2
 2*3
2*4
2*5
2*6
2*n 		D3d
D4d
D5d
D6d
Dnd 		±D6
DD16
±D10
DD24
DD4n / ±D2n 		[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]



12
16
20
24
4n 		D12
D16
D20
D24
D4n
6m2
4/mmm
10m2
6/mmm 		32
42
52
62
n2 		*223
*224
*225
*226
*22n 		D3h
D4h
D5h
D6h
Dnh 		DD12
±D8
DD20
±D12
±D2n / DD4n 		[2,3]
[2,4]
[2,5]
[2,6]
[2,n]



12
16
20
24
4n 		D12
Z2×D8
D20
Z2×D12
Z2×D2n
≅D4n (odd n)

Polyhedral symmetry

Further information: Polyhedral groups

There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.

Tetrahedral symmetry
Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
23 3.3 332 T T [3,3]+
 12 A4
m3 43 3*2 Th ±T [4,3+]
 24 2×A4
43m 33 *332 Td TO [3,3]
 24 S4
Octahedral symmetry
Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
432 4.3 432 O O [4,3]+
 24 S4
m3m 43 *432 Oh ±O [4,3]
 48 2×S4
Icosahedral symmetry
Intl Geo
 Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
532 5.3 532 I I [5,3]+
 60 A5
532/m 53 *532 Ih ±I [5,3]
 120 2×A5

Continuous symmetries

All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1. SO(1) is just the identity. Half turns, C2, are needed to complete.

Rank 3 groups Other names Example geometry Example finite subgroups
O(3) Full symmetry of the sphere [3,3] = , [4,3] = , [5,3] =
[4,3+] =
SO(3) Sphere group Rotational symmetry [3,3]+ = , [4,3]+ = , [5,3]+ =
O(2)×O(1)
O(2)⋊C2		 Dih∞×Dih1
Dih∞⋊C2		 Full symmetry of a spheroid, torus, cylinder, bicone or hyperboloid
Full circular symmetry with half turn		 [p,2] = [p]×[ ] =
[2p,2+] = , [2p+,2+] =
SO(2)×O(1) C∞×Dih1 Rotational symmetry with reflection [p+,2] = [p]+×[ ] =
SO(2)⋊C2 C∞⋊C2 Rotational symmetry with half turn [p,2]+ =
O(2)×SO(1) Dih∞
Circular symmetry		 Full symmetry of a hemisphere, cone, paraboloid
or any surface of revolution		 [p,1] = [p] =
SO(2)×SO(1) C∞
Circle group		 Rotational symmetry [p,1]+ = [p]+ =

See also

References

  1. ^ Johnson, 2015
  2. ^ Conway, John H. (2008). The symmetries of things. Wellesley, Mass: A.K. Peters. ISBN 978-1-56881-220-5. OCLC 181862605.
  3. ^ Conway, John; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, Mass: A.K. Peters. ISBN 978-1-56881-134-5. OCLC 560284450.
  4. ^ Sands, "Introduction to Crystallography", 1993

Further reading

  • Peter R. Cromwell, Polyhedra (1997), Appendix I
  • Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3.
  • On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3-space