List of finite spherical symmetry groups

From HandWiki
Short description: None

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 ][
[ ]+
CDel node h2.png 1 Z1 Sphere symmetry group c1.png
2 2 22 D1
= C2
D2
= C2
[2]+ CDel node h2.pngCDel 2x.pngCDel node h2.png 2 Z2 Sphere symmetry group c2.png
1 22 × Ci
= S2
CC2 [2+,2+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png 2 Z2 Sphere symmetry group ci.png
2
= m
1 * Cs
= C1v
= C1h
±C1
= CD2
[ ] CDel node.png 2 Z2 Sphere symmetry group cs.png

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 S4 CC4 [2+,4+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png 4 Z4 Sphere symmetry group s4.png
2/m 22 2* C2h
= D1d
±C2
= ±D2
[2,2+]
[2+,2]
CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png
4 Z4 Sphere symmetry group c2h.png
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]+
CDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 3.pngCDel node h2.png
CDel node h2.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 5.pngCDel node h2.png
CDel node h2.pngCDel 6.pngCDel node h2.png
2
3
4
5
6
n
Z2
Z3
Z4
Z5
Z6
Zn
Sphere symmetry group c2.png
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]
CDel node.pngCDel 2.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.png
4
6
8
10
12
2n
D4
D6
D8
D10
D12
D2n
Sphere symmetry group c2v.png
3
8
5
12
-
62
82
10.2
12.2
2n.2




S6
S8
S10
S12
S2n
±C3
CC8
±C5
CC12
CC2n / ±Cn
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 6.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 8.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 10.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 12.pngCDel node h2.png
6
8
10
12
2n
Z6
Z8
Z10
Z12
Z2n
Sphere symmetry group s6.png
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+]
CDel node.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 4.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 5.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 6.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
6
8
10
12
2n
Z6
Z2×Z4
Z10
Z2×Z6
Z2×Zn
≅Z2n (odd n)
Sphere symmetry group c3h.png

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]+
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
4 D4 Sphere symmetry group d2.png
42m 42 2*2 D2d DD8 [2+,4]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node.png
8 D4 Sphere symmetry group d2d.png
mmm 22 *222 D2h ±D4 [2,2]
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
8 Z2×D4 Sphere symmetry group d2h.png
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]+
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 5.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 6.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
6
8
10
12
2n
D6
D8
D10
D12
D2n
Sphere symmetry group d3.png
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]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 6.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 8.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 10.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 12.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png
12
16
20
24
4n
D12
D16
D20
D24
D4n
Sphere symmetry group d3d.png
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]
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png
12
16
20
24
4n
D12
Z2×D8
D20
Z2×D12
Z2×D2n
≅D4n (odd n)
Sphere symmetry group d3h.png

Polyhedral symmetry

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]+
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png
12 A4 Sphere symmetry group t.png
m3 43 3*2 Th ±T [4,3+]
CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
24 A4 Sphere symmetry group th.png
43m 33 *332 Td TO [3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
24 S4 Sphere symmetry group td.png
Octahedral symmetry
Intl Geo
Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
432 4.3 432 O O [4,3]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
24 S4 Sphere symmetry group o.png
m3m 43 *432 Oh ±O [4,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
48 S4 Sphere symmetry group oh.png
Icosahedral symmetry
Intl Geo
Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
532 5.3 532 I I [5,3]+
CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png
60 A5 Sphere symmetry group i.png
532/m 53 *532 Ih ±I [5,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
120 A5 Sphere symmetry group ih.png

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 Blender-meta-ball.png [3,3] = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, [4,3] = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, [5,3] = CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
[4,3+] = CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
SO(3) Sphere group Rotational symmetry [3,3]+ = CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png, [4,3]+ = CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png, [5,3]+ = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png
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
Spheroid.svg100px45px48pxHyperboloid1.png [p,2] = [p]×[ ] = CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
[2p,2+] = CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.png, [2p+,2+] = CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
SO(2)×O(1) C×Dih1 Rotational symmetry with reflection [p+,2] = [p]+×[ ] = CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node.png
SO(2)⋊C2 C⋊C2 Rotational symmetry with half turn [p,2]+ = CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
O(2)×SO(1) Dih
Circular symmetry
Full symmetry of a hemisphere, cone, paraboloid
or any surface of revolution
Hemisphere (1).png85px70pxParaboloid of Revolution.svg [p,1] = [p] = CDel node.pngCDel p.pngCDel node.png
SO(2)×SO(1) C
Circle group
Rotational symmetry [p,1]+ = [p]+ = CDel node h2.pngCDel p.pngCDel node h2.png

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, 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

External links