List of spherical symmetry groups

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Selected 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

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.

Contents

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
1111C1C1][
[ ]+
CDel node h2.png1Z1 Sphere symmetry group c1.png
2222D1
= C2
D2
= C2
[2]+CDel node h2.pngCDel 2x.pngCDel node h2.png2 Z2 Sphere symmetry group c2.png
122×Ci
= S2
CC2[2+,2+]CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png2Z2 Sphere symmetry group ci.png
2
= m
1*Cs
= C1v
= C1h
±C1
= CD2
[ ]CDel node.png2Z2 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
442S4CC4[2+,4+]CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png4 Z4 Sphere symmetry group s4.png
2/m222*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
4Z4 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
2222.2222D2D4[2,2]+
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
4 D4 Sphere symmetry group d2.png
42m422*2D2dDD8[2+,4]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node.png
8 D4 Sphere symmetry group d2d.png
mmm22*222D2h±D4[2,2]
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
8Z2×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
233.3332TT[3,3]+
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png
12 A4 Sphere symmetry group t.png
m3433*2Th±T[4,3+]
CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
24A4 Sphere symmetry group th.png
43m33*332TdTO[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
4324.3432OO[4,3]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
24 S4 Sphere symmetry group o.png
m3m43*432Oh±O[4,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
48S4 Sphere symmetry group oh.png
Icosahedral symmetry
Intl Geo
Orbifold Schönflies Conway Coxeter Order Abstract Fund.
domain
5325.3532II[5,3]+
CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png
60 A5 Sphere symmetry group i.png
532/m53*532Ih±I[5,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
120A5 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 groupsOther namesExample geometryExample 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.svg Simple Torus.svg Cylinder geometry.svg Bicone.svg Hyperboloid1.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×Dih1Rotational symmetry with reflection[p+,2] = [p]+×[ ] = CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node.png
SO(2)⋊C2C⋊C2Rotational 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).png Truncated Cone (PSF).png Infinite-gonal pyramid1.png Paraboloid 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

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<span class="mw-page-title-main">5-demicube</span>

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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