List of spherical symmetry groups

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Point groups in three dimensions
Sphere symmetry group cs.png
Involutional symmetry
Cs, [1], (*)
Sphere symmetry group c3v.png
Cyclic symmetry
Cnv, [n], (*nn)
Sphere symmetry group d3h.png
Dihedral symmetry
Dnh, [n,2], (*n22)
Polyhedral group, [n,3], (*n32)
Sphere symmetry group td.png
Tetrahedral symmetry
Td, [3,3], (*332)
Sphere symmetry group oh.png
Octahedral symmetry
Oh, [4,3], (*432)
Sphere symmetry group ih.png
Icosahedral symmetry
Ih, [5,3], (*532)

Spherical symmetry groups are also called point groups in three dimensions, however this article is limited to the finite symmetries. 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.[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[edit]

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

Intl Geo
[5]
Orbifold Schönflies Conway Coxeter Order Fundamental
domain
1 1 11 C1 C1 ][
[ ]+
1 Sphere symmetry group c1.png
2 2 22 D1
= C2
D2
= C2
[1,2]+
=[2,1]+
2 Sphere symmetry group c2.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
1 22 × Ci
= S2
CC2 [2+,2+] 2 Sphere symmetry group ci.png
2
= m
1 * Cs
= C1v
= C1h
±C1
= CD2
[ ]
[1+,2]
[2,1+]
2 Sphere symmetry group cs.png

Cyclic symmetry[edit]

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 Fundamental
domain
2 2 22 C2
= D1
C2
= D2
[1,2]+
[2,1]+
2 Sphere symmetry group c2.png
mm2 2 *22 C2v
= D1h
CD4
= DD4
[1,2]
[2,1]
4 Sphere symmetry group c2v.png
4 42 S4 CC4 [2+,4+] 4 Sphere symmetry group s4.png
2/m 22 2* C2h
= D1d
±C2
= ±D2
[2,2+]
[2+,2]
4 Sphere symmetry group c2h.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
3
4
5
6
n
3
4
5
6
n
33
44
55
66
nn
C3
C4
C5
C6
Cn
C3
C4
C5
C6
Cn
[1,3]+
[1,4]+
[1,5]+
[1,6]+
[1,n]+
3
4
5
6
n
Sphere symmetry group c3.png
3m
4mm
5m
6mm
-
3
4
5
6
n
*33
*44
*55
*66
*nn
C3v
C4v
C5v
C6v
Cnv
CD6
CD8
CD10
CD12
CD2n
[1,3]
[1,4]
[1,5]
[1,6]
[1,n]
6
8
10
12
2n
Sphere symmetry group c3v.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+]
6
8
10
12
2n
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+]
6
8
10
12
2n
Sphere symmetry group c3h.png

Dihedral symmetry[edit]

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

Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
222 2.2 222 D2 D4 [2,2]+ 4 Sphere symmetry group d2.png
42m 42 2*2 D2d DD8 [2+,4] 8 Sphere symmetry group d2d.png
mmm 22 *222 D2h ±D4 [2,2] 8 Sphere symmetry group d2h.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
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
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]
12
16
20
24
4n
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]
12
16
20
24
4n
Sphere symmetry group d3h.png

Polyhedral symmetry[edit]

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

[3,3]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
23 3.3 332 T T [3,3]+
= [1+,4,3]+
12 Sphere symmetry group t.png
m3 43 3*2 Th ±T [4,3+]
=[[3,3]+]
24 Sphere symmetry group th.png
43m 33 *332 Td TO [3,3]
= [1+,4,3]
24 Sphere symmetry group td.png
[4,3]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
432 4.3 432 O O [4,3]+
= [[3,3]]+
24 Sphere symmetry group o.png
m3m 43 *432 Oh ±O [4,3]
= [[3,3]]
48 Sphere symmetry group oh.png
[5,3]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
532 5.3 532 I I [5,3]+ 60 Sphere symmetry group i.png
532/m 53 *532 Ih ±I [5,3] 120 Sphere symmetry group ih.png

See also[edit]

Notes[edit]

  1. ^ Johnson, 2011
  2. ^ Conway, 2008
  3. ^ Conway, 2003
  4. ^ Sands, 1993
  5. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1]

References[edit]

  • 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-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 [2]
    • (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, Manuscript, (2011) Chapter 11: Finite symmetry groups

External links[edit]