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List of finite spherical symmetry groups

  (Redirected from List of spherical symmetry groups)
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.

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]

Contents

Involutional symmetryEdit

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

Intl Geo
[5]
Orb. Schön. Con. Cox. Ord. Fund.
domain
1 1 11 C1 C1 ][
[ ]+
1  
2 2 22 D1
= C2
D2
= C2
[2]+ 2  
1 22 × Ci
= S2
CC2 [2+,2+] 2  
2
= m
1 * Cs
= C1v
= C1h
±C1
= CD2
[ ] 2  

Cyclic symmetryEdit

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
Orb. Schön. Con. Cox. Ord. Fund.
domain
4 42 S4 CC4 [2+,4+] 4  
2/m 22 2* C2h
= D1d
±C2
= ±D2
[2,2+]
[2+,2]
4  
Intl Geo
Orb. Schön. Con. Cox. Ord. 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
 
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
 
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
 
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
 

Dihedral symmetryEdit

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

Intl Geo
Orb. Schön. Con. Cox. Ord. Fund.
domain
222 2.2 222 D2 D4 [2,2]+ 4  
42m 42 2*2 D2d DD8 [2+,4] 8  
mmm 22 *222 D2h ±D4 [2,2] 8  
Intl Geo
Orb. Schön. Con. Cox. Ord. 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
 
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
 
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
 

Polyhedral symmetryEdit

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
Orb. Schön. Con. Cox. Ord. Fund.
domain
23 3.3 332 T T [3,3]+
= [4,3+]+
12  
m3 43 3*2 Th ±T [4,3+] 24  
43m 33 *332 Td TO [3,3]
= [1+,4,3]
24  
Octahedral symmetry
Intl Geo Orb. Schön. Con. Cox. Ord. Fund.
domain
432 4.3 432 O O [4,3]+
= [[3,3]]+
24  
m3m 43 *432 Oh ±O [4,3]
= [[3,3]]
48  
Icosahedral symmetry
Intl Geo Orb. Schön. Con. Cox. Ord. Fund.
domain
532 5.3 532 I I [5,3]+ 60  
532/m 53 *532 Ih ±I [5,3] 120  

See alsoEdit

NotesEdit

  1. ^ Johnson, 2015
  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]

ReferencesEdit

  • 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, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3-space

External linksEdit