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List of spherical symmetry groups

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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,Johnson, 2015 orbifold notation,{{cite book | last=Conway | first=John H. | title=The symmetries of things | publisher=A.K. Peters | publication-place=Wellesley, Mass | year=2008 | isbn=978-1-56881-220-5 | oclc=181862605}} 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.{{cite book | last1=Conway | first1=John | last2=Smith | first2=Derek A. | title=On quaternions and octonions: their geometry, arithmetic, and symmetry | publisher=A.K. Peters | publication-place=Natick, Mass | year=2003 | isbn=978-1-56881-134-5 | oclc=560284450}}

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.Sands, "Introduction to Crystallography", 1993

Involutional symmetry

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

class="wikitable"
Intl

! Geo

! Orbifold

! Schönflies

! Conway

! colspan=2|Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| 1

| {{overline|1}}

| 11

| C1

| C1

| ][
[ ]+

{{CDD|node_h2}}

| 1

| Z1

| 100px

align=center

| 2

| {{overline|2}}

| 22

| D1
= C2

| D2
= C2

| [2]+

{{CDD|node_h2|2x|node_h2}}

| 2

| Z2

| 100px

align=center

| {{overline|1}}

| {{overline|22}}

| ×

| Ci
= S2

| CC2

| [2+,2+]

{{CDD|node_h2|2x|node_h4|2x|node_h2}}

| 2

| Z2

| 100px

align=center

| {{overline|2}}
= m

| 1

| *

| Cs
= C1v
= C1h

| ±C1
= CD2

| [ ]

{{CDD|node}}

| 2

| Z2

| 100px

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)

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Intl

! Geo

! Orbifold

! Schönflies

! Conway

! colspan=2|Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| {{overline|4}}

| {{overline|42}}

| 2×

| S4

| CC4

| [2+,4+]

{{CDD|node_h2|2x|node_h4|2x|node_h2}}

| 4

| Z4

| 100px

align=center

| 2/m

| {{overline|2}}2

| 2*

| C2h
= D1d

| ±C2
= ±D2

| [2,2+]
[2+,2]

{{CDD|node|2|node_h2|2x|node_h2}}
{{CDD|node_h2|2x|node_h2|2|node}}

| 4

| Z4

| 100px

class="wikitable"
Intl

! Geo

! Orbifold

! Schönflies

! Conway

! colspan=2|Coxeter

! Order

! Abstract

! Fund.
domain

align=center valign=top

| 2
3
4
5
6
n

| {{overline|2}}
{{overline|3}}
{{overline|4}}
{{overline|5}}
{{overline|6}}
{{overline|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]+

{{CDD|node_h2|2x|node_h2}}
{{CDD|node_h2|3|node_h2}}
{{CDD|node_h2|4|node_h2}}
{{CDD|node_h2|4|node_h2}}
{{CDD|node_h2|5|node_h2}}
{{CDD|node_h2|6|node_h2}}

| 2
3
4
5
6
n

| Z2
Z3
Z4
Z5
Z6
Zn

| 100px

align=center valign=top

| 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]

{{CDD|node|2|node}}
{{CDD|node|3|node}}
{{CDD|node|4|node}}
{{CDD|node|4|node}}
{{CDD|node|5|node}}
{{CDD|node|6|node}}

| 4
6
8
10
12
2n

| D4
D6
D8
D10
D12
D2n

| 100px

align=center valign=top

| {{overline|3}}
{{overline|8}}
{{overline|5}}
{{overline|12}}
-

| {{overline|62}}
{{overline|82}}
{{overline|10.2}}
{{overline|12.2}}
{{overline|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+]

{{CDD|node_h2|2x|node_h4|6|node_h2}}
{{CDD|node_h2|2x|node_h4|8|node_h2}}
{{CDD|node_h2|2x|node_h4|10|node_h2}}
{{CDD|node_h2|2x|node_h4|12|node_h2}}

| 6
8
10
12
2n

| Z6
Z8
Z10
Z12
Z2n

|100px

align=center valign=top

| 3/m={{overline|6}}
4/m
5/m={{overline|10}}
6/m
n/m

| {{overline|3}}2
{{overline|4}}2
{{overline|5}}2
{{overline|6}}2
{{overline|n}}2

| 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+]

{{CDD|node|2|node_h2|3|node_h2}}
{{CDD|node|2|node_h2|4|node_h2}}
{{CDD|node|2|node_h2|5|node_h2}}
{{CDD|node|2|node_h2|6|node_h2}}
{{CDD|node|2|node_h2|n|node_h2}}

| 6
8
10
12
2n

| Z6
Z2×Z4
Z10
Z2×Z6
Z2×Zn
≅Z2n (odd n)

|100px

Dihedral symmetry

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

class="wikitable"
Intl

! Geo

! Orbifold

! Schönflies

! Conway

! Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| 222

| {{overline|2}}.{{overline|2}}

| 222

| D2

| D4

| [2,2]+
{{CDD|node_h2|2x|node_h2|2x|node_h2}}

| 4

| D4

|100px

align=center

| {{Overline|4}}2m

| 4{{overline|2}}

| 2*2

| D2d

| DD8

| [2+,4]
{{CDD|node_h2|2x|node_h2|4|node}}

| 8

| D4

| 100px

align=center

| mmm

| 22

| *222

| D2h

| ±D4

| [2,2]
{{CDD|node|2|node|2|node}}

| 8

| Z2×D4

| 100px

class="wikitable"
Intl

! Geo

! Orbifold

! Schönflies

! Conway

! colspan=2|Coxeter

! Order

! Abstract

! Fund.
domain

align=center valign=top

| 32
422
52
622

| {{overline|3}}.{{overline|2}}
{{overline|4}}.{{overline|2}}
{{overline|5}}.{{overline|2}}
{{overline|6}}.{{overline|2}}
{{overline|n}}.{{overline|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]+

{{CDD|node_h2|2xnode_h2|3|node_h2}}
{{CDD|node_h2|2x		node_h2|4|node_h2}}
{{CDD|node_h2|2x		node_h2|5node_h2}}
{{CDD|node_h2|2x		node_h2|6|node_h2}}
{{CDD|node_h2|2x		node_h2|n|node_h2}}

| 6
8
10
12
2n

| D6
D8
D10
D12
D2n

|100px

align=center valign=top

| {{Overline|3}}m
{{Overline|8}}2m
{{Overline|5}}m
{{Overline|12}}.2m

| 6{{overline|2}}
8{{overline|2}}
10.{{overline|2}}
12.{{overline|2}}
n{{overline|2}}

| 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]

{{CDD|node_h2|2xnode_h2|6|node}}
{{CDD|node_h2|2x|node_h2|8|node}}
{{CDD|node_h2|2x		node_h2|10node}}
{{CDD|node_h2|2x		node_h2|12|node}}
{{CDD|node_h2|2x		node_h2|2x|n|node}}

| 12
16
20
24
4n

| D12
D16
D20
D24
D4n

| 100px

align=center valign=top

| {{overline|6}}m2
4/mmm
{{overline|10}}m2
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]

{{CDD|node|2node|3|node}}
{{CDD|node|2		node|4|node}}
{{CDD|node|2		node|5node}}
{{CDD|node|2		node|6|node}}
{{CDD|node|2		node|n|node}}

| 12
16
20
24
4n

| D12
Z2×D8
D20
Z2×D12
Z2×D2n
≅D4n (odd n)

| 100px

Polyhedral symmetry

{{See|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.

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|+ Tetrahedral symmetry

Intl

! Geo

! Orbifold

! Schönflies

! Conway

! Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| 23

| {{overline|3}}.{{overline|3}}

| 332

| T

| T

| [3,3]+
{{CDD|node_h2|3|node_h2|3|node_h2}}

| 12

| A4

| 100px

align=center

| m{{overline|3}}

| 4{{overline|3}}

| 3*2

| Th

| ±T

| [4,3+]
{{CDD|node|4|node_h2|3|node_h2}}

| 24

| 2×A4

| 100px

align=center

| {{overline|4}}3m

| 33

| *332

| Td

| TO

| [3,3]
{{CDD|node|3|node|3|node}}

| 24

| S4

| 100px

class="wikitable"

|+ Octahedral symmetry

Intl

! Geo

! Orbifold

! Schönflies

! Conway

! Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| 432

| {{overline|4}}.{{overline|3}}

| 432

| O

| O

| [4,3]+
{{CDD|node_h2|4|node_h2|3|node_h2}}

| 24

| S4

| 100px

align=center

| m{{overline|3}}m

| 43

| *432

| Oh

| ±O

| [4,3]
{{CDD|node|4|node|3|node}}

| 48

| 2×S4

| 100px

class="wikitable"

|+ Icosahedral symmetry

Intl

! Geo

! Orbifold

! Schönflies

! Conway

! Coxeter

! Order

! Abstract

! Fund.
domain

align=center

| 532

| {{overline|5}}.{{overline|3}}

| 532

| I

| I

| [5,3]+
{{CDD|node_h2|5|node_h2|3|node_h2}}

| 60

| A5

| 100px

align=center

| {{overline|53}}2/m

| 53

| *532

| Ih

| ±I

| [5,3]
{{CDD|node|5|node|3|node}}

| 120

| 2×A5

| 100px

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.

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!Rank 3 groups

Other namescolspan=2|Example geometryExample finite subgroups
align=center

|O(3)

Full symmetry of the sphererowspan=2|100px[3,3] = {{CDD|node|3|node|3|node}}, [4,3] = {{CDD|node|4|node|3|node}}, [5,3] = {{CDD|node|5|node|3|node}}
[4,3+] = {{CDD|node|4|node_h2|3|node_h2}}
align=center bgcolor="#ffe0e0"

|SO(3)

Sphere groupRotational symmetry[3,3]+ = {{CDD|node_h2|3|node_h2|3|node_h2}}, [4,3]+ = {{CDD|node_h2|4|node_h2|3|node_h2}}, [5,3]+ = {{CDD|node_h2|5|node_h2|3|node_h2}}
align=center

|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		rowspan=3|100px100px45px48px100px[p,2] = [p]×[ ] = {{CDD|node|p|node|2|node}}
[2p,2+] = {{CDD|node|2x|p|node_h2|2x|node_h2}}, [2p+,2+] = {{CDD|node_h2|2x|p|node_h4|2x|node_h2}}
align=center bgcolor="#ffffe0"

|SO(2)×O(1)

C∞×Dih1Rotational symmetry with reflection[p+,2] = [p]+×[ ] = {{CDD|node_h2|p|node_h2|2|node}}
align=center bgcolor="#ffe0e0"

|SO(2)⋊C2

C∞⋊C2Rotational symmetry with half turn[p,2]+ = {{CDD|node_h2|p|node_h2|2x|node_h2}}
align=center

|O(2)×SO(1)

Dih∞
Circular symmetry		Full symmetry of a hemisphere, cone, paraboloid
or any surface of revolution		rowspan=2|130px85px70px100px[p,1] = [p] = {{CDD|node|p|node}}
align=center bgcolor="#ffe0e0"

|SO(2)×SO(1)

C∞
Circle group		Rotational symmetry[p,1]+ = [p]+ = {{CDD|node_h2|p|node_h2}}

See also

References

{{reflist}}

Further reading

  • Peter R. Cromwell, Polyhedra (1997), Appendix I
  • {{cite book|last=Sands |first=Donald E. |title=Introduction to Crystallography |year=1993 |publisher=Dover Publications, Inc. |location=Mineola, New York |isbn=0-486-67839-3 |chapter=Crystal Systems and Geometry |page=165 }}
  • 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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

  • [http://www.geom.uiuc.edu/~math5337/Orbifolds/costs.html Finite spherical symmetry groups]
  • {{MathWorld | urlname=SchoenfliesSymbol | title=Schoenflies symbol}}
  • {{MathWorld | urlname=CrystallographicPointGroups | title=Crystallographic point groups}}
  • [https://web.archive.org/web/20080316083237/http://homepage.mac.com/dmccooey/polyhedra/Simplest.html Simplest Canonical Polyhedra of Each Symmetry Type], by David I. McCooey

{{DEFAULTSORT:Finite spherical symmetry groups}}

Category:Polyhedra

Category:Symmetry

Category:Group theory

Spherical symmetry groups, Finite