Point group

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|240px<br/>The Bauhinia blakeana flower on the Hong Kong region flag has C₅ symmetry; the star on each petal has D₅ symmetry.

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In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to . Each element of a point group is either a rotation (determinant of ), or it is a reflection or improper rotation (determinant of ).

The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups.

Chiral and achiral point groups, reflection groups

Point groups can be classified into chiral (or purely rotational) groups and achiral groups.

The chiral groups are subgroups of the special orthogonal group SO(d): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group, the orientation-preserving transformations form a (chiral) subgroup of index 2.

Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for the trivial group containing only the identity element).

List of point groups

One dimension

There are only two one-dimensional point groups, the identity group and the reflection group.

{| class=wikitable

!Group

!Coxeter

!Coxeter diagram

!Order

!Description

|- align=center

||C₁||[&nbsp;]⁺|| ||1||identity

|- align=center

||D₁||[&nbsp;]||||2||reflection group

|}

Two dimensions

Point groups in two dimensions, sometimes called rosette groups.

They come in two infinite families:

1. Cyclic groups C_n of n-fold rotation groups
2. Dihedral groups D_n of n-fold rotation and reflection groups

Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.

{| class="wikitable"

|-

! Group

! Intl

! Orbifold

! Coxeter

! Order

! Description

|- align=center

| C_n

| n

| <span style="color:blue;">n•</span>

| [n]⁺

| n

|align=left|cyclic: n-fold rotations; abstract group Z_n, the group of integers under addition modulo n

|- align=center

| D_n

| nm

| <span style="color:red;">*n•</span>

| [n]

| 2n

|align=left|dihedral: cyclic with reflections; abstract group Dih_n, the dihedral group

|}

thumb|Finite isomorphism and correspondences

The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.

{| class=wikitable style="text-align:center;"

! colspan=6 | Reflective

! colspan=3 | Rotational

! rowspan=2 | Related <br/>polygons

|-

! Group

! colspan=2 | Coxeter group

! colspan=2 | Coxeter diagram

! Order

! Subgroup

! Coxeter

! Order

|-

| D₁|| A₁||[&nbsp;]||||||2||C₁||[]⁺||1

| digon

|-

| D₂|| A₁²||[2]||||||4||C₂||[2]⁺||2

| rectangle

|-

| D₃|| A₂||[3]||||||6||C₃||[3]⁺||3

| equilateral triangle

|-

| D₄|| BC₂||[4]||||||8||C₄||[4]⁺||4

| square

|-

| D₅|| H₂||[5]||||||10||C₅||[5]⁺||5

| regular pentagon

|-

| D₆|| G₂||[6]||||||12||C₆||[6]⁺||6

| regular hexagon

|-

| D_n|| I₂(n)||[n]||||||2n||C_n||[n]⁺||n

| regular polygon

|-

| D₂×2|| A₁²×2|| = [4]|||| = ||8

|-

| D₃×2|| A₂×2|| = [6]|||| = ||12

|-

| D₄×2|| BC₂×2|| = [8]|||| = ||16

|-

| D₅×2|| H₂×2|| = [10]|||| = ||20

|-

| D₆×2|| G₂×2|| = [12]|||| = ||24

|-

| D_n×2|| I₂(n)×2|| = [2n]|||| = ||4n

|}

Three dimensions

Point groups in three dimensions, sometimes called molecular point groups after their wide use in studying symmetries of molecules.

They come in 7 infinite families of axial groups (also called prismatic), and 7 additional polyhedral groups (also called Platonic). In Schoenflies notation,

• Axial groups: C_n, S_2n, C_nh, C_nv, D_n, D_nd, D_nh
• Polyhedral groups: T, T_d, T_h, O, O_h, I, I_h

Applying the crystallographic restriction theorem to these groups yields the 32 crystallographic point groups.

{|class="wikitable"

|+ Even/odd colored fundamental domains of the reflective groups

|-

! C_1v<br/>Order 2

! C_2v<br/>Order 4

! C_3v<br/>Order 6

! C_4v<br/>Order 8

! C_5v<br/>Order 10

! C_6v<br/>Order 12

!...

|-

| 80px

| 80px

| 80px

| 80px

| 80px

| 80px

|-

|-

! D_1h<br/>Order 4

! D_2h<br/>Order 8

! D_3h<br/>Order 12

! D_4h<br/>Order 16

! D_5h<br/>Order 20

! D_6h<br/>Order 24

! ...

|-

|-

| 80px

| 80px

| 80px

| 80px

| 80px

| 80px

|-

! T_d<br/>Order 24

! O_h<br/>Order 48

! I_h<br/>Order 120

|-

| 80px

| 80px

| 80px

|}

{| class="wikitable"

|- valign=top

|

{| class="wikitable"

|-

! Intl^*

! Geo<br/>

! Orbifold

! Schoenflies

! Coxeter

! Order

|- align=center

| 1

|

| 1

| C₁

| [&nbsp;]⁺

| 1

|- align=center

|

|

| ×1

| C_i = S₂

| [2⁺,2⁺]

| 2

|- align=center

| = m

| 1

| *1

| C_s = C_1v = C_1h

| [&nbsp;]

| 2

|- align=center valign=top

| 2<br/>3<br/>4<br/>5<br/>6<br/>n

| <br/><br/><br/><br/><br/>

| 22<br/>33<br/>44<br/>55<br/>66<br/>nn

| C₂<br/>C₃<br/>C₄<br/>C₅<br/>C₆<br/>C_n

| [2]⁺<br/>[3]⁺<br/>[4]⁺<br/>[5]⁺<br/>[6]⁺<br/>[n]⁺<br/>

| 2<br/>3<br/>4<br/>5<br/>6<br/>n

|- align=center valign=top

| mm2<br/>3m<br/>4mm<br/>5m<br/>6mm<br/>nmm<br/>nm

| 2<br/>3<br/>4<br/>5<br/>6<br/>n

| *22<br/>*33<br/>*44<br/>*55<br/>*66<br/>*nn

| C_2v<br/>C_3v<br/>C_4v<br/>C_5v<br/>C_6v<br/>C_nv

| [2]<br/>[3]<br/>[4]<br/>[5]<br/>[6]<br/>[n]

| 4<br/>6<br/>8<br/>10<br/>12<br/>2n

|- align=center valign=top

| 2/m<br/><br/>4/m<br/><br/>6/m<br/>n/m<br/>

| 2<br/> 2<br/> 2<br/> 2<br/> 2<br/> 2

| 2*<br/>3*<br/>4*<br/>5*<br/>6*<br/>n*

| C_2h<br/>C_3h<br/>C_4h<br/>C_5h<br/>C_6h<br/>C_nh

| [2,2⁺]<br/>[2,3⁺]<br/>[2,4⁺]<br/>[2,5⁺]<br/>[2,6⁺]<br/>[2,n⁺]

| 4<br/>6<br/>8<br/>10<br/>12<br/>2n

|- align=center valign=top

| <br/><br/><br/><br/><br/><br/>

| <br/><br/><br/><br/><br/>

| 2×<br/>3×<br/>4×<br/>5×<br/>6×<br/>n×

| S₄<br/>S₆<br/>S₈<br/>S₁₀<br/>S₁₂<br/>S_2n

| [2⁺,4⁺]<br/>[2⁺,6⁺]<br/>[2⁺,8⁺]<br/>[2⁺,10⁺]<br/>[2⁺,12⁺]<br/>[2⁺,2n⁺]

| 4<br/>6<br/>8<br/>10<br/>12<br/>2n

|}

|

{| class="wikitable"

|-

! Intl

! Geo

! Orbifold

! Schoenflies

! Coxeter

! Order

|- align=center valign=top

| 222<br/>32<br/>422<br/>52<br/>622<br/>n22<br/>n2

| <br/> <br/> <br/> <br/> <br/>

| 222<br/>223<br/>224<br/>225<br/>226<br/>22n

| D₂<br/>D₃<br/>D₄<br/>D₅<br/>D₆<br/>D_n

| [2,2]⁺<br/>[2,3]⁺<br/>[2,4]⁺<br/>[2,5]⁺<br/>[2,6]⁺<br/>[2,n]⁺

| 4<br/>6<br/>8<br/>10<br/>12<br/>2n

|- align=center valign=top

| mmm<br/>m2<br/>4/mmm<br/>m2<br/>6/mmm<br/>n/mmm<br/>m2

| 2 2<br/>3 2<br/>4 2<br/>5 2<br/>6 2<br/>n 2

| *222<br/>*223<br/>*224<br/>*225<br/>*226<br/>*22n

| D_2h<br/>D_3h<br/>D_4h<br/>D_5h<br/>D_6h<br/>D_nh

| [2,2]<br/>[2,3]<br/>[2,4]<br/>[2,5]<br/>[2,6]<br/>[2,n]

| 8<br/>12<br/>16<br/>20<br/>24<br/>4n

|- align=center valign=top

| 2m<br/>m<br/>2m<br/>m<br/>2m<br/>2m<br/>m

| 4 <br/>6 <br/>8 <br/>10 <br/>12 <br/>n <br/>

| 2*2<br/>2*3<br/>2*4<br/>2*5<br/>2*6<br/>2*n

| D_2d<br/>D_3d<br/>D_4d<br/>D_5d<br/>D_6d<br/>D_nd

| [2⁺,4]<br/>[2⁺,6]<br/>[2⁺,8]<br/>[2⁺,10]<br/>[2⁺,12]<br/>[2⁺,2n]

| 8<br/>12<br/>16<br/>20<br/>24<br/>4n

|- align=center

| 23

|

| 332

| T

| [3,3]⁺

| 12

|- align=center

| m

| 4

| 3*2

| T_h

| [3⁺,4]

| 24

|- align=center

| 3m

| 3 3

| *332

| T_d

| [3,3]

| 24

|- align=center

| 432

|

| 432

| O

| [3,4]⁺

| 24

|- align=center

| mm

| 4 3

| *432

| O_h

| [3,4]

| 48

|- align=center

| 532

|

| 532

| I

| [3,5]⁺

| 60

|- align=center

| m

| 5 3

| *532

| I_h

| [3,5]

| 120

|}

|-

|colspan=2|(*) When the Intl entries are duplicated, the first is for even n, the second for odd n.

|}

Reflection groups

thumb|Finite isomorphism and correspondences

The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The [3,3] group can be doubled, written as , mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.

{| class=wikitable

! Schoenflies

! colspan=2 | Coxeter group

! colspan=3 | Coxeter diagram

! Order

! Related regular and <br/>prismatic polyhedra

|- align=center

||T_d|| A₃||[3,3]

|rowspan=2|

||||||24||tetrahedron

|- align=center

||T_d×Dih₁ = O_h|| A₃×2 = BC₃|| = [4,3]

||||= ||48||stellated octahedron

|- align=center

||O_h|| BC₃||[4,3]||

||||||48||cube, octahedron

|- align=center

||I_h||H₃||[5,3]||

||||||120||icosahedron, dodecahedron

|- align=center

||D_3h|| A₂×A₁||[3,2]

|rowspan=2|

||||||12||triangular prism

|- align=center

||D_3h×Dih₁ = D_6h|| A₂×A₁×2||[[3],2]

||||= ||24||hexagonal prism

|- align=center

||D_4h|| BC₂×A₁||[4,2]

|rowspan=2|

||||||16||square prism

|- align=center

||D_4h×Dih₁ = D_8h|| BC₂×A₁×2||[[4],2] = [8,2]

||||= ||32||octagonal prism

|- align=center

||D_5h|| H₂×A₁||[5,2]||

||||||20||pentagonal prism

|- align=center

||D_6h|| G₂×A₁||[6,2]||

||||||24||hexagonal prism

|- align=center

||D_nh|| I₂(n)×A₁||[n,2]

|rowspan=2|

||||||4n||n-gonal prism

|- align=center

||D_nh×Dih₁ = D_2nh|| I₂(n)×A₁×2||[[n],2]

||||= ||8n

|- align=center

||D_2h|| A₁³||[2,2]

|rowspan=3|

|||||8

|rowspan=3|cuboid

|- align=center

||D_2h×Dih₁|| A₁³×2||[[2],2] = [4,2]

||||= ||16

|- align=center

||D_2h×Dih₃ = O_h|| A₁³×6||[3[2,2]] = [4,3]

||||= ||48

|- align=center

||C_3v|| A₂||[1,3]||

||||||6

|rowspan=9|hosohedron

|- align=center

||C_4v|| BC₂||[1,4]||

||||||8

|- align=center

||C_5v|| H₂||[1,5]||

||||||10

|- align=center

||C_6v|| G₂||[1,6]||

||||||12

|- align=center

||C_nv|| I₂(n)||[1,n]

|rowspan=2|

||||||2n

|- align=center

||C_nv×Dih₁ = C_2nv|| I₂(n)×2||[1,[n]] = [1,2n]

||||= ||4n

|- align=center

||C_2v|| A₁²||[1,2]

|rowspan=2|

||||||4

|- align=center

||C_2v×Dih₁|| A₁²×2||[1,[2]]

||||= ||8

|- align=center

||C_s|| A₁||[1,1]||

||||||2

|}

Four dimensions

The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,