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Point group
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 The Bauhinia blakeana flower on the Hong Kong region flag has C5 symmetry; the star on each petal has D5 symmetry. |
 The Yin and Yang symbol has C2 symmetry of geometry with inverted colors |
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 y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).
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
[edit]Point groups can be classified into chiral (or purely rotational) groups and achiral groups.[1] 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
[edit]One dimension
[edit]There are only two one-dimensional point groups, the identity group and the reflection group.
| Group | Coxeter | Coxeter diagram | Order | Description |
|---|---|---|---|---|
| C1 | [ ]+ | 1 | identity | |
| D1 | [ ] |  |
2 | reflection group |
Two dimensions
[edit]Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
- Cyclic groups Cn of n-fold rotation groups
- Dihedral groups Dn 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.
| Group | Intl | Orbifold | Coxeter | Order | Description |
|---|---|---|---|---|---|
| Cn | n | n• | [n]+ | n | cyclic: n-fold rotations; abstract group Zn, the group of integers under addition modulo n |
| Dn | nm | *n• | [n] | 2n | dihedral: cyclic with reflections; abstract group Dihn, the dihedral group |
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.
| Reflective | Rotational | Related polygons | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Group | Coxeter group | Coxeter diagram | Order | Subgroup | Coxeter | Order | |||
| D1 | A1 | [ ] | |
 |
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 |
| Dn | I2(n) | [n] |  |
|
2n | Cn | [n]+ | n | regular polygon |
| D2×2 | A12×2 | [[2]] = [4] | |
= |
8 | ||||
| D3×2 | A2×2 | [[3]] = [6] | |
= |
12 | ||||
| D4×2 | BC2×2 | [[4]] = [8] | |
=  |
16 | ||||
| D5×2 | H2×2 | [[5]] = [10] | |
=  |
20 | ||||
| D6×2 | G2×2 | [[6]] = [12] | |
=  |
24 | ||||
| Dn×2 | I2(n)×2 | [[n]] = [2n] | |
=  |
4n | ||||
Three dimensions
[edit]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: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
- Polyhedral groups: T, Td, Th, O, Oh, I, Ih
Applying the crystallographic restriction theorem to these groups yields the 32 crystallographic point groups.
| C1v Order 2 |
C2v Order 4 |
C3v Order 6 |
C4v Order 8 |
C5v Order 10 |
C6v Order 12 |
... |
|---|---|---|---|---|---|---|
|
|
|
|
|
| |
| D1h Order 4 |
D2h Order 8 |
D3h Order 12 |
D4h Order 16 |
D5h Order 20 |
D6h Order 24 |
... |
|
|
|
|
|
| |
| Td Order 24 |
Oh Order 48 |
Ih Order 120 | ||||
|
|
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (*) When the Intl entries are duplicated, the first is for even n, the second for odd n. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Reflection groups
[edit]
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 [[3,3]], mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.
| Schoenflies | Coxeter group | Coxeter diagram | Order | Related regular and prismatic polyhedra | |||
|---|---|---|---|---|---|---|---|
| Td | A3 | [3,3] |
|
 |
24 | tetrahedron | |
| Td×Dih1 = Oh | A3×2 = BC3 | [[3,3]] = [4,3] | |
= |
48 | stellated octahedron | |
| Oh | BC3 | [4,3] |
|
|
48 | cube, octahedron | |
| Ih | H3 | [5,3] |
|
|
120 | icosahedron, dodecahedron | |
| D3h | A2×A1 | [3,2] |
|
|
12 | triangular prism | |
| D3h×Dih1 = D6h | A2×A1×2 | [[3],2] | |
= |
24 | hexagonal prism | |
| D4h | BC2×A1 | [4,2] |
|
|
16 | square prism | |
| D4h×Dih1 = D8h | BC2×A1×2 | [[4],2] = [8,2] | |
= |
32 | octagonal prism | |
| D5h | H2×A1 | [5,2] |
|
|
20 | pentagonal prism | |
| D6h | G2×A1 | [6,2] |
|
|
24 | hexagonal prism | |
| Dnh | I2(n)×A1 | [n,2] |
|
|
4n | n-gonal prism | |
| Dnh×Dih1 = D2nh | I2(n)×A1×2 | [[n],2] | |
= |
8n | ||
| D2h | A13 | [2,2] |
|
|
8 | cuboid | |
| D2h×Dih1 | A13×2 | [[2],2] = [4,2] | |
= |
16 | ||
| D2h×Dih3 = Oh | A13×6 | [3[2,2]] = [4,3] | |
= |
48 | ||
| C3v | A2 | [1,3] |
|
|
6 | hosohedron | |
| C4v | BC2 | [1,4] |
|
|
8 | ||
| C5v | H2 | [1,5] |
|
|
10 | ||
| C6v | G2 | [1,6] |
|
|
12 | ||
| Cnv | I2(n) | [1,n] |
|
|
2n | ||
| Cnv×Dih1 = C2nv | I2(n)×2 | [1,[n]] = [1,2n] | |
= |
4n | ||
| C2v | A12 | [1,2] |
|
|
4 | ||
| C2v×Dih1 | A12×2 | [1,[2]] | |
= |
8 | ||
| Cs | A1 | [1,1] |
|
|
2 | ||
Four dimensions
[edit]The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,[1] Section 4, Tables 4.1–4.3.
The following list gives the four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3]+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example [[3,3,3]] with its order doubled to 240.
| Coxeter group/notation | Coxeter diagram | Order | Related polytopes | ||
|---|---|---|---|---|---|
| A4 | [3,3,3] |  |
120 | 5-cell | |
| A4×2 | [[3,3,3]] | |
240 | 5-cell dual compound | |
| BC4 | [4,3,3] | |
384 | 16-cell / tesseract | |
| D4 | [31,1,1] |   |
192 | demitesseractic | |
| D4×2 = BC4 | <[3,31,1]> = [4,3,3] |  |
= |
384 | |
| D4×6 = F4 | [3[31,1,1]] = [3,4,3] | |
= |
1152 | |
| F4 | [3,4,3] | |
1152 | 24-cell | |
| F4×2 | [[3,4,3]] | |
2304 | 24-cell dual compound | |
| H4 | [5,3,3] | |
14400 | 120-cell / 600-cell | |
| A3×A1 | [3,3,2] | |
48 | tetrahedral prism | |
| A3×A1×2 | [[3,3],2] = [4,3,2] | |
= |
96 | octahedral prism |
| BC3×A1 | [4,3,2] | |
96 | ||
| H3×A1 | [5,3,2] | |
240 | icosahedral prism | |
| A2×A2 | [3,2,3] | |
36 | duoprism | |
| A2×BC2 | [3,2,4] | |
48 | ||
| A2×H2 | [3,2,5] | |
60 | ||
| A2×G2 | [3,2,6] | |
72 | ||
| BC2×BC2 | [4,2,4] | |
64 | ||
| BC22×2 | [[4,2,4]] | |
128 | ||
| BC2×H2 | [4,2,5] | |
80 | ||
| BC2×G2 | [4,2,6] | |
96 | ||
| H2×H2 | [5,2,5] | |
100 | ||
| H2×G2 | [5,2,6] | |
120 | ||
| G2×G2 | [6,2,6] | |
144 | ||
| I2(p)×I2(q) | [p,2,q] |   |
4pq | ||
| I2(2p)×I2(q) | [[p],2,q] = [2p,2,q] | |
= |
8pq | |
| I2(2p)×I2(2q) | [[p]],2,[[q]] = [2p,2,2q] | |
= |
16pq | |
| I2(p)2×2 | [[p,2,p]] | |
8p2 | ||
| I2(2p)2×2 | [[[p]],2,[p]]] = [[2p,2,2p]] | |
= |
32p2 | |
| A2×A1×A1 | [3,2,2] | |
24 | ||
| BC2×A1×A1 | [4,2,2] | |
32 | ||
| H2×A1×A1 | [5,2,2] | |
40 | ||
| G2×A1×A1 | [6,2,2] | |
48 | ||
| I2(p)×A1×A1 | [p,2,2] | |
8p | ||
| I2(2p)×A1×A1×2 | [[p],2,2] = [2p,2,2] | |
= |
16p | |
| I2(p)×A12×2 | [p,2,[2]] = [p,2,4] | |
= |
16p | |
| I2(2p)×A12×4 | [[p]],2,[[2]] = [2p,2,4] | |
= |
32p | |
| A1×A1×A1×A1 | [2,2,2] | |
16 | 4-orthotope | |
| A12×A1×A1×2 | [[2],2,2] = [4,2,2] | |
= |
32 | |
| A12×A12×4 | [[2]],2,[[2]] = [4,2,4] | |
= |
64 | |
| A13×A1×6 | [3[2,2],2] = [4,3,2] | |
= |
96 | |
| A14×24 | [3,3[2,2,2]] = [4,3,3] | |
= |
384 | |
Five dimensions
[edit]
The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3]+ has four 3-fold gyration points and symmetry order 360.
| Coxeter group/notation | Coxeter diagrams |
Order | Related regular and prismatic polytopes | ||
|---|---|---|---|---|---|
| A5 | [3,3,3,3] |
|
 |
720 | 5-simplex |
| A5×2 | [[3,3,3,3]] |
|
|
1440 | 5-simplex dual compound |
| BC5 | [4,3,3,3] |
|
|
3840 | 5-cube, 5-orthoplex |
| D5 | [32,1,1] | 
|
|
1920 | 5-demicube |
| D5×2 | <[3,3,31,1]> |
|
= |
3840 | |
| A4×A1 | [3,3,3,2] |
|
|
240 | 5-cell prism |
| A4×A1×2 | [[3,3,3],2] |
|
|
480 | |
| BC4×A1 | [4,3,3,2] |
|
|
768 | tesseract prism |
| F4×A1 | [3,4,3,2] |
|
|
2304 | 24-cell prism |
| F4×A1×2 | [[3,4,3],2] |
|
|
4608 | |
| H4×A1 | [5,3,3,2] |
|
|
28800 | 600-cell or 120-cell prism |
| D4×A1 | [31,1,1,2] |
|
|
384 | demitesseract prism |
| A3×A2 | [3,3,2,3] |
|
|
144 | duoprism |
| A3×A2×2 | [[3,3],2,3] |
|
|
288 | |
| A3×BC2 | [3,3,2,4] |
|
|
192 | |
| A3×H2 | [3,3,2,5] |
|
|
240 | |
| A3×G2 | [3,3,2,6] |
|
|
288 | |
| A3×I2(p) | [3,3,2,p] |
|
|
48p | |
| BC3×A2 | [4,3,2,3] |
|
|
288 | |
| BC3×BC2 | [4,3,2,4] |
|
|
384 | |
| BC3×H2 | [4,3,2,5] |
|
|
480 | |
| BC3×G2 | [4,3,2,6] |
|
|
576 | |
| BC3×I2(p) | [4,3,2,p] |
|
|
96p | |
| H3×A2 | [5,3,2,3] |
|
|
720 | |
| H3×BC2 | [5,3,2,4] |
|
|
960 | |
| H3×H2 | [5,3,2,5] |
|
|
1200 | |
| H3×G2 | [5,3,2,6] |
|
|
1440 | |
| H3×I2(p) | [5,3,2,p] |
|
240p | ||
| A3×A12 | [3,3,2,2] |
|
96 | ||
| BC3×A12 | [4,3,2,2] |
|
192 | ||
| H3×A12 | [5,3,2,2] |
|
480 | ||
| A22×A1 | [3,2,3,2] |
|
72 | duoprism prism | |
| A2×BC2×A1 | [3,2,4,2] |
|
96 | ||
| A2×H2×A1 | [3,2,5,2] |
|
120 | ||
| A2×G2×A1 | [3,2,6,2] |
|
144 | ||
| BC22×A1 | [4,2,4,2] |
|
128 | ||
| BC2×H2×A1 | [4,2,5,2] |
|
160 | ||
| BC2×G2×A1 | [4,2,6,2] |
|
192 | ||
| H22×A1 | [5,2,5,2] |
|
200 | ||
| H2×G2×A1 | [5,2,6,2] |
|
240 | ||
| G22×A1 | [6,2,6,2] |
|
288 | ||
| I2(p)×I2(q)×A1 | [p,2,q,2] |
|
8pq | ||
| A2×A13 | [3,2,2,2] |
|
48 | ||
| BC2×A13 | [4,2,2,2] |
|
64 | ||
| H2×A13 | [5,2,2,2] |
|
80 | ||
| G2×A13 | [6,2,2,2] |
|
96 | ||
| I2(p)×A13 | [p,2,2,2] |
|
16p | ||
| A15 | [2,2,2,2] |
|
|
32 | 5-orthotope |
| A15×(2!) | [[2],2,2,2] |
|
= |
64 | |
| A15×(2!×2!) | [[2]],2,[2],2] |
|
= |
128 | |
| A15×(3!) | [3[2,2],2,2] |
|
= |
192 | |
| A15×(3!×2!) | [3[2,2],2,[[2]] |
|
= |
384 | |
| A15×(4!) | [3,3[2,2,2],2]] |
|
= |
768 | |
| A15×(5!) | [3,3,3[2,2,2,2]] |
|
= |
3840 | |
Six dimensions
[edit]
The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3]+ has five 3-fold gyration points and symmetry order 2520.
| Coxeter group | Coxeter diagram |
Order | Related regular and prismatic polytopes | |
|---|---|---|---|---|
| A6 | [3,3,3,3,3] | |
5040 (7!) | 6-simplex |
| A6×2 | [[3,3,3,3,3]] |   |
10080 (2×7!) | 6-simplex dual compound |
| BC6 | [4,3,3,3,3] | |
46080 (26×6!) | 6-cube, 6-orthoplex |
| D6 | [3,3,3,31,1] |   |
23040 (25×6!) | 6-demicube |
| E6 | [3,32,2] | |
51840 (72×6!) | 122, 221 |
| A5×A1 | [3,3,3,3,2] | |
1440 (2×6!) | 5-simplex prism |
| BC5×A1 | [4,3,3,3,2] | |
7680 (26×5!) | 5-cube prism |
| D5×A1 | [3,3,31,1,2] | |
3840 (25×5!) | 5-demicube prism |
| A4×I2(p) | [3,3,3,2,p] | |
240p | duoprism |
| BC4×I2(p) | [4,3,3,2,p] | |
768p | |
| F4×I2(p) | [3,4,3,2,p] | |
2304p | |
| H4×I2(p) | [5,3,3,2,p] | |
28800p | |
| D4×I2(p) | [3,31,1,2,p] | |
384p | |
| A4×A12 | [3,3,3,2,2] | |
480 | |
| BC4×A12 | [4,3,3,2,2] | |
1536 | |
| F4×A12 | [3,4,3,2,2] | |
4608 | |
| H4×A12 | [5,3,3,2,2] | |
57600 | |
| D4×A12 | [3,31,1,2,2] | |
768 | |
| A32 | [3,3,2,3,3] | |
576 | |
| A3×BC3 | [3,3,2,4,3] | |
1152 | |
| A3×H3 | [3,3,2,5,3] | |
2880 | |
| BC32 | [4,3,2,4,3] | |
2304 | |
| BC3×H3 | [4,3,2,5,3] | |
5760 | |
| H32 | [5,3,2,5,3] | |
14400 | |
| A3×I2(p)×A1 | [3,3,2,p,2] | |
96p | duoprism prism |
| BC3×I2(p)×A1 | [4,3,2,p,2] | |
192p | |
| H3×I2(p)×A1 | [5,3,2,p,2] | |
480p | |
| A3×A13 | [3,3,2,2,2] | |
192 | |
| BC3×A13 | [4,3,2,2,2] | |
384 | |
| H3×A13 | [5,3,2,2,2] | |
960 | |
| I2(p)×I2(q)×I2(r) | [p,2,q,2,r] |  |
8pqr | triaprism |
| I2(p)×I2(q)×A12 | [p,2,q,2,2] | |
16pq | |
| I2(p)×A14 | [p,2,2,2,2] | |
32p | |
| A16 | [2,2,2,2,2] | |
64 | 6-orthotope |
Seven dimensions
[edit]The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3]+ has six 3-fold gyration points and symmetry order 20160.
| Coxeter group | Coxeter diagram | Order | Related polytopes | |
|---|---|---|---|---|
| A7 | [3,3,3,3,3,3] | |
40320 (8!) | 7-simplex |
| A7×2 | [[3,3,3,3,3,3]] |  |
80640 (2×8!) | 7-simplex dual compound |
| BC7 | [4,3,3,3,3,3] | |
645120 (27×7!) | 7-cube, 7-orthoplex |
| D7 | [3,3,3,3,31,1] | |
322560 (26×7!) | 7-demicube |
| E7 | [3,3,3,32,1] | |
2903040 (8×9!) | 321, 231, 132 |
| A6×A1 | [3,3,3,3,3,2] | |
10080 (2×7!) | |
| BC6×A1 | [4,3,3,3,3,2] | |
92160 (27×6!) | |
| D6×A1 | [3,3,3,31,1,2] | |
46080 (26×6!) | |
| E6×A1 | [3,3,32,1,2] | |
103680 (144×6!) | |
| A5×I2(p) | [3,3,3,3,2,p] | |
1440p | |
| BC5×I2(p) | [4,3,3,3,2,p] | |
7680p | |
| D5×I2(p) | [3,3,31,1,2,p] | |
3840p | |
| A5×A12 | [3,3,3,3,2,2] | |
2880 | |
| BC5×A12 | [4,3,3,3,2,2] | |
15360 | |
| D5×A12 | [3,3,31,1,2,2] | |
7680 | |
| A4×A3 | [3,3,3,2,3,3] | |
2880 | |
| A4×BC3 | [3,3,3,2,4,3] | |
5760 | |
| A4×H3 | [3,3,3,2,5,3] | |
14400 | |
| BC4×A3 | [4,3,3,2,3,3] | |
9216 | |
| BC4×BC3 | [4,3,3,2,4,3] | |
18432 | |
| BC4×H3 | [4,3,3,2,5,3] | |
46080 | |
| H4×A3 | [5,3,3,2,3,3] | |
345600 | |
| H4×BC3 | [5,3,3,2,4,3] | |
691200 | |
| H4×H3 | [5,3,3,2,5,3] | |
1728000 | |
| F4×A3 | [3,4,3,2,3,3] | |
27648 | |
| F4×BC3 | [3,4,3,2,4,3] | |
55296 | |
| F4×H3 | [3,4,3,2,5,3] | |
138240 | |
| D4×A3 | [31,1,1,2,3,3] | |
4608 | |
| D4×BC3 | [3,31,1,2,4,3] | |
9216 | |
| D4×H3 | [3,31,1,2,5,3] | |
23040 | |
| A4×I2(p)×A1 | [3,3,3,2,p,2] | |
480p | |
| BC4×I2(p)×A1 | [4,3,3,2,p,2] | |
1536p | |
| D4×I2(p)×A1 | [3,31,1,2,p,2] | |
768p | |
| F4×I2(p)×A1 | [3,4,3,2,p,2] | |
4608p | |
| H4×I2(p)×A1 | [5,3,3,2,p,2] | |
57600p | |
| A4×A13 | [3,3,3,2,2,2] | |
960 | |
| BC4×A13 | [4,3,3,2,2,2] | |
3072 | |
| F4×A13 | [3,4,3,2,2,2] | |
9216 | |
| H4×A13 | [5,3,3,2,2,2] | |
115200 | |
| D4×A13 | [3,31,1,2,2,2] | |
1536 | |
| A32×A1 | [3,3,2,3,3,2] | |
1152 | |
| A3×BC3×A1 | [3,3,2,4,3,2] | |
2304 | |
| A3×H3×A1 | [3,3,2,5,3,2] | |
5760 | |
| BC32×A1 | [4,3,2,4,3,2] | |
4608 | |
| BC3×H3×A1 | [4,3,2,5,3,2] | |
11520 | |
| H32×A1 | [5,3,2,5,3,2] | |
28800 | |
| A3×I2(p)×I2(q) | [3,3,2,p,2,q] | |
96pq | |
| BC3×I2(p)×I2(q) | [4,3,2,p,2,q] | |
192pq | |
| H3×I2(p)×I2(q) | [5,3,2,p,2,q] | |
480pq | |
| A3×I2(p)×A12 | [3,3,2,p,2,2] | |
192p | |
| BC3×I2(p)×A12 | [4,3,2,p,2,2] | |
384p | |
| H3×I2(p)×A12 | [5,3,2,p,2,2] | |
960p | |
| A3×A14 | [3,3,2,2,2,2] | |
384 | |
| BC3×A14 | [4,3,2,2,2,2] | |
768 | |
| H3×A14 | [5,3,2,2,2,2] | |
1920 | |
| I2(p)×I2(q)×I2(r)×A1 | [p,2,q,2,r,2] | |
16pqr | |
| I2(p)×I2(q)×A13 | [p,2,q,2,2,2] | |
32pq | |
| I2(p)×A15 | [p,2,2,2,2,2] | |
64p | |
| A17 | [2,2,2,2,2,2] | |
128 | |
Eight dimensions
[edit]The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3,3]+ has seven 3-fold gyration points and symmetry order 181440.
| Coxeter group | Coxeter diagram | Order | Related polytopes | |
|---|---|---|---|---|
| A8 | [3,3,3,3,3,3,3] | |
362880 (9!) | 8-simplex |
| A8×2 | [[3,3,3,3,3,3,3]] | |
725760 (2×9!) | 8-simplex dual compound |
| BC8 | [4,3,3,3,3,3,3] | |
10321920 (288!) | 8-cube, 8-orthoplex |
| D8 | [3,3,3,3,3,31,1] | |
5160960 (278!) | 8-demicube |
| E8 | [3,3,3,3,32,1] | |
696729600 (192×10!) | 421, 241, 142 |
| A7×A1 | [3,3,3,3,3,3,2] | |
80640 | 7-simplex prism |
| BC7×A1 | [4,3,3,3,3,3,2] | |
645120 | 7-cube prism |
| D7×A1 | [3,3,3,3,31,1,2] | |
322560 | 7-demicube prism |
| E7×A1 | [3,3,3,32,1,2] | |
5806080 | 321 prism, 231 prism, 142 prism |
| A6×I2(p) | [3,3,3,3,3,2,p] | |
10080p | duoprism |
| BC6×I2(p) | [4,3,3,3,3,2,p] | |
92160p | |
| D6×I2(p) | [3,3,3,31,1,2,p] | |
46080p | |
| E6×I2(p) | [3,3,32,1,2,p] | |
103680p | |
| A6×A12 | [3,3,3,3,3,2,2] | |
20160 | |
| BC6×A12 | [4,3,3,3,3,2,2] | |
184320 | |
| D6×A12 | [33,1,1,2,2] | |
92160 | |
| E6×A12 | [3,3,32,1,2,2] | |
207360 | |
| A5×A3 | [3,3,3,3,2,3,3] | |
17280 | |
| BC5×A3 | [4,3,3,3,2,3,3] | |
92160 | |
| D5×A3 | [32,1,1,2,3,3] | |
46080 | |
| A5×BC3 | [3,3,3,3,2,4,3] | |
34560 | |
| BC5×BC3 | [4,3,3,3,2,4,3] | |
184320 | |
| D5×BC3 | [32,1,1,2,4,3] | |
92160 | |
| A5×H3 | [3,3,3,3,2,5,3] | |
||
| BC5×H3 | [4,3,3,3,2,5,3] | |
||
| D5×H3 | [32,1,1,2,5,3] | |
||
| A5×I2(p)×A1 | [3,3,3,3,2,p,2] | |
||
| BC5×I2(p)×A1 | [4,3,3,3,2,p,2] | |
||
| D5×I2(p)×A1 | [32,1,1,2,p,2] | |
||
| A5×A13 | [3,3,3,3,2,2,2] | |
||
| BC5×A13 | [4,3,3,3,2,2,2] | |
||
| D5×A13 | [32,1,1,2,2,2] | |
||
| A4×A4 | [3,3,3,2,3,3,3] | |
||
| BC4×A4 | [4,3,3,2,3,3,3] | |
||
| D4×A4 | [31,1,1,2,3,3,3] | |
||
| F4×A4 | [3,4,3,2,3,3,3] | |
||
| H4×A4 | [5,3,3,2,3,3,3] | |
||
| BC4×BC4 | [4,3,3,2,4,3,3] | |
||
| D4×BC4 | [31,1,1,2,4,3,3] | |
||
| F4×BC4 | [3,4,3,2,4,3,3] | |
||
| H4×BC4 | [5,3,3,2,4,3,3] | |
||
| D4×D4 | [31,1,1,2,31,1,1] | |
||
| F4×D4 | [3,4,3,2,31,1,1] | |
||
| H4×D4 | [5,3,3,2,31,1,1] | |
||
| F4×F4 | [3,4,3,2,3,4,3] | |
||
| H4×F4 | [5,3,3,2,3,4,3] | |
||
| H4×H4 | [5,3,3,2,5,3,3] | |
||
| A4×A3×A1 | [3,3,3,2,3,3,2] | |
duoprism prisms | |
| A4×BC3×A1 | [3,3,3,2,4,3,2] | |
||
| A4×H3×A1 | [3,3,3,2,5,3,2] | |
||
| BC4×A3×A1 | [4,3,3,2,3,3,2] | |
||
| BC4×BC3×A1 | [4,3,3,2,4,3,2] | |
||
| BC4×H3×A1 | [4,3,3,2,5,3,2] | |
||
| H4×A3×A1 | [5,3,3,2,3,3,2] | |
||
| H4×BC3×A1 | [5,3,3,2,4,3,2] | |
||
| H4×H3×A1 | [5,3,3,2,5,3,2] | |
||
| F4×A3×A1 | [3,4,3,2,3,3,2] | |
||
| F4×BC3×A1 | [3,4,3,2,4,3,2] | |
||
| F4×H3×A1 | [3,4,2,3,5,3,2] | |
||
| D4×A3×A1 | [31,1,1,2,3,3,2] | |
||
| D4×BC3×A1 | [31,1,1,2,4,3,2] | |
||
| D4×H3×A1 | [31,1,1,2,5,3,2] | |
||
| A4×I2(p)×I2(q) | [3,3,3,2,p,2,q] | |
triaprism | |
| BC4×I2(p)×I2(q) | [4,3,3,2,p,2,q] | |
||
| F4×I2(p)×I2(q) | [3,4,3,2,p,2,q] | |
||
| H4×I2(p)×I2(q) | [5,3,3,2,p,2,q] | |
||
| D4×I2(p)×I2(q) | [31,1,1,2,p,2,q] | |
||
| A4×I2(p)×A12 | [3,3,3,2,p,2,2] | |
||
| BC4×I2(p)×A12 | [4,3,3,2,p,2,2] | |
||
| F4×I2(p)×A12 | [3,4,3,2,p,2,2] | |
||
| H4×I2(p)×A12 | [5,3,3,2,p,2,2] | |
||
| D4×I2(p)×A12 | [31,1,1,2,p,2,2] | |
||
| A4×A14 | [3,3,3,2,2,2,2] | |
||
| BC4×A14 | [4,3,3,2,2,2,2] | |
||
| F4×A14 | [3,4,3,2,2,2,2] | |
||
| H4×A14 | [5,3,3,2,2,2,2] | |
||
| D4×A14 | [31,1,1,2,2,2,2] | |
||
| A3×A3×I2(p) | [3,3,2,3,3,2,p] | |
||
| BC3×A3×I2(p) | [4,3,2,3,3,2,p] | |
||
| H3×A3×I2(p) | [5,3,2,3,3,2,p] | |
||
| BC3×BC3×I2(p) | [4,3,2,4,3,2,p] | |
||
| H3×BC3×I2(p) | [5,3,2,4,3,2,p] | |
||
| H3×H3×I2(p) | [5,3,2,5,3,2,p] | |
||
| A3×A3×A12 | [3,3,2,3,3,2,2] | |
||
| BC3×A3×A12 | [4,3,2,3,3,2,2] | |
||
| H3×A3×A12 | [5,3,2,3,3,2,2] | |
||
| BC3×BC3×A12 | [4,3,2,4,3,2,2] | |
||
| H3×BC3×A12 | [5,3,2,4,3,2,2] | |
||
| H3×H3×A12 | [5,3,2,5,3,2,2] | |
||
| A3×I2(p)×I2(q)×A1 | [3,3,2,p,2,q,2] | |
||
| BC3×I2(p)×I2(q)×A1 | [4,3,2,p,2,q,2] | |
||
| H3×I2(p)×I2(q)×A1 | [5,3,2,p,2,q,2] | |
||
| A3×I2(p)×A13 | [3,3,2,p,2,2,2] | |
||
| BC3×I2(p)×A13 | [4,3,2,p,2,2,2] | |
||
| H3×I2(p)×A13 | [5,3,2,p,2,2,2] | |
||
| A3×A15 | [3,3,2,2,2,2,2] | |
||
| BC3×A15 | [4,3,2,2,2,2,2] | |
||
| H3×A15 | [5,3,2,2,2,2,2] | |
||
| I2(p)×I2(q)×I2(r)×I2(s) | [p,2,q,2,r,2,s] |  |
16pqrs | |
| I2(p)×I2(q)×I2(r)×A12 | [p,2,q,2,r,2,2] | |
32pqr | |
| I2(p)×I2(q)×A14 | [p,2,q,2,2,2,2] | |
64pq | |
| I2(p)×A16 | [p,2,2,2,2,2,2] | |
128p | |
| A18 | [2,2,2,2,2,2,2] | |
256 | |
See also
[edit]References
[edit]- ^ a b Conway, John H.; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters. ISBN 978-1-56881-134-5.
- ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages)
Further reading
[edit]- H. S. M. Coxeter (1995), F. Arthur Sherk; Peter McMullen; Anthony C. Thompson; Asia Ivic Weiss (eds.), Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Wiley-Interscience Publication, ISBN 978-0-471-01003-6
- (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- H. S. M. Coxeter; W. O. J. Moser (1980), Generators and Relations for Discrete Groups (4th ed.), New York: Springer-Verlag
- N. W. Johnson (2018), "Chapter 11: Finite symmetry groups", Geometries and Transformations
External links
[edit]
Point group
View on Grokipediafrom Grokipedia
A point group is a collection of symmetry operations—such as rotations, reflections, and inversions—that leave a molecule or other finite object indistinguishable from its original form, with all such operations intersecting at a single fixed point, typically the center of mass or symmetry center.[1] This fixed point distinguishes point groups from space groups, which account for translational symmetries in infinite lattices like crystals.[2] In chemistry and physics, point groups provide a systematic way to classify the geometric symmetry of molecules, enabling predictions of physical and chemical properties.[3]
The concept of point groups arises from group theory in mathematics, where a group is defined by properties including closure under operation composition, the presence of an identity operation, associativity, and inverses for each element.[2] For molecular symmetry, chemists primarily use the Schoenflies notation, which categorizes groups based on principal rotation axes and additional elements like mirror planes or inversion centers; common classes include Cn (cyclic rotations), Cnv (rotations with vertical planes), Dnh (dihedral with horizontal planes), and high-symmetry groups like Td (tetrahedral) and Oh (octahedral).[4] In contrast, crystallography employs the Hermann-Mauguin notation for the 32 distinct point groups compatible with three-dimensional periodic lattices, as higher symmetries would violate translational invariance.[5] These 32 groups are distributed across seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.[6]
Point groups are essential for understanding molecular behavior, as symmetry dictates properties such as dipole moments, vibrational spectra, electronic transitions, and reactivity; for instance, molecules in groups without inversion centers (Cs, Cn, Cnv) may exhibit permanent dipoles, while high-symmetry groups like Ih (icosahedral) describe fullerenes.[3] They also inform stereochemistry, including chirality, where groups lacking improper rotations (e.g., Cn, Dn) indicate potential optical activity.[2] In practice, assigning a point group involves identifying symmetry elements via visual inspection or computational tools, often using flowcharts or character tables to match the object's operations.[1] Applications extend to spectroscopy, crystal engineering, and materials science, where symmetry analysis optimizes designs for properties like conductivity or luminescence.[4]
The three N-H σ orbitals span a representation reducible to A_1 + E; the A_1 SALC (totally symmetric, with equal coefficients) bonds with the nitrogen 2s/2p_z hybrid, while the degenerate E SALCs (with coefficients like (2, -1, -1) and (0, √3, -√3) up to normalization) interact with the 2p_x and 2p_y orbitals, yielding bonding, non-bonding, and antibonding molecular orbitals. This approach predicts the molecule's pyramidal geometry and lone pair directionality./Advanced_Inorganic_Chemistry_(Wikibook)/01:_Chapters/1.08:_NH3_Molecular_Orbitals)
Recent developments as of 2025 leverage machine learning, such as graph neural networks, to predict molecular point groups directly from atomic coordinates or chemical compositions, accelerating symmetry analysis in high-throughput screening for drug design where molecular symmetry informs receptor binding and pharmacokinetic properties. External magnetic fields can further modify effective point group symmetries by introducing time-reversal considerations, potentially lowering symmetry (e.g., from C_{nv} to C_n) and activating forbidden transitions in spectroscopy, with implications for studying magnetically perturbed molecular dynamics.[46][47]
Regarding chirality, the pure cyclic groups for are chiral, as they contain no reflections and thus admit enantiomorphic forms not superimposable on their mirror images, while and all dihedral groups are achiral due to the presence of mirror symmetries.
Introduction
Definition
A point group is defined as a finite subgroup of the orthogonal group in -dimensional Euclidean space, comprising isometries that preserve distances and fix a specific point, conventionally the origin.[7] These groups capture the discrete rotational and reflectional symmetries of objects around a central point, distinguishing them from space groups that also include translations.[8] The elements of a point group are represented by orthogonal matrices , which act linearly on position vectors to yield , satisfying the orthogonality condition to ensure preservation of the Euclidean norm .[8] Within , proper rotations form the special orthogonal group , consisting of matrices with determinant , while improper isometries, such as reflections and rotoinversions, have .[9] Basic symmetry operations in point groups include the identity operation, which leaves all points unchanged; rotations by an angle around an axis passing through the origin; reflections across hyperplanes containing the origin; and the inversion operation, which maps each point to .[10] While infinite point groups exist, such as the full , the focus in applications like molecular and crystal symmetry is on finite point groups, which describe discrete symmetry structures.[7]Historical Overview
The concept of point groups emerged in the early 19th century through efforts to classify crystal symmetries using geometric approaches. In 1830, Johann Friedrich Christian Hessel systematically enumerated the 32 possible finite symmetry classes for crystals based on morphological observations, laying the groundwork for what would later be formalized as point groups.[11] This work, though initially overlooked, provided a descriptive framework for the external symmetries of crystals without invoking abstract algebraic structures.[11] Nearly four decades later, Axel Gadolin independently rediscovered these 32 classes in 1867, employing stereographic projections to visualize and identify the positions of symmetry elements on crystal forms.[12] Gadolin's method offered a more graphical and reproducible technique, confirming Hessel's count while associating specific mineral examples with most classes, though some classes had not yet been observed in minerals at the time.[13] Building on this, late 19th-century mathematicians began recasting these symmetries within emerging group-theoretic frameworks. In 1884, Bernhard Minnigerode applied group theory to link elastic properties of crystals to their symmetry classes, recognizing the algebraic structure underlying physical phenomena.[14] Arthur Schönflies advanced this in 1891 by deriving point group notations and integrating them into broader spatial transformation groups, introducing the Schönflies notation still used today.[15] Heinrich Weber's 1896 algebraic treatise further solidified the abstract definition of groups, enabling rigorous classifications of finite symmetry operations relevant to point groups.[16] The early 20th century saw point groups incorporated into the study of space groups, which extend finite symmetries to periodic lattices. Evgraf Fedorov and Arthur Schönflies independently enumerated the 230 space groups in 1891, explicitly building upon point group foundations to describe full crystal symmetries.[17] Ludwig Bieberbach's theorems from 1911–1912 generalized crystallographic groups to higher dimensions, proving that translation subgroups form lattices and establishing finiteness for linear parts, thus connecting point groups to infinite discrete symmetries.[18] Post-1900 developments integrated point groups deeper into abstract group theory; H.S.M. Coxeter's work on finite reflection groups from the 1930s onward provided geometric realizations and classifications, treating point groups as Coxeter groups generated by reflections.[19] In 2003, John H. Conway and Derek A. Smith extended these ideas to higher dimensions using quaternions and octonions, classifying 4D rotation groups and their finite subgroups as point group analogs.[20] By the 21st century, point group theory found applications in computational materials science, particularly through machine learning for symmetry prediction. Recent advancements from 2020 to 2025 have developed ML models to accurately predict crystal point groups from chemical compositions, achieving high precision for ternary materials and accelerating discovery in complex systems.[21] These tools, often leveraging graph neural networks, integrate point group symmetries into broader materials informatics pipelines, enabling efficient screening of novel compounds with desired properties.[22]Mathematical Framework
Symmetry Operations and Elements
Symmetry operations in point groups are distance-preserving transformations, or isometries, that fix a central point, typically the origin, and map the object onto itself. These operations form the geometric foundation of point group symmetry and can be represented as orthogonal matrices in a Cartesian basis, ensuring the preservation of lengths and angles relative to the origin. In three dimensions, relevant to crystallography and molecular symmetry, the operations are linear maps that act on position vectors while maintaining the Euclidean metric.[2] The operations are classified into proper and improper types based on whether they preserve or reverse orientation (handedness). Proper rotations involve turning an object around an axis by an angle of , where (the order) is a positive integer greater than or equal to 1; for , this is the identity operation. Reflections occur across a mirror plane, effectively inverting coordinates perpendicular to that plane. Inversions pass points through the central origin, mapping to . Rotoinversions combine a proper rotation with an inversion, while improper rotations (also called rotoreflections) combine a proper rotation with a reflection perpendicular to the rotation axis. Note that is equivalent to a reflection (), and is equivalent to an inversion (). For higher orders, improper rotations () and rotoinversions remain distinct symmetry operations, though related through the group structure in specific point groups. These classifications ensure all operations are elements of the orthogonal group , with proper rotations in the special orthogonal subgroup .[23][24][10] Associated with these operations are symmetry elements, which are the geometric features invariant under the operation. A rotation axis (denoted ) is a line through the origin about which rotations occur. Mirror planes () are flat surfaces through the origin that act as reflection boundaries, subdivided into horizontal (), vertical (), or dihedral () based on orientation relative to principal axes. The inversion center () is the origin itself. Improper rotation axes () serve as combined loci for rotations and reflections or inversions, with , , and higher orders like or appearing in more complex symmetries. These elements provide the loci where the operations are defined and executed.[23][24] In crystallographic contexts, the crystallographic restriction theorem limits possible rotation orders to 1, 2, 3, 4, or 6 in two- and three-dimensional periodic lattices, as higher orders like 5 or 7 would disrupt translational periodicity. The proof sketch in two dimensions considers a primitive lattice vector rotated by angle to and by to ; since the lattice is invariant, must be an integer multiple of , yielding where is an integer with . The solutions correspond to , hence orders 4, 3, 6, 2, 1. In three dimensions, a similar argument applies to the plane perpendicular to the rotation axis, reducing to the two-dimensional case for a basis vector in that plane. This restriction ensures compatibility between rotational symmetries and discrete translations in crystal lattices.[25][24] As orthogonal transformations, these operations are exemplified by matrix representations in a suitable basis. In two dimensions, a 180° rotation (order 2) about the origin is given by the matrix which is the negative identity and maps to . A reflection across the x-axis uses inverting the y-coordinate while preserving x. In three dimensions, extending to the z-axis, the 180° rotation about z becomes and reflection across the xz-plane is These matrices illustrate how operations act linearly on coordinates while fixing the origin.[2]Group Theory Basics
Point groups are finite groups consisting of isometries that fix a common point, such as the origin in Euclidean space, with the group operation defined by composition of these isometries.[26] These groups satisfy the standard axioms of a group: closure under composition, associativity of the operation, the existence of an identity element (the trivial isometry), and the presence of inverses for each element (since isometries are invertible). As finite groups, point groups are discrete and compact, distinguishing them from continuous symmetry groups like the full orthogonal group , which includes all orthogonal transformations in -dimensions without finiteness restrictions.[9] The order of a point group is the number of distinct elements it contains, which corresponds to the finite number of symmetry operations preserving the fixed point.[27] For example, the cyclic group , generated by a single rotation by radians, has order , reflecting its distinct powers under composition.[28] Subgroups of point groups are subsets that themselves form groups under the same operation, and cosets partition the group into equal-sized translates of a subgroup. In achiral point groups, the chiral (rotation-only) subgroup often has index 2, meaning the full group has twice as many elements, with the coset consisting of improper rotations or reflections.[29] Representations of point groups provide an algebraic framework for analyzing symmetries by mapping group elements to linear transformations on a vector space, specifically as homomorphisms from the group to the general linear group , where is the dimension of the space.[30] Irreducible representations, which cannot be decomposed into simpler non-trivial representations, form the building blocks for understanding how symmetries act on physical systems, such as molecular orbitals or crystal vibrations.[31] Character theory extends this by associating to each representation a function (the character) that traces the diagonal of the matrix representations, enabling the decomposition of reducible representations into irreducibles via orthogonality relations.[32] Point groups are often isomorphic to abstract groups studied in group theory, meaning there exists a bijective homomorphism preserving the group structure. For instance, the dihedral group , symmetries of an equilateral triangle, is isomorphic to the symmetric group on three elements, both of order 6 and non-abelian.[33] These isomorphisms highlight that point groups realize geometric symmetries through algebraic structures, facilitating computations without reference to specific geometries.[34]Classification
Chiral and Achiral Point Groups
Point groups are classified as chiral or achiral based on whether they contain only orientation-preserving symmetries or include orientation-reversing elements. Chiral point groups consist exclusively of proper rotations, which are linear transformations with determinant +1, forming subgroups of the special orthogonal group SO(d) in d dimensions.[35] These groups lack any improper isometries, such as reflections, inversions, or improper rotations (roto-reflections), ensuring that no symmetry operation maps an object to its mirror image.[36] In contrast, achiral point groups incorporate improper isometries with determinant -1, allowing symmetries that reverse handedness. Every achiral point group contains a normal chiral subgroup of index 2, which is the set of its proper rotations, and adjoining an improper isometry generates the full group.[37] This structure implies that achiral groups can superimpose an object onto its mirror image through internal symmetry operations.[37] The primary criterion for a point group to be chiral is the absence of any S_n axis (improper rotation axis, where n ≥ 1, including mirror planes as S_1 and inversion as S_2) or mirror planes. This absence prevents the group from containing elements that would make a symmetric object achiral, leading to implications in molecular chemistry where chiral point groups permit the existence of enantiomers—non-superimposable mirror-image forms that cannot interconvert without breaking bonds. For example, representative chiral point groups include the cyclic groups C_n (pure rotations about one axis), dihedral groups D_n (rotations about a principal axis and perpendicular axes), and the polyhedral groups T (tetrahedral rotations, isomorphic to A_4), O (octahedral rotations, isomorphic to S_4), and I (icosahedral rotations, isomorphic to A_5).[36] In three dimensions, particularly for crystallographic point groups compatible with lattice translations, there are 32 such groups in total, of which 11 are chiral: 1, 2, 3, 4, 6 (cyclic), 222, 422, 32, 622 (dihedral), 23 (tetrahedral), and 432 (octahedral).[38] These chiral groups exhibit handedness in that their symmetry operations cannot produce a mirror image internally; achieving the enantiomorphic form requires an external improper operation outside the group. Achiral groups, comprising the remaining 21, often include reflection-generated subgroups as a special case.[39]Reflection Groups
Reflection groups, in the context of point groups, are finite Coxeter groups generated by reflections across hyperplanes passing through the origin in Euclidean space. These groups act as subgroups of the orthogonal group and capture the symmetries produced by such reflections, forming the basis for many finite symmetry groups in geometry and algebra. The generators are involutions corresponding to reflections, satisfying specific relations that define the group's structure.[40] Coxeter-Dynkin diagrams provide a graphical representation of these groups, where nodes correspond to the reflecting hyperplanes (or mirrors), and edges between nodes indicate the order of the product of the corresponding reflections, related to the dihedral angle between the hyperplanes. For example, the diagram for the series is a chain of nodes connected by single edges, representing the symmetry group of an -simplex, such as the alternating group or symmetric group in lower dimensions. These diagrams classify all irreducible finite reflection groups into series like , , , , , , , , and .[40][41] Weyl groups are a significant class of reflection groups, arising as the reflection subgroups generated by simple roots in the Lie groups associated with semisimple Lie algebras. In three dimensions, examples include the Weyl group of type , which is the full tetrahedral group of order 24, and type , corresponding to the octahedral or cubic group of order 48. The fundamental domains of these groups, known as chambers, are simplices bounded by the reflecting hyperplanes, tiling the space under the group action.[40][42] In three dimensions, the irreducible finite reflection groups are closely tied to the symmetries of the Platonic solids: the tetrahedral group () has order 24, the octahedral/cubic group () has order 48, and the icosahedral/dodecahedral group () has order 120. These groups encompass both rotations and reflections, distinguishing them from pure rotation subgroups. Reflection groups are fundamentally linked to root systems in Lie theory, where the roots define the reflecting hyperplanes, and the Weyl group acts by permuting the roots while preserving their lengths and angles, facilitating the classification of semisimple Lie algebras.[40][41][42]Applications
In Chemistry
In chemistry, point groups classify the symmetry of discrete molecules using Schoenflies notation, which denotes the principal rotation axis and additional symmetry elements such as mirror planes or inversion centers. For example, the water molecule (H₂O) belongs to the C_{2v} point group, characterized by a twofold rotation axis (C_2) bisecting the H-O-H angle and two vertical mirror planes (σ_v), one containing the molecular plane and the other perpendicular to it. In contrast, buckminsterfullerene (C₆₀) exhibits I_h symmetry, the full icosahedral point group with 120 symmetry operations, including fivefold (C_5), threefold (C_3), and twofold (C_2) rotation axes, inversion, and mirror planes, reflecting its highly spherical structure./03:_An_Introduction_to_Group_Theory/3.03:_Determining_the_Point_Group_for_a_Molecule-_the_Schoenflies_notation)[43] This notation facilitates the analysis of molecular geometries and electronic structures without relying on crystallographic constraints. Point group symmetry underpins applications in spectroscopy by dictating selection rules for vibrational transitions in infrared (IR) and Raman spectra, based on the transformation properties of normal modes under group operations. A vibrational mode is IR active if its irreducible representation matches one of the translational coordinates (x, y, or z), indicating a change in dipole moment, while it is Raman active if it matches quadratic forms like x² + y² or xy, corresponding to polarizability changes. For octahedral molecules such as sulfur hexafluoride (SF₆, O_h point group), the symmetric stretching mode (A_{1g}) preserves the dipole moment and is thus IR inactive but Raman active, appearing at approximately 775 cm⁻¹. Conversely, in water (C_{2v}), the asymmetric O-H stretch (B_2) changes the dipole moment along the molecular plane and is both IR and Raman active, contributing to its characteristic absorption bands around 3750 cm⁻¹./CHEM_431_Readings/07:_Vibrational_Spectroscopy/7.02:_Identifying_all_IR-_and_Raman-active_vibrational_modes_in_a_molecule)[44]/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes/Symmetry_Adapted_Linear_Combinations) These rules enable the assignment of spectral features and the exclusion of inactive modes, streamlining experimental interpretation. In reactivity and stereochemistry, point groups distinguish chiral molecules—those without improper rotation axes (S_n)—which belong exclusively to pure rotational groups like C_n, D_n, T, O, or I and exhibit optical activity by rotating the plane of polarized light due to their non-superimposable mirror images. Such chirality influences enantioselective reactions and biological interactions, as seen in pharmaceuticals where one enantiomer may be therapeutic while the other is inactive or harmful. During SN2 nucleophilic substitutions, point group symmetry evolves: for methyl halides (e.g., CH₃Cl, C_{3v}), the backside attack forms a trigonal bipyramidal transition state with D_{3h} symmetry, where the nucleophile, carbon, and leaving group align collinearly, resulting in inversion of configuration for chiral substrates and altering the overall molecular handedness./03:_Introduction_to_Molecular_Symmetry/3.08:_Chiral_Molecules)[45] Character tables summarize the irreducible representations of a point group, enabling the construction of symmetry-adapted linear combinations (SALCs) of atomic orbitals to form molecular orbitals that respect the group's symmetry. For ammonia (NH₃, C_{3v} point group), the character table is:| C_{3v} | E | 2C_3 | 3σ_v | Linear functions, rotations | Quadratic functions | |
|---|---|---|---|---|---|---|
| A_1 | 1 | 1 | 1 | z | (x² + y², z²) | |
| A_2 | 1 | 1 | -1 | R_z | ||
| E | 2 | -1 | 0 | (x, y), (R_x, R_y) | (xy, x² - y²) |
In Physics and Crystallography
In physics and crystallography, point groups describe the rotational and reflection symmetries compatible with the periodic lattice of crystals, limiting the possible symmetries to 32 distinct classes in three dimensions that align with the 14 Bravais lattices. These crystallographic point groups arise from the requirement that symmetry operations must preserve the translational periodicity of the lattice, excluding infinite rotations or translations while allowing combinations of rotations, reflections, and inversions. The Hermann-Mauguin notation is commonly used to denote these groups, where symbols like 4/mmm indicate a tetragonal symmetry with a fourfold rotation axis, perpendicular mirror planes, and additional mirrors parallel to the axis.[5][48] These point groups classify crystal forms and habits, linking symmetry to observable morphologies; for instance, the cubic point group (Hermann-Mauguin: ) governs the isotropic octahedral habit of sodium chloride (NaCl) crystals, where equal-length edges and faces reflect the high symmetry. Physical properties such as piezoelectricity are directly tied to point group symmetry: it is absent in the 11 centrosymmetric groups due to the inversion center canceling electric dipole moments under strain, while the 21 non-centrosymmetric groups (excluding class 432) exhibit this effect, enabling applications in sensors and actuators.[49][50][51] In solid-state physics, point groups underpin the analysis of electronic and vibrational properties, dictating band structure degeneracies, phonon mode classifications, and selection rules for optical transitions; for example, symmetry-imposed degeneracies at high-symmetry k-points simplify computations of material responses. Space groups, which fully describe crystal symmetries, number 230 in three dimensions and form as semidirect products of these point groups with the lattice translation subgroup, incorporating glide planes and screw axes for non-symmorphic cases.[52]/03%3A_Space_Groups/3.04%3A_Group_Properties) Recent advancements leverage point groups in topological materials, where symmetry indicators—computed from band representations under the 230 space groups—diagnose insulating states with protected surface modes, as in bismuth-based compounds. In two-dimensional materials like graphene, the point group enforces symmetry in phonon dispersions, leading to degenerate acoustic and optical branches at the K-point and linear crossings that influence thermal and electronic transport.[53][54][55]Lists of Point Groups
In One Dimension
In one dimension, point groups are finite subgroups of the orthogonal group O(1), which consists of transformations that preserve distances and fix the origin on a line. These groups capture the possible symmetries of objects like line segments, limited to the identity operation and reflection through the origin, as no proper rotations other than the identity exist in this setting.[56] The trivial point group, denoted , contains only the identity element (or ), which maps every point to itself. This group has order 1 and is isomorphic to the trivial group . Geometrically, represents a line segment with no non-trivial symmetries, such as an asymmetric interval lacking any mirror plane.[56] The second point group in one dimension is (equivalently denoted ), which includes the identity and a single reflection through the origin, mapping to . This group has order 2 and is isomorphic to the cyclic group . Geometrically, it describes the symmetries of a line segment invariant under mirroring, such as a symmetric interval centered at the origin. In one dimension, no rotations are possible beyond 180°, as such a rotation is equivalent to the reflection operation.[56]In Two Dimensions
In two dimensions, the crystallographic point groups are the finite symmetry groups compatible with periodic lattices, restricted by the crystallographic restriction theorem to rotation orders of 1, 2, 3, 4, or 6. This yields exactly ten such groups, divided into two types: cyclic groups consisting only of proper rotations and dihedral groups that incorporate both rotations and reflections across lines through the fixed point.[57][58] The cyclic groups, denoted in Schoenflies notation, are generated by rotations by multiples of about the origin, resulting in group order . The allowed values are (trivial identity, order 1), (180° rotation, order 2), (120° and 240° rotations, order 3), (90°, 180°, and 270° rotations, order 4), and (60°, 120°, 180°, 240°, and 300° rotations, order 6). In international (Hermann-Mauguin) notation, these correspond to 1, 2, 3, 4, and 6, respectively. For instance, captures the pure rotational symmetries of a square, excluding reflections.[57][59] The dihedral groups, denoted in Schoenflies notation, extend the rotations by including reflections, yielding group order . For the allowed , these are (single reflection, order 2), (180° rotation plus two perpendicular reflections, order 4), (order 6), (order 8), and (order 12). Their international notations are m (for ), 2mm (for ), 3m, 4mm, and 6mm, respectively, where "m" indicates mirror lines—vertical along the principal axis and others at angles of . Geometric examples include the full symmetry of an equilateral triangle under (3m) and a square under (4mm), while a regular hexagon exhibits (6mm).[57][59] The ten groups are summarized in the following table:| Schoenflies | International | Type | Order | Description |
|---|---|---|---|---|
| 1 | Cyclic | 1 | Trivial (identity only) | |
| 2 | Cyclic | 2 | 180° rotation | |
| 3 | Cyclic | 3 | 120° and 240° rotations | |
| 4 | Cyclic | 4 | 90°, 180°, 270° rotations | |
| 6 | Cyclic | 6 | 60°, 120°, 180°, 240°, 300° rotations | |
| m | Dihedral | 2 | Single reflection () | |
| 2mm | Dihedral | 4 | 180° rotation + 2 perpendicular mirrors | |
| 3m | Dihedral | 6 | rotations + 3 mirrors | |
| 4mm | Dihedral | 8 | rotations + 4 mirrors | |
| 6mm | Dihedral | 12 | rotations + 6 mirrors |
In Three Dimensions
In three dimensions, finite point groups classify the symmetries of bounded objects under rotations, reflections, inversions, and their combinations, all fixing a central point. These groups form the finite subgroups of the orthogonal group O(3), and their classification traces back to the work of Felix Klein, who identified the possible finite rotation groups as cyclic, dihedral, tetrahedral, octahedral, and icosahedral. While infinitely many such groups exist in principle—arising from arbitrary rotation orders—the crystallographic restriction theorem limits those compatible with periodic crystal lattices to rotations of order 1, 2, 3, 4, or 6, along with associated reflections and inversions, yielding exactly 32 distinct point groups. These 32 groups, known as the crystallographic point groups, are essential for describing the external symmetry of crystals across the seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Two primary notations describe these groups: the Schoenflies notation, commonly used in molecular spectroscopy and chemistry to emphasize rotational axes and mirror planes, and the Hermann-Mauguin notation (also called international notation), preferred in crystallography for its focus on symmetry elements like rotation axes and mirrors. For example, the group with a 4-fold rotation axis and perpendicular mirrors is denoted D_{4h} in Schoenflies and 4/mmm in Hermann-Mauguin. The groups are organized into families based on principal symmetry elements: axial (cyclic and dihedral) and polyhedral (tetrahedral, octahedral). The axial groups form the majority and are built around a principal n-fold rotation axis, with n limited to 1–6 by the crystallographic restriction. The cyclic groups C_n consist solely of rotations about this axis (order n; e.g., C_1 has order 1 with only the identity, C_2 order 2 with a 180° rotation). Adding a horizontal mirror plane perpendicular to the axis yields C_{nh} (order 2n; e.g., C_{2h} order 4). Vertical mirror planes containing the axis produce C_{nv} (order 2n; e.g., C_{2v} order 4). Improper rotation groups S_{2n} (or C_{ni} for odd n) involve roto-inversions (order 2n; e.g., S_4 order 4). Dihedral groups D_n include the C_n rotations plus n perpendicular 2-fold axes (order 2n; e.g., D_2 order 4, D_3 order 6, up to D_6 order 12). With a horizontal mirror, they become D_{nh} (order 4n; e.g., D_{4h} order 16, D_{6h} order 24); with alternating vertical dihedral planes, D_{nd} (order 4n; e.g., D_{2d} order 8, D_{3d} order 12). The polyhedral groups correspond to the symmetries of regular polyhedra and appear only in the cubic crystal system. The tetrahedral groups are T (pure rotations: four 3-fold and three 2-fold axes, order 12), T_d (with mirrors, order 24), and T_h (with inversion, order 24). The octahedral groups are O (pure rotations: four 3-fold, three 4-fold, and six 2-fold axes, order 24) and O_h (full symmetry including mirrors and inversion, order 48). These account for the high-symmetry cubic classes like 23 (T), m\bar{3} (T_h), 432 (O), \bar{4}3m (T_d), and m\bar{3}m (O_h). Beyond the 32 crystallographic groups, non-crystallographic finite point groups exist, notably the icosahedral family, which violates the restriction due to 5-fold rotations incompatible with lattice periodicity. The icosahedral rotation group I has order 60 (twelve 5-fold, twenty 3-fold, and fifteen 2-fold axes), while the full icosahedral group I_h, including mirrors, has order 120; these describe the symmetries of icosahedra and dodecahedra but do not occur in crystalline solids.| Family | Schoenflies | Hermann-Mauguin Example | Order | Principal Elements |
|---|---|---|---|---|
| Cyclic | C_n (n=1–6) | n (e.g., 2, 3, 4, 6) | n | n-fold rotation |
| Cyclic with horizontal mirror | C_{nh} | n/m (e.g., 2/m, 4/m, 6/m) | 2n | n-fold rotation, horizontal mirror |
| Cyclic with vertical mirrors | C_{nv} | nm (e.g., m, mm2, 4mm, 6mm) | 2n | n-fold rotation, n vertical mirrors |
| Improper rotation | S_{2n} or C_{ni} | \bar{n} (e.g., \bar{1}, \bar{4}, \bar{6}) | 2n | n-fold roto-inversion |
| Dihedral | D_n (n=2–6) | nn2 (e.g., 222, 32, 422, 622) | 2n | n-fold + n 2-fold axes |
| Dihedral with horizontal mirror | D_{nh} | n/mmm (e.g., mmm, 4/mmm, 6/mmm) | 4n | As D_n + horizontal mirror + vertical mirrors |
| Dihedral with dihedral planes | D_{nd} | \bar{n}2m (e.g., \bar{4}2m, \bar{6}2m, \bar{3}m) | 4n | As D_n + 2n dihedral mirrors |
| Tetrahedral | T, T_d, T_h | 23, \bar{4}3m, m\bar{3} | 12, 24, 24 | 3-fold/2-fold axes, mirrors/inversion |
| Octahedral | O, O_h | 432, m\bar{3}m | 24, 48 | 4-fold/3-fold/2-fold axes, mirrors/inversion |
| Icosahedral (non-cryst.) | I, I_h | - | 60, 120 | 5-fold/3-fold/2-fold axes, mirrors |