Point group

 ( Crystallography ) 

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 (isometry 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 are described by , 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.

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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 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).

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List of point groups

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

identity

reflection group

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Two dimensions Point groups in two dimensions, sometimes called rosette groups.

They come in two infinite families:

1. C _n of n-fold rotation groups
2. 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.

C n	n	n•	n+	n	cyclic: n-fold rotations; abstract group Z n, the group of integers under addition modulo n
D n	nm	* n•	n	2 n	dihedral: cyclic with reflections; abstract group Dih n, 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.

1	digon
2	rectangle
3	equilateral triangle
4	square
5	regular pentagon
6	regular hexagon
n	regular polygon
8
12
16
20
24
4 n

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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_{2 n}, C _nh, C _nv, D _n, D _nd, D _nh
• : 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.

+ Even/odd colored fundamental domains of the reflective groups

{ class="wikitable"
1		1	C1	+	1
		×1	Ci = S2	2+,2+	2
= m	1	*1	Cs = C1v = C1h		2
2 3 4 5 6 n		22 33 44 55 66 nn	C2 C3 C4 C5 C6 C n	2+ 3+ 4+ 5+ 6+ n+	2 3 4 5 6 n
mm2 3m 4mm 5m 6mm nmm nm	2 3 4 5 6 n	*22 *33 *44 *55 *66 * nn	C2v C3v C4v C5v C6v C nv	2 3 4 5 6 n	4 6 8 10 12 2 n
2/m 4/m 6/m n/m	2 2 2 2 2 2	2* 3* 4* 5* 6* n*	C2h C3h C4h C5h C6h C nh	2,2+ 2,3+ 2,4+ 2,5+ 2,6+ 2,n+	4 6 8 10 12 2 n
		2× 3× 4× 5× 6× n×	S4 S6 S8 S10 S12 S2 n	2+,4+ 2+,6+ 2+,8+ 2+,10+ 2+,12+ 2+,2 n+	4 6 8 10 12 2 n

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222 32 422 52 622 n22 n2		222 223 224 225 226 22 n	D2 D3 D4 D5 D6 D n	2,2+ 2,3+ 2,4+ 2,5+ 2,6+ 2, n+	4 6 8 10 12 2 n
mmm m2 4/mmm m2 6/mmm n/mmm m2	2 2 3 2 4 2 5 2 6 2 n 2	*222 *223 *224 *225 *226 *22 n	D2h D3h D4h D5h D6h D nh	2,2 2,3 2,4 2,5 2,6 2, n	8 12 16 20 24 4 n
2m m 2m m 2m 2m m	4 6 8 10 12 n	2*2 2*3 2*4 2*5 2*6 2* n	D2d D3d D4d D5d D6d D nd	2+,4 2+,6 2+,8 2+,10 2+,12 2+,2 n	8 12 16 20 24 4 n
23		332	T	3,3+	12
m	4	3*2	Th	3+,4	24
3m	3 3	*332	Td	3,3	24
432		432	O	3,4+	24
mm	4 3	*432	Oh	3,4	48
532		532	I	3,5+	60
m	5 3	*532	Ih	3,5	120

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

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Reflection groups 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.

3,3			tetrahedron
= 4,3			stellated octahedron
		cube, octahedron
		icosahedron, dodecahedron
3,2			triangular prism
[3,2]			hexagonal prism
4,2			square prism
[4,2] = 8,2			octagonal prism
		pentagonal prism
		hexagonal prism
n,2			n-gonal prism
[ n,2]			8 n
2,2			8	cuboid
[2,2] = 4,2			16
3[2,2] = 4,3			48
		6	hosohedron
		8
		10
		12
1, n				2 n
1,[ n] = 1,2 n				4 n
1,2				4
1,[2]				8
		2

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Four dimensions The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith, 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 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 with its order doubled to 240.

5-cell
5-cell dual compound
16-cell / tesseract
192	demihypercube
384
1152
24-cell
24-cell dual compound
120-cell / 600-cell
tetrahedral prism
96	octahedral prism
96
icosahedral prism
36	duoprism
48
60
72
64
128
80
96
100
120
144
4 pq
8 pq
16 pq
8 p2
32 p2
24
32
40
48
8 p
16 p
16 p
32 p
16	4-orthotope
32
64
96
384

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Five dimensions

The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as . 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.

		5-simplex
		5-simplex dual compound
		5-cube, 5-orthoplex
		5-demicube
	=|3840
		5-cell prism
		480
		tesseract prism
		2304	24-cell prism
		4608
		600-cell or 120-cell prism
		demitesseract prism
		144	duoprism
		288
		192
		240
		288
		48 p
		288
		384
		480
		576
		96 p
		720
		960
		1200
		1440
		240 p
		96
		192
		480
		72	duoprism prism
		96
		120
		144
		128
		160
		192
		200
		240
		288
		8 pq
		48
		64
		80
		96
		16 p
		32	5-orthotope
	=|64
	=|128
	=|192
	=|384
	=|768
	=|3840

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Six dimensions

The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as . 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.

6-simplex
6-simplex dual compound
6-cube, 6-orthoplex
6-demicube
122, 221
5-simplex prism
5-cube prism
5-demicube prism
240 p	duoprism
768 p
2304 p
28800 p
384 p
480
1536
4608
57600
768
576
1152
2880
2304
5760
14400
96 p	duoprism prism
192 p
480 p
192
384
960
8pqr''	triaprism
16 pq
32 p
6-orthotope

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Seven dimensions

The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as . 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.

7-simplex
7-simplex dual compound
7-cube, 7-orthoplex
7-demicube
321, 231, 132
10080 (2×7!)
92160 (27×6!)
46080 (26×6!)
103680 (144×6!)
1440 p
7680 p
3840 p
2880
15360
7680
2880
5760
14400
9216
18432
46080
345600
691200
1728000
27648
55296
138240
4608
9216
23040
480 p
1536 p
768 p
4608 p
57600 p
960
3072
9216
115200
1536
1152
2304
5760
4608
11520
28800
96 pq
192 pq
480 pq
192 p
384 p
960 p
384
768
1920
16 pqr
32 pq
64 p
128

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Eight dimensions

The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as . 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.

8-simplex
8-simplex dual compound
8-cube, 8-orthoplex
8-demicube
421, 241, 142
7-simplex prism
7-cube prism
7-demicube prism
321 prism, 231 prism, 142 prism
10080 p	duoprism
92160 p
46080 p
103680 p
20160
184320
92160
207360
17280
92160
46080
34560
184320
92160
	duoprism prisms
	triaprism
16 pqrs
32 pqr
64 pq
128 p
256

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See also

• Point groups in two dimensions
• Point groups in three dimensions
• Point groups in four dimensions
• Crystallography
• Crystallographic point group
• Molecular symmetry
• Space group
• X-ray diffraction
• Bravais lattice
• Infrared spectroscopy of metal carbonyls

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Further reading

• (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, Math.

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External links

• Web-based point group tutorial (needs Java and Flash)
• Subgroup enumeration (needs Java)
• The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)

Categories: Crystallography, Euclidean Symmetries, Group Theory, Molecular Geometry

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