A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. It is a subgroup of the symmetry group of 3D space itself which leaves the origin fixed, which is orthogonal group O(3). In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... The symmetry group of an object (e. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

All isometries of a bounded 3D object have one or more common fixed points. We choose the origin at one of them. The symmetry group may or may not be a discrete group; in this article we use, as is often done, the term "point group" in the sense of "discrete point group". Infinite discrete groups as in the case of translational symmetry and glide reflectional symmetry do not apply for bounded objects. In mathematics, a discrete group is a group G equipped with the discrete topology. ...

The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group iff the object is chiral. In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...

When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H₁, H₂ of a group G are conjugate, if there exists g ∈ G such that H₁=g^-1H₂g ). For example: In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

• two 3D objects have mirror symmetry, but with respect to a different mirror plane
• two 3D objects have 3-fold rotational symmetry, but with respect to a different axis

Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others). The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ...

The seven infinite series

The infinite series have an index n, which can be any integer; in each series, the nth symmetry group contains a rotation by an angle 360°/n about an axis. They are, using Schönflies notation, and explained below: Arthur Moritz Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ...

• C_n of order n (abstract group C_n)
• C_nh of order 2n (for odd n abstract group C_2n, for even n abstract group C_n × C₂)
• C_nv of order 2n (abstract group D_n)
• D_n of order 2n (abstract group D_n)
• S_2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C_2n)
• D_nh of order 4n (abstract group D_n × C₂)
• D_nd (or D_nv) of order 4n (abstract group D_2n)

The terms horizontal (h) and vertical (v) are used with respect to a vertical axis of rotation. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

The situation in 3-D is more complicated than in 2D, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group C₁, and then there are three groups of order 2, called C_s (or C_1h), C_i, and C₂. These have the single symmetry operation of reflection in a plane, in a point, and in a line (equivalent to a rotation of 180°), respectively.

The last of these is the first of the uniaxial groups (cyclic groups) C_n of order n (also applicable in 2D), which are generated by a single rotation of angle 360°/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group C_nh of order 2n, or a set of n mirror planes containing the axis, giving the group C_nv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... Geometric shape created by connecting a polygonal base to an apex A pyramid is a geometric shape formed by connecting a polygonal base and a point, called the apex, by triangular faces. ...

If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4n is called D_nh. Its subgroup of rotations is the dihedral group D_n of order 2n which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D D_n includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections. In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ...

There is one more group in this family, called D_nd (or D_nv), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane it has an isometry which is the combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. D_nh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. D_nd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. D_n is the symmetry group of a partially rotated prism. In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ...

S_n is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For n odd this is equal to the group generated by the two separately, C_nh of order 2n, and therefore the notation S_n is not needed; however, for n even it is distinct, and of order n. Like D_nd it contains a number of improper rotations without containing the corresponding rotations. In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...

All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:

• C_1h and C_1v: group of order 2 with a single reflection (C_s)
• C₂ and D₁: group of order 2 with a single 180° rotation
• D_1h and C_2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
• D_1d and C_2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane

S₂ is the group of order 2 with a single inversion (C_i)

"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. S_2n is algebraically isomorphic with C_2n.

The seven remaining point groups

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using C_n to denote an axis of rotation through 360°/n and S_n to denote an axis of improper rotation through the same, the groups are: A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ...

• T (tetrahedral). There are four C₃ axes, directed through the corners of a cube, and three C₂ axes, directed through the centers of the cube's faces. There are no other symmetry operations, giving the group an order of 12. This group is isomorphic to A₄, the alternating group on 4 letters, and is the rotation group for a regular tetrahedron.

• T_d. This group has the same rotation axes as T, but with six mirror planes, each containing a single C₂ axis and four C₃ axes. The C₂ axes are now actually S₄ axes. This group has order 24, and is the symmetry group for a regular tetrahedron. T_d is isomorphic to S₄, the symmetric group on 4 letters.

• T_h. This group has the same rotation axes as T, but with mirror planes, each containing two C₂ axes and no C₃ axes. The C₃ axes become S₆ axes, and a center of inversion appears. Again, group has order 24. T_h is isomorphic to A₄ × C₂.

• O (octahedral). This group is similar to T, but the C₂ axes are now C₄ axes, and a new set of 12 C₂ axes appear, directed towards the edges of the original cube. This group of order 24 is also isomorphic to S₄, and is the rotation group of the cube and octahedron.

• O_h. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group has order 48, is isomorphic to S₄ × C₂, and is the symmetry group of the cube and octahedron.

• I (icosahedral) of order 60 is the rotation group of the icosahedron and the dodecahedron. It is a normal subgroup of index 2 in the full group of symmetries I_h. The group I is isomorphic to A₅, the alternating group on 5 letters. The group contains 10 versions of D₃ and 6 versions of D₅ (rotational symmetries like prisms and antiprisms).

• I_h of order 120 is the symmetry group of the icosahedron and the dodecahedron. The group I_h is isomorphic to to A₅ × C₂. The group contains 10 versions of D_3d and 6 versions of D_5d (symmetries like antiprisms).

This completes the classification (up to conjugacy) of the finite subgroups of O(3). Cube may denote one of the following. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces. ... A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ... 5 (five) is the natural number following 4 and preceding 6. ... An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces. ... A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

Rotation groups

The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups C_n (the rotation group of a regular pyramid), the dihedral groups D_n (the rotation group of a regular prism, or regular bipyramid), and the rotation groups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron. Geometric shape created by connecting a polygonal base to an apex A pyramid is a geometric shape formed by connecting a polygonal base and a point, called the apex, by triangular faces. ... See: Prism (geometry) Prism (optics) Prism (band) PRISM is an abbreviation for Probabilistic Symbolic Model Checker PRISM was an aborted RISC processor effort at DEC, see DEC PRISM This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Cube may denote one of the following. ... An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces. ... A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ...

In particular, the dihedral groups D₃, D₄ etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.

• An object with symmetry group C_n, C_nh, C_nv or S_2n has rotation group C_n.
• An object with symmetry group D_n, D_nh, or D_nd has rotation group D_n.
• An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: T, O or I.

The rotation group of an object is equal to its full symmetry group iff the object is chiral. In other words, the chiral objects are those with their symmetry group in the list of rotation groups. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...

Center of symmetry

For C₁ there is no symmetry, hence no center of symmetry.

For C_s there is only a center plane.

For C_n (n>1) and C_nv (n>1) there is only a center axis.

In all other cases there is a specific center of symmetry. In the case of a physical body, if for the symmetry not only the shape but also the density is taken into account, this is the center of mass. The center of mass or center of inertia of an object is a point at which the objects mass can be assumed, for many purposes, to be concentrated. ...

The groups arranged by abstract group type

Below the groups explained above are arranged by abstract group type.

The smallest abstract groups which are not any symmetry group in 3D, are the quaternion group (of order 8), the dicyclic group Dic₃ (of order 12), and 10 of the 14 groups of order 16. Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... In group theory, a dicyclic group is a member of a class of groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic). ...

Cyclic symmetry groups in 3D

The symmetry group for n-fold rotational symmetry is algebraically of the type called cyclic group, denoted by C_n, and itself also denoted by C_n. However, there are two more infinite series of cyclic symmetry groups: The symmetry group of an object (e. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ...

• For even order 2n there is the group S_2n (Schoenflies notation) generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For S₂ the notation C_i is used; it is generated by inversion.
• For any order 2n where n is odd, we have C_nh; it has an n-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360°/n about the axis, combined with the reflection. For C_1h the notation C_s is used; it is generated by reflection in a plane.

Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the crystallographic restriction applies: In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...

• order 1: C₁
• order 2: C₂, C_s, C_i
• order 3: C₃
• order 4: C₄, S₄
• order 5: C₅
• order 6: C₆, S₆, C_3h
• order 7: C₇
• order 8: C₈, S₈
• order 9: C₉
• order 10: C₁₀, S₁₀, C_5h

etc.

Dihedral symmetry groups in 3D

In 2D dihedral group D_n includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside. In 3D the two operations are distinguished: the symmetry group denoted by D_n contains "flipping over", not reflections. In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ...

In addition to this infinite series of symmetry groups, there are two more which are algebraically of type D_n:

• C_nv of order 2n, the symmetry group of a regular, n-sided prism with one base colored differently, and also of a regular n-sided pyramid
• D_nd of order 4n, the symmetry group of a regular, n-sided antiprism

Thus we have, with bolding of the 11 crystallographic point groups: See: Prism (geometry) Prism (optics) Prism (band) PRISM is an abbreviation for Probabilistic Symbolic Model Checker PRISM was an aborted RISC processor effort at DEC, see DEC PRISM This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Geometric shape created by connecting a polygonal base to an apex A pyramid is a geometric shape formed by connecting a polygonal base and a point, called the apex, by triangular faces. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

• order 4: D₂, C_2v, D_1d
• order 6: D₃, C_3v
• order 8: D₄, C_4v, D_2d
• order 10: D₅, C_5v
• order 12: D₆, C_6v, D_3d
• order 14: D₇, C_7v
• order 16: D₈, C_8v, D_4d

etc.

Other

C_2n,h of order 4n is of abstract group type C_2n × C₂. For n=1 we get D₂, already covered above, so n≥2.

Thus we have, with bolding of the 2 cyclic crystallographic point groups:

• order 8: C_4h
• order 12: C_6h
• order 16: C_8h
• order 20: C_10h

etc.

D_nh of order 4n is of abstract group type D_n × C₂. For n=1 we get D₂, already covered above, so n≥2.

Thus we have, with bolding of the 4 dihedral crystallographic point groups:

• order 8: D_2h
• order 12: D_3h
• order 16: D_4h
• order 20: D_5h
• order 24: D_6h
• order 28: D_7h
• order 32: D_8h

etc.

The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):

• order 12: of type A₄ (alternating group): T
• order 24:
  • of type S₄ (symmetric group, not to be confused with the symmetry group with this notation): T_d, O
  • of type A₄ × C₂: T_h .
• order 48, of type S₄ × C₂: O_h
• order 60, of type A₅: I
• order 120, of type A₅ × C₂: I_h

In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

Related topics

• symmetry
• Euclidean plane isometry
• group action
• point group
• crystallographic group
  • crystallographic point group
  • space group

Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. ... In mathematics, groups are often used to describe symmetries of objects. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... A crystallographic group is a topologically discrete subgroup of the group of isometries of some geometric space (typically, not necessarily a Euclidean space) with a compact fundamental domain. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

External links

• Overview of the 32 crystallographic point groups - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups