page_title: Point groups in three dimensions
section_title: The seven infinite series of axial groups
hierarchical_section_title: Point groups in three dimensions / The seven infinite series of axial groups
context_page_description: In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E of all isometries.
  Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.
  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 of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.
context_section_description: The infinite series of axial or prismatic groups have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry (see cyclic symmetries) and three with additional axes of 2-fold symmetry (see dihedral symmetry). They can be understood as point groups in two dimensions extended with an axial coordinate and reflections in it. They are related to the frieze groups; they can be interpreted as frieze-group patterns repeated n times around a cylinder.
  The following table lists several notations for point groups: Hermann–Mauguin notation (used in crystallography), Schönflies notation (used to describe molecular symmetry), orbifold notation, and Coxeter notation. The latter three are not only conveniently related to its properties, but also to the order of the group. It is a unified notation, also applicable for wallpaper groups and frieze groups. The crystallographic groups have n restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer.
  The series are:
  For odd n we have Z₂ₙ = Zₙ × Z₂ and Dih₂ₙ = Dihₙ × Z₂.
  The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).
  The simplest nontrivial ones have involutional symmetry (abstract group Z₂ or Dih₁):
  Cᵢ – inversion symmetry
  C₂ – 2-fold rotational symmetry
  Cₛ – reflection symmetry, also called bilateral symmetry.
  The second of these is the first of the uniaxial groups (cyclic groups) Cₙ 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 Cₙₕ of order 2n, or a set of n mirror planes containing the axis, giving the group Cₙᵥ, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cₙ or Dₙ is a  propeller.
  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 Dₙₕ.  Its subgroup of rotations is the dihedral group Dₙ 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 Dₙ 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.
  There is one more group in this family, called Dnd (or Dₙᵥ), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/n.  Dₙₕ is the symmetry group for a regular (n+2)-sided prisms and also for a regular (2n)-sided bipyramid. Dnd is the symmetry group for a regular (n+2)-sided antiprism, and also for a regular (2n)-sided trapezohedron. Dₙ is the symmetry group of a partially rotated prism.
  The groups D₂ and D₂ₕ are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular 2-fold axes. D₂ is a subgroup of all the polyhedral symmetries (see below), and D₂ₕ is a subgroup of the polyhedral groups Tₕ and Oₕ. D₂ can occur in homotetramers such as Concanavalin A, in tetrahedral coordination compounds with four identical chiral ligands, or in a molecule such as tetrakis(chlorofluoromethyl)methane if all the chlorofluoromethyl groups have the same chirality. The elements of D₂ are in 1-to-2 correspondence with the rotations given by the unit Lipschitz quaternions.
  The group Sₙ is generated by the combination of a reflection in the horizontal plane and a rotati
caption: Patterns on a cylindrical band illustrating the case n = 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group.
exif: {}