Symmetry element

A symmetry element is a point of reference about which symmetry operations can take place. In particular, symmetry elements can be identities, mirror planes, axes of rotation (both proper and improper), and centers of inversion.[1] A symmetry element corresponds to a symmetry operation that generates the same representation of an object.


The identity symmetry element is found in all objects and is denoted E.[2] It corresponds to an operation of doing nothing to the object.

Mirror planes

Mirror planes are denoted by σ. In particular, a vertical mirror plane is denoted σv.

Cyclic symmetry

Cyclic symmetry, also known as radial symmetry, is represented by an axis about which the object rotates in its corresponding symmetry operation. For proper axes of rotation, cyclic symmetry is denoted as Cn, where n is the order of rotation.[3] Improper rotation, also known as screw axis symmetry, is denoted by Sn. Improper rotation is the composition of a rotation axis and a mirror plane. In its symmetry operation, the object is rotated about the axis, then reflected across a mirror plane that is normal to the axis of rotation.


For inversion, denoted i, there must be a point in the center of an object that is the inversion center. In the inversion operation for 3D coordinates, the inversion center is the origin (0,0,0). When an object is inverted, the position vector of a point in an object, ⟨x,y,z⟩, is inverted to ⟨-x,-y,-z⟩.

See also


  1. Robert G. Mortimer (10 June 2005). Mathematics for Physical Chemistry. Academic Press. pp. 276–. ISBN 978-0-08-049288-9.
  2. Glazer, Michael; Burns, Gerald; Glazer, Alexander. Space groups for solid state scientists.
  3. Glazer, Michael; Burns, Gerald; Glazer, Alexander. Space groups for solid state scientists.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.