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Concept
Symmetry element
Summary
In chemistry and crystallography, a symmetry element is a point, line, or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane, an axis of rotation (either proper and improper), or a center of inversion. For an object such as a molecule or a crystal, a symmetry element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the object unchanged. The set containing these operations form one of the symmetry groups of the object. The elements of this symmetry group should not to be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.
The identity symmetry element is found in all objects and is denoted E. It corresponds to an operation of doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. An object having no symmetry elements other than E is called asymmetric. Such an object is necessarily chiral.
Reflection (mathematics)
Mirror planes are denoted by σ. In a molecule that also has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane and is labeled σ_v , while one perpendicular to the axis is called a horizontal mirror plane and is labeled σ_h . A vertical mirror plane that bisects the angle between two C2 axes is called a dihedral mirror plane, σ_d .
Rotational symmetry
Rotational symmetry, also known as radial symmetry, is represented by an axis about which the object rotates in its corresponding symmetry operation. A group of proper rotations is denoted as C_n, where the degrees of rotation that restore the object is 360/n (C_2= 180o rotation, C_3= 120o rotation, C_4= 90o rotation, C_5= 72o rotation).
Official source
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