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An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetries is called centrosymmetric. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their physical ...
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. [1] In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have inversion symmetry. [2] Point reflection is a similar term used in geometry.
Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor . This symmetry becomes Poincaré symmetry if we set = When = the second condition requires that is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique ...
If k = m, then such a transformation is known as a point reflection, or an inversion through a point. On the plane (m = 2), a point reflection is the same as a half-turn (180°) rotation; see below. Antipodal symmetry is an alternative name for a point reflection symmetry through the origin. [14]
Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center; Inversion of a circle is another circle; or it is a line if the original circle contains the center; Inversion of a parabola is a cardioid; Inversion of hyperbola is a lemniscate of Bernoulli
The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, can be achieved through the use of the appropriate permutation-inversion group (called the molecular symmetry group), as introduced by Longuet-Higgins.
In particular, a symmetry element can be a mirror plane, an axis of rotation (either proper and improper), or a center of inversion. [1] [2] [3] 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 ...