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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.
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.
Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. In other words, a molecule has a center of symmetry when the points (x,y,z) and (−x,−y,−z) of the molecule always look identical.
In 2011, the Magnetic Space Groups data compiled from H.T. Stokes & B.J. Campbell's [4] and D. Litvin's [5] 's works general positions/symmetry operations and Wyckoff positions for different settings, along with systematic absence rules have also been incorporated into the server and a new shell has been dedicated to the related tools (MGENPOS, MWYCKPOS, MAGNEXT).
The Bauhinia blakeana flower on the Hong Kong region flag has C 5 symmetry; the star on each petal has D 5 symmetry. The Yin and Yang symbol has C 2 symmetry of geometry with inverted colors In geometry , a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space ) that have a fixed point in common.
Inversion symmetry plays a major role in the properties of materials, as also do other symmetry operations. [2] Some molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In many cases they can be considered as polyhedra, categorized by their coordination number and bond angles.
A center of symmetry of a figure is a point , such that is invariant under the mapping (,,) (,,), when is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry.
The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere.