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. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect.
The following space groups have inversion symmetry: the triclinic space group 2, the monoclinic 1015, the orthorhombic 4774, the tetragonal 8388 and 123142, the trigonal 147, 148 and 162167, the hexagonal 175, 176 and 191194, the cubic 200206 and 221230.^{[3]}
Point groups lacking an inversion center (noncentrosymmetric) can be polar, chiral, both, or neither.
A polar point group is one whose symmetry operations leave more than one common point unmoved. A polar point group has no unique origin because each of those unmoved points can be chosen as one. One or more unique polar axes could be made through two such collinear unmoved points. Polar crystallographic point groups include 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm.
A chiral (often also called enantiomorphic) point group is one containing only proper (often called "pure") rotation symmetry. No inversion, reflection, rotoinversion or rotoreflection (i.e., improper rotation) symmetry exists in such point group. Chiral crystallographic point groups include 1, 2, 3, 4, 6, 222, 422, 622, 32, 23, and 432. Chiral molecules such as proteins crystallize in chiral point groups.
The remaining noncentrosymmetric crystallographic point groups 4, 42m, 6, 6m2, 43m are neither polar nor chiral.
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References
 ^ Tilley, Richard (2006). "4". Crystals and Crystal Structures. John Wiley. pp. 80–83. ISBN 9780470018217.
 ^ Fu, Liang; Kane, C. "Topological insulators with inversion symmetry". Physical Review B. 76 (4). arXiv:condmat/0611341. Bibcode:2007PhRvB..76d5302F. doi:10.1103/PhysRevB.76.045302.
 ^ Cockcroft, Jeremy Karl. "The 230 3Dimensional Space Groups". Birkbeck College, University of London. Retrieved 18 August 2014.