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  2. Inversion transformation - Wikipedia

    en.wikipedia.org/wiki/Inversion_transformation

    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 ...

  3. Point reflection - Wikipedia

    en.wikipedia.org/wiki/Point_reflection

    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.

  4. Symmetry operation - Wikipedia

    en.wikipedia.org/wiki/Symmetry_operation

    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.

  5. Inversive geometry - Wikipedia

    en.wikipedia.org/wiki/Inversive_geometry

    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

  6. Crystal structure - Wikipedia

    en.wikipedia.org/wiki/Crystal_structure

    Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point; Improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 ...

  7. Centrosymmetry - Wikipedia

    en.wikipedia.org/wiki/Centrosymmetry

    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.

  8. Symmetry (geometry) - Wikipedia

    en.wikipedia.org/wiki/Symmetry_(geometry)

    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]

  9. Parity (physics) - Wikipedia

    en.wikipedia.org/wiki/Parity_(physics)

    In physics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection ):