Symmetry operation

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Opération de symétrie (Fr). Symmetrieoperation (Ge). Operación de simetría (Sp). Operazione di simmetria (It). 対称操作 (Ja).

A symmetry operation is an isometry, i.e. a transformation under which two objects, or two configurations or an object, are brought to coincide. A symmetry operation is a Euclidean mapping: to each point of the first configuration there corresponds a point of the second configuration. The distances between two points are kept by the transformation, as are the angles.

The two configurations/objects can be either congruent or enantiomorphous. Correspondingly, the symmetry operations are classed into two kinds:

  • symmetry operations of first kind: they relate congruent objects and consist of translations, rotations and screw rotations;
  • symmetry operations of second kind: they relate enantiomorphous objects and consist of inversion, reflections, rotoinversions, and glide reflections. There exists a 1:1 correspondence between rotoinversion and rotoreflections: the latter are more used in Schoenflies notation, whereas rotoinversions are preferred in Hermann-Mauguin notation.

A symmetry operation can be performed about a symmetry element. Exceptions are the identity and the pure translations, for which a symmetry element is not defined.