Algebraic representation of crystallographic symmetry

Considering conventional crystallographic symmetry, the elements representing rotations, i.e. ry, are restricted to 1, 0, or -1, and the elements representing translations, i.e. /, in Eq. 1.48, are restricted to 1/2, 1/3, 1/4, 1/6, 1/8 including their integer multiples, and 0. In this way, all possible transformations of atoms by symmetry operations are represented by the multiplication of matrices and vectors. 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

Symmetry Transformed First Second Third Fourth 

Symmetry element and orientation Transformed coordinates First symmetry operation Second symmetry operation Third symmetry operation Fourth symmetry operation 

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