Symmetry Operation

Symmetry operations include the improper rotation, inversion operation, mirror plane, and rotation. Together, these operations create 32 crystal classes corresponding to the 32 point groups.

The inversion operation takes


and is denoted i. When used in conjunction with a rotation, it becomes an improper rotation. An improper rotation by 360 degrees/n is denoted n^_ (or S_n). For periodic crystals, the crystallography restriction allows only the improper rotations 1^_, 2^_, 3^_, 4^_, and 6^_.

The mirror plane symmetry operation takes


etc., which is equivalent to 2^_. Invariance under reflection can be denoted nsigma_v or nsigma_h. The rotation symmetry operation for 360 degrees/n is denoted n (or C_n). For periodic crystals, crystallography restriction allows only 1, 2, 3, 4, and 6.

Symmetry operations can be indicated with symbols such as C_n, S_n, E, i, nsigma_v, and nsigma_h.

1. C_n indicates rotation about an n-fold symmetry axis.

2. S_n indicates improper rotation about an n-fold symmetry axis.

3. E (or I) indicates invariance under translation.

4. i indicates a center of symmetry under inversion.

5. nsigma_v indicates invariance under n vertical reflections.

6. nsigma_h indicates invariance under n horizontal reflections.

See also

Crystallography Restriction, Glide, Improper Rotation, Inversion Operation, Mirror Plane, Point Groups, Rotation, Symmetry

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Addington, S. "The Four Types of Symmetry in the Plane."

Referenced on Wolfram|Alpha

Symmetry Operation

Cite this as:

Weisstein, Eric W. "Symmetry Operation." From MathWorld--A Wolfram Web Resource.

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