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10.1.5: Inversion Centers

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    Figure 10.15: Inversion centers

    We just looked at reflection and rotation, two kinds of symmetry. Inversion, a third type of symmetry, is symmetry with respect to a point. We call the point the inversion center and often designate it with the lowercase letter i. As with mirror planes, inversion relates identical faces on a crystal. But, while mirror planes “reflect” faces and change their “handedness,” inversion centers invert them.

    Inversion produces faces related in the same way that a lens may yield an upside-down and backward image. So, the two cats in Figure 10.15a are related by inversion – they are upside down and backward images of each other. In two dimensions, inversion centers give the same results as 2-fold axes of symmetry. For example, in Figure 10.15a, a cat has been inverted. The cat could have been rotated 180o and the result would have been the same.

    In three dimensions, inversion symmetry is different from rotation. The crystal in Figure 10.15b has an inversion center and no other symmetry. It does not have any rotational symmetry or reflection (mirror plane) symmetry. It contains four different shaped faces (a, b, c, and e); each of them has an inverted matching face on the back and left sides of the crystal. That is all they symmetry that is present.

    In contrast, the crystal in Figure 10.15c contains an inversion center, but other symmetry is present, too. 90o rotation relates the four a-faces. Inversion and rotation relate the eight e-faces. Additionally, many mirror planes are present.

    We use small letters on crystal faces, such as those seen in these two crystal drawings, to distinguish faces of different shapes and to show those related by symmetry. The a-faces in Figure 10.15c, for example, are the same shape and are related by a 4-fold rotation axis. Conventions, not worth discussing here, dictate which letters we use for different kinds of faces.

    For a video discussion of inversion centers, click on the link below:

    Video 10-3: Inversion centers (6 minutes)

    This page titled 10.1.5: Inversion Centers is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Dexter Perkins via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.