Here one of more than 30 examples of two dimensional objects periodically repeated in space according to the rectangular centered group c2mm can be observed every time this page is uploaded. Each straight line segment in red represents a mirror plane, but binary axes and glide planes given in the reference below are not represented in this page. It means that those symmetry elements are still operating and can be detected, as suggested by exercises 1-4. Interaction succeeds a click on any gray rectangle.
Letter "M" is the label for multiplicity, letters "P.s." mean point symmetry and "S.o." for symmetry operations.
When the mouse pointer is moved close to the rectangular cell, the pointer's coordinates are displayed in the status bar displayed in the browser.
The periodic structure of the plane group c2mm will exhibit at least one object characterized by any of the point groups: 1, m, 2, and 2mm. When at least one object exists at position 1 it will be classified as in a general position. Any object in any of the positions m, 2, and 2mm will be classified as in a special position.
Differently from objects in group p2mm, here in group c2mm all the environment on the origin will be repeated on the center of the rectangular cell. For this reason objects in comparable positions in cell p2mm will have here twice the multiplicity of the former.

Exercises

1) Write the coordinates of the center of a disk  in general position in a copy of a selected example, close to the cell origin and write the coordinates of the next disk center obtained after:

1a] a reflection by a glide plane with Miller indices (4 0).

1b] a reflection by a mirror plane with Miller indices (0 2).

1c] a rotation about a binary axis at coordinates x = 0.25 and y = 0.25.

2) Apply the operations given in 1a], 1b], and 1c] for the coordinates of a disk in special position in a copy of one of the examples of this page. Write the significant differences and similarities observed.

5) Why it does not exist any location for an object with multiplicity M = 1 in the group c2mm ?

Reference

International Tables for Crystallography (2005). Vol. A, edited by T. Hahn, Dordrecht: Springer.

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