Two Dimensional Periodic Systems:

While most crystalline structures are three-dimensional, periodic systems can exist in any dimensions. In particular, two-dimensional crystals are interesting in systems such as graphene, and concepts such as rotations, inversions, and mirror planes are easily shown in a two-dimensional medium such as this web page.

In the following we present a brief tutorial about crystallography in two dimensions. A more formal version can be found in D. Hicks et al., Comp. Mat. Sci. 161, S1-1011 (2019), doi=10.1016/j.commatsci.2018.10.043.

This discussion is divided into four parts:

1. Lattices and Translational Symmetry
2. Lattices and Symmetries
3. Possible Two Dimensional Lattices
4. Two Dimensional Plane Groups