For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an mmm-fold symmetry is compatible with translational symmetry is the same as whether φ(m)≤2\varphi(m)\le2φ(m)≤2.
https://ulysseszh.github.io/math/2023/11/09/lattice-algebraic.html
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an mmm-fold symmetry is compatible with translational symmetry is the same as whether φ(m)≤2\varphi(m)\le2φ(m)≤2.