The following is a simple general proof that, on the hypothesis that crystals have a homogeneous cellular structure, the only possible axes of symmetry are those with cyclic numbers 2, 3, 4, or 6.

According to this hypothesis, every crystal is entirely made up of minute space-partitions or ceils, which (apart from those of purely surface layers) have all the same form, size, internal configuration, orientation, and external relations. Accordingly, every point in one cell has an equivalent point exactly corresponding to it in every other cell.