Explain the Crystallographic Theorem from a non-euclidian geometric perspective.?

Answer 1

Euclidian geometries permit only certain symmetries, while non-Euclidian geometries may permit others than cannot exist in Euclidian geometries. For example, regular tessellation of the plane by pentagons is impossible in Euclidian space, but permissible in some hyperbolic spaces. The wiki article below gives examples of both Euclidian and Non-Euclidian tessellations.