For general lattices in Lie groups, the cells are simply called fundamental domains.

There are many ways to choose a fundamental domain.

The images of a chosen fundamental domain under the group action then tile the space.

The fundamental domain of a point group is a conic solid.

They fit anyway if the fundamental domain is bounded by reflection planes.

For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane.

For example, in the disdyakis triacontahedron one full face is a fundamental domain.

Also the surface in the fundamental domain may be composed of multiple faces.

In the case of additional symmetries a fundamental domain is smaller.

A set of such representatives forms a fundamental domain.