If a symmetry group applies for a pattern, then within each orbit the color is the same.
They appear on the right, again coloured according to their symmetry groups.
Since the projective plane has a large enough symmetry group, they are in no way special, though.
A similar fact holds true for the symmetry group of an icosahedron.
It is, loosely speaking, the symmetry group of the object.
The orders of the full symmetry groups are twice as much again (24, 48, and 120).
Various other 4-dimensional symmetry groups can be constructed from 2I.
However, the symmetry group of constant functions is G itself.
The symmetry group isn't clear, let alone the detailed dynamics.
Their symmetry group has two elements, the identity and the 180 rotation.