S_α is always closed, nonempty, and infinite

S_α is also eventually constant if for no other reason than that eventually you run out of points. call the resulting set T

pretty sure because T is perfect and this is a nice enough topological space T has cardinality continuum

i think this reasoning still works if we're talking tiles where there's only a finite number of ways to glue 2 tiles together sensibly
you don't quite get the group action but you do still get a compact topology on all the ways to build a tesselation around some base tile

and one point in this space always gives you an infinite number because aperiodic

(note this reasoning does not hold if there exists a periodic tiling, and I feel like it's not too difficult to construct counterexamples here)