Determine whether spacetime quasicrystals admit the proposed tiling construction

Determine whether a G-symmetric spacetime quasi-tiling with a finite number of equivalence classes of tiles, each having infinitely many members related by an infinite symmetry group G, exists, and, if so, whether it can be constructed using the symmetric cut-and-project method.

Background

The thesis proves a no-go theorem excluding promotion of a G-invariant spacetime quasicrystal to a G-invariant quasi-tiling with finitely many polygonal tiles when G is an infinite non-crystallographic symmetry group. It then notes that tilings with infinitely many rotated tile copies, such as pinwheel tilings, provide a possible analogue.

The unresolved issue is whether a spacetime analogue with finitely many tile equivalence classes can exist despite having infinitely many symmetry-related members, and whether such a structure can arise from the symmetric cut-and-project framework.