Quasicrystals are based on the symmetry system of the icosahedron/dodecahedron, in which it it is impossible to periodically tile space.

“Proper” crystal structures are periodic, and are not necessarily related to any Platonic solid. See https://en.wikipedia.org/wiki/Crystal_system (but I’m not quite sure where the 5 comes from in TFA).

Not all of those correspond to patterns of crystal structure (although some do: tetrahedron, cube, and octahedron, IIRC).