Basically, as the title says. Maybe this is trivially true or false, but I have not enough intuitions about topological surfaces or aperiodic tilings.
To make it a bit more precise - I mean the kind of aperiodic tilings such as Penrose tilings or the tiling generated by the recently found hat tile.
By "gluing" the patch into a torus (or some other surface) I mean essentially just like you fold a net into some geometric shape by connecting edges. And the edges should be glued in such a way that all matching rules of the tiling are satisfied.
If this is possible - would the resulting "tiling of the surface" also be aperiodic, or could it be made to be? Here I would consider "aperiodic" to mean that there are no non-trivial symmetries between any two tiles on the surface.