# Can a patch of an aperiodic tiling of the plane be mapped onto / glued into a closed surface such as a torus?

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.

• I don't think thats possible, not because of a periodicity in the end result, but because you cannot find a patch of an aperiodic tiling, that can be seamlessly roled into /glued onto a closed surface. Finding "matching edges" implies, the surface is not aperiodic, doesn't it?
– Max
Commented May 9, 2023 at 9:26
• Max's point is correct and the most important one. I also think your definition of an 'aperiodic' tiling on the torus is a bit naïve. By that definition, there are periodic tilings of the plane that have patches that can tile the torus aperiodically (pretty trivially even, just take a single tile from a standard square tiling and deform the squares slightly to break rotational symmetry). Commented May 9, 2023 at 11:16
• Periodic plane tilings are equivalent to tilings on a torus. A periodic tiling repeats in two different directions, two vectors. If you take a parallelogram with those vectors as its sides then the tiling pattern can be seen as made up from identical copies of that parallelogram. By identifying opposite edges of that parallelogram you turn that into a tiling on a torus. Conversely, a tiling on a torus is a tiling of a parallelogram with opposite edges glued, so you can unglue the edges and duplicate that parallelogram to tile the plane periodically. So an aperiodic tiling cannot tile a torus. Commented May 9, 2023 at 12:31
• Okay so according to the third comment this it is not possible (requires cutting out a parallelogram that can be repeated, but the tiling is strongly aperiodic, so such a parallelogram cannot exist in the tiling). Thanks! Commented May 11, 2023 at 7:38