I was just wondering something about non-periodic tilings (of the plane, though I imagine the dimension is irrelevant for finite dimensions; would be interesting if it wasn't!).
I assume we know what a tiling of the plane is; suppose it's tiled using only a finite number of different shapes. For me, a 'non-periodic' tiling is one so that there is no non-identity isometry of the plane that carries every tile exactly onto a tile (it's just what you'd imagine). See, for example, the Penrose tiling.
I am wondering, is it true that there will be a connected, unbounded union of tiles (closed tiles, so corners touching counts as connected, though again I imagine it doesn't matter) that has at least one isometric copy of itself elsewhere in the tiling?
I don't really know anything about tilings, it was just a question that came to mind because someone was showing one to me. Thanks!