Unbounded, Repeated Figures in Non-periodic Tilings

Question

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!

Answer 1

Consider this:

        .....
---------------------
 |     |     |     |
---------------------
 |     |  |  |     |
-------|  |  |-------
 |     |  |  |     |
---------------------
 |     |     |     |
---------------------
        .....

En entire plane is tiled with horizontal bricks, except the two which are placed vertically. According to your definition, it is an aperiodic tiling. And here are plenty of isometric copies.

I think your definition needs some refinement.