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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!

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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.

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  • $\begingroup$ 180 degree rotation, it's not aperiodic. Anyway, that is one example of a tiling. The question asks whether there are connected, unbounded unions of tiles appearing at least twice for ANY aperiodic tiling. You want to ask for periodic tilings, as well? It's trivially true, so I didn't bother. My definition of aperiodic tiling is the same as everyone uses, I think your answer needs some refinement since it's an off-topic example to a question that asks about universal existence. $\endgroup$ – John Samples Feb 3 '18 at 0:46
  • $\begingroup$ @JohnSamples Rotation and other symmetries are easy to address: a single L-shaped tile, and the rest are 2x1 tiles in the horizontal direction. $\endgroup$ – user58697 Feb 3 '18 at 0:58
  • $\begingroup$ Then it is aperiodic and trivially satisfies the desired property. You have a proof or disproof that this happens for every tiling? Lemme know, because then you actually have an "answer." $\endgroup$ – John Samples Feb 3 '18 at 0:59

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