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How to prove or disprove that if a polyomino tiles the plane, it must also be able to perfectly tile some larger polyomino, which also tiles the plane?

A polyomino is finite set of unit squares connected side to side. Allowed to rotate when tiling. Tiles must be disjoint. Perfect tiling=Exact cover.

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    Telling people that you need an answer doesn't make anyone want to give you an answer.2012-01-19
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    If you are not looking for an answer, why ask the question?2012-01-19
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    Apparently, the first known aperiodic tiling using just one shape was found about two years ago, see http://blog.makezine.com/2010/03/26/worlds-first-aperiodic-tiling-with/ It's not a polyomino. There are sets of three polyominos that tile only aperiodically, see http://mathworld.wolfram.com/PolyominoTiling.html2012-01-19
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    Apparently there exist a set of two polyominos that tile only aperiodically aswell. 2nd last page: http://www.univ-orleans.fr/lifo/Members/Nicolas.Ollinger/talks/2011/04/greyc.pdf I couldnt find further reference for this2012-01-19
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    here http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Goodman3.pdf2012-01-19
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    @gerdur I am interested what led you down this line of questioning and what kind of research you may have already done. Some of the proofs in this area are notoriously difficult (Hilbert was misled regarding anisohedral tilings) and there are many unproven conjectures.2012-01-19
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    When I asked it seemed kind of obvious that a counterexample cant exist, and I thought there was a simple proof, but now I am not so sure anymore.2012-01-19
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    @GerryMyerson I think you can find it here: http://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling . The Ammann 4 tile consists of two polyominos2012-08-21

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