The polyomino '□□ □□' (two blocks of two squares with a gap) does not tile any rectangle, how do I prove/disprove that it tiles the plane?
The "□□ □□" polyomino
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$\begingroup$
geometry
tiling
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0For people who don't know the definition: http://en.wikipedia.org/wiki/Polyomino – 2011-07-05
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2My first inclination was to try to prove that it doesn't, but I've now managed to fill up quite a bit of graph paper and there's no indication how I'd run into trouble if I go on, so my guess is now that it does. – 2011-07-05
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1My guess too is that it does tile the plane. But I'm curious too look at the proof that it doesn't tile any rectangle. – 2011-07-05
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1@Student73: I think you can do the proof for the rectangle by enumeration, starting in a corner. – 2011-07-05
1 Answers
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You can fill the plane, by forming rows repeating horizontally the following figure:
then attaching the rows with a shift of $1$ square.