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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?

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    @Student73: I think you can do the proof for the rectangle by enumeration, starting in a corner.2011-07-05

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You can fill the plane, by forming rows repeating horizontally the following figure:
enter image description here
then attaching the rows with a shift of $1$ square.