TITLE: The "□□ □□" polyomino
QUESTION [12 upvotes]: 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?

REPLY [24 votes]: You can fill the plane, by forming rows repeating horizontally the following figure:

then attaching the rows with a shift of $1$ square.