The tile is just a 1x2 rectangle. Mostly they are all placed vertically, but there is a line along which horizontal tiles are placed. One could interpret the pattern of absence or presence of a horizontal tile in the sequence of columns as expressing a fraction in base 2. If the fraction is irrational, the pattern will be aperiodic. Hmmm, even if there was just one horizontal tile in the middle, the pattern would be aperiodic!
There must be some trickier definition in play, of what aperiodic means. But anyway, now it doesn't seem so impossible!