Comment by danbruc

There exists no translation that leaves the tiling of the entire plane unchanged. The same local pattern [1] will of course occur repeatedly, there are only so many different patterns of any given size, but if you look at large enough scales, any translated copy will eventually diverge.

Maybe someone can make a version of this - if it not already exists - where you can move a semitransparent copy of the tiling around, maybe with a score for the alignment achieved.

[1] I am not sure if this is true, but as we have to tile an infinite plane, I could imagine that any pattern of finite but arbitrary size will occur infinitely often.