The plane is infinite, so the special property that mathematicians want is that the tiling doesn't contain arbitrarily large repeating patches. Any tiling that necessarily repeats itself will contain arbitrary large repeated (translated) chunks when the whole infinite plane is considered. The tricky thing is to create a set of one or more tiles that can only tile the plane without repetitions.
https://en.m.wikipedia.org/wiki/Aperiodic_tiling