> Does aperiodicity have any cool properties that help in specific domains?

well, it helps as the basis for all life on the planet as we know it (DNA, RNA, et al.). which is pithy, but i think actually has fairly deep implications (ie is not _just_ pithy).

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.

> Does it mean simply lack of pattern? But that doesn't seem to be the case, at least visually.

Aperiodic tiling means that the whole thing doesn't repeat. If you overlay a copy of the tiling and move it around, you'll never find it match perfectly everywhere except for the one position you cloned it as. A grid of squares is periodic, you can translate it one unit to the side and it's the exact same, everywhere.

This is of course a different kind of periodic than is meant with The Periodic Table.