Those configurations appear often, but they don't appear with a period. Look at a chessboard.. i can make it out of individual black and white tiles, or out of pre-assembled units of 1 white and one black tile (or a strip of length 4 or squares of size 4, etc). I can make a chess board by grouping the base units into a pattern, then only using that pattern. I can put that pattern on a wheel and roll it along a surface forever to get an infinite chessboard pattern.

There's not a way to do that sort of grouping for these tiles.

Compare pi. I can find the numbers representing my name in ascii an infinite number of times in the digits of pi, but if i find it once, there's no information in where to find it again, i can't just move forward n digits to find it, then another n digits to find it again, and ao on.