Is there an infinite binary array, so that the distribution of all possible binary $N\times N$ arrays on it will be uniform?
In other words, randomly picked $N\times N$ square will be $\sim\mathrm{Uniform}\left(\{0,1\}^{N\times N}\right)$.
The only direction I can think of is a tesselation with some finite pattern, such that distribution of possible arrays on the pattern itself will be uniform. Plus, all junctions must be ok.
Came up with this problem out of the blue, so any thoughts or pointers are appreciated.