Depending on the infinite grid filling scheme even these properties may not be sufficient to guarantee that every two dimensional pattern is initially generated because the grid is two-dimensional, but the number property is "one-dimensional". A spiral pattern for example may always make it line up in a way such that certain 2d patterns are never generated.
This has not been proven yet: https://math.stackexchange.com/a/216348/575868
(or more generally: https://en.wikipedia.org/wiki/Disjunctive_sequence)
Depending on the infinite grid filling scheme even these properties may not be sufficient to guarantee that every two dimensional pattern is initially generated because the grid is two-dimensional, but the number property is "one-dimensional". A spiral pattern for example may always make it line up in a way such that certain 2d patterns are never generated.