But isn't it enough to show that any point in the plane will eventually get tiled after 1 - 3 run for long enough?
Because surely all points do get tiled eventually by the method above.
Because surely all points do get tiled eventually by the method above.
Finding progressively larger, finite tilings is not the same as having a single infinite tiling, just like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers (which wouldn't be a natural number).
König's lemma implies that for tilings both statements are in fact equivalent.
Very nice analogy!
> König's lemma implies that for tilings both statements are in fact equivalent.
Exactly, and instead of working by shifting the tiling (which would produce a sequence of incompatible tilings) it works by finding a sequence of finite tilings where each is a subset of the next, so it makes sense to take the union of the sequence.