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That's what I meant by attempting to find a constructive proof. 2 doesn't work because no matter how much you shift down and to the left you'll always have a bo
by ot 9y ago
That's what I meant by attempting to find a constructive proof. 2 doesn't work because no matter how much you shift down and to the left you'll always have a boundary.
> our link proves something stronger
No, it proves that the 3 statements are equivalent, so it is not stronger.
- soVeryTired 9y agoBut 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.
- foobar__ 9y agoThis falls short of the original goal: While your method shows that for every point, you can find a tiling large enough that covers it, the original question was to find a single tiling covering everything. 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.
- deleted 9y ago[deleted]
- nightcracker 9y agoWith the caveat that if your tiling is inductive in the sense that the tiling of n+1 is an extension of n, a series of finite tilings will tile the plane.
- ot 9y ago> like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers 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.