Actually, it's really easy to mess up(i.e. build a non-still-life) with only tiling. If you restrict the tiles so that tiling up makes only still lifes,dvgrn wrote:Now, if we're allowed to just stick 3x3 tiles next to each other, we can still build lots of things that aren't still lifes. For example, one parity of a glider can be divided into two Still-Life Compatible 3x3s. It doesn't do any good to go to N>3, because you could still build a glider with two SLC tiles for any size N. So how do we make sure that we can build all still lifes and nothing but still lifes using this set?
i) Obviously you'll need to include the blank tile.
ii) If a tile T is in the set, then tile T surrounded by blank tiles should be stable, so tile T should be stable itself.
iii) If a tile T is in the set, then the plane tiled by T should be stable.
ii) and iii) restricts the tiles a LOT. I don't think most of the still lifes can be covered by these tiles at all. i.e. Can you build a Long^n table with sufficiently big n with NxN tiles that satisfy ii) and iii)?
EDIT: And that means you should restrict the neighbor tiles a tile can have.