Axiomatizing Rectangular Grids with no Extra Non-unary Relations

We construct a first-order formula phi such that all finite models of phi are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A subseteq N is a spectrum of a formula which has only planar models if numbers n in A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) <= n and $(n),s(n) = Omega(log(n)).