We construct a formula \(\phi\) which axiomatizes non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set \(A \subseteq \mathbb{N}\) 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 \(t(n)s(n) \leq n\) and \(t(n),s(n) = \Omega(\log(n))\).