Summary: In the discrete threshold model for crystal growth in the plane we begin with some set \(A_{0} \subset{\mathbf Z}^{2}\) of seed crystals and observe crystal growth over time by generating a sequence of subsets \( A_{0} \subset A_{1} \subset A_{2} \subset \dotsb\) of \({\mathbf Z}^{2}\) by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site’s prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if \(A_{0}\) is finite and \(A_{i+1} \neq A_{i} \forall i\) imply \( \bigcup_{i=0}^{\infty} A_{i} = {\mathbf Z}^{2}\). We prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed \(\rho\), the neighborhood of \(z\) is \( \{ x \in{\mathbf Z}^{2} : \| x-z\| _{\infty} \leq \rho\})\). This result has important implications in the study of the first passage time when \(A_{0}\) is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which \( n^{-1}A_{n} \) converges.