A variety of nonlinear chemical models, such as the Selkov-Schnakenberg, exhibit Turing patterns. The Templator, which is based on a minimal autocatalytic monomer-dimer system, is a newer two-variable model also able to show Turing patterns. Here we find that the dynamic behavior of the Templator is quite similar to other models with cubic nonlinearities. This is demonstrated through a series of computer simulations in two dimensions utilizing the cellular automata approach. The selection of parameter values is based on linear stability analysis, which provides a relatively simple method of predicting Turing pattern formation. The simulations reveal spot, labyrinth, and striped patterns, in agreement with the predictions of the analysis. Other behaviors, such as honeycomb patterns, are also observed. For some parameter values, we study transient spot replication. Our findings strongly suggest that the Templator may belong to the same class of models previously studied by Pearson.