We prove that any Turmite, except for those who behave the same on every symbol, can simulate arbitrary computation in specific periodic configurations which are periodic, except for a finite input area. We also prove the P-completeness of predicting their future behavior by providing an explicit log-space reduction from the Topological Circuit Value Problem. A similar result was already established for Langton's ant: here we use a similar technique but prove a stronger notion of simulation, and for a more general family. These results are obtained by using gadgets similar to single-use logic gates; each gate performs its computation as the Turmite travels through it, therefore simulating arbitrary computation.