Symmetry in Self-Correcting Cellular Automata

We study a class of cellular automata that are capable of correcting finite configurations of errors within a finite amount of time. Subject to certain natural conditions, we determine the geometric symmetries such automata may possess. In three dimensions the answer is particularly simple: such an automaton may be invariant under all proper rotations that leave the underlying lattice invariant, but it cannot be invariant under the inversion that takes each configuration into its mirror image.