Symmetries [and state networks]

Many of the networks above contain large numbers of identical pieces. Typically the reason is that the states in each piece are shifted copies of each other, and in such cases the number of pieces will be a divisor of n. (See page 950.) If the underlying cellular automaton rule exhibits an invariance—say under reflection in space or permutation of colors—this will also often lead to the presence of identical pieces in the final network, corresponding to cosets of the symmetry transformation.