If the network is a regular 1d or 2d CA (i.e. with homogeneous regular wiring and rule) rotational symmetries of the periodic array result in equivalent dynamics (Wuensche 1992). A given network state (periodic pattern) A is embedded in a particular past and future made up of other connected network states. If the given state is translated around its circle (1d) or torus (2d) by an arbitrary number of moves, and transformed to state B, the states in B's past and future must be identical transformations of the states in A's past and future. The two sets of states are equivalent, and their connection topology is identical.
A complication arises because some states are made up of repeating pattern segments, for example 011,011,011. If the repeating segment size=s, there will be only s equivalent states irrespective of the network size n. whereas with no repeating segments (always the case if n is prime), the number of equivalents is n.
Note also that states with a given repeating segment size s, cannot be upstream of a state with greater s. In an attractor cycle the value of s for all the states must be equal. The uniform states (all 0s and all 1s) with the shortest repeating segment size (s=1) are often powerful attractors.
For regular 1d and 2d CA, ``compression`` algorithms automatically come into play (unless suppressed, see #20.1.1) to compute equivalent basins, transient trees, and sub-trees (from the uniform states), from each prototype. This considerably speeding up the program. For equivalent basins, only the prototype is shown. Equivalent trees and subtrees are shown, but may be suppressed to varying degrees (see 20.1.2)