The neighborhood
The Margolus neighborhood partitions both space and time into even and odd components.
First, a mapping is defined for a 
neighborhood
into itself; in the CAM context, each cell has four states, producing 256 neighborhoods
for each cell. The mapping is therefore defined by a 
matrix.
Next, non overlapping neighborhoods are used to tile the plane. It is convenient to locate the lower left hand corners of the neighborhoods at lattice points both of whose coordinates are even, leaving the mutually odd coordinates to occupy neighborhood centers.
Since the neighborhoods do not overlap, there is no interaction between them; to create an interaction, time is also partitioned into even and odd moments. As time progresses, neighborhoods are alternately selected from even and odd coordinates, so that the neighbors which a given cell sees vary according the parity of the moment.
In principle, larger neighborhoods with longer timing cycles could be constructed; however such complexity lies beyond what CAM hardware permits. Furthermore, the hardware switching arrangements permit ignoring either the horizontal or the vertical parity, but this should probably be considered as a pathology which would not normally arise.
Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx