Isometric cellular systems
Isometric neighbourhoods are defined as being those where the domain
and range contain the same number of cells.
Isometric cellular systems are defined as being those where the
domain and range contain the same quantity of information.
Isometric cellular systems are usually based on Isometric
neighbourhoods  which are designed to facilitate their construction.
In an isometric cellular system, reversibility of the local map
translates directly into reversibility of the whole system. This helps with
designing reversible cellular systems which exhibit particular classes of
behaviour.
Examples of isometric cellular systems
 Bijective neighbourhoods
 Isometric neighbourhoods
 Partitioning neighbourhoods
 Stellated neighbourhoods
Partitioning schemes
Partitioning cellular systems form a subcategory of isometric cellular systems.
Space is divided into tesselated, noninteracting regions. Each region is then
bijectively mapped onto itself. Then space is repartitioned  and another
bijective map is repeated. This process is iterated.
P.119 in Cellular Automata Machines defines partitioning as follows:
[...] let us introduce a new style of cellular automaton, called a
partitioning cellular automaton.
 The array of cells is partitioned into a collection of finite,
disjoint and uniformly arranged pieces called blocks.
 A block rule is given that looks at the contents of a block and
updates the whole block (rather than a single cell as in an ordinary
cellular automaton); an example is given [...] below. The same rule
is applied to every block. Note that blocks do not overlap, and no
information is exchanged between adjacent blocks.
 The partition is changed from one step to the next, so as to have
some overlap between the blocks used at one step and those used at the
next one.

One of the simplest partitioning schemes is the Margolus neighbourhood, named after
Norman Margolus and studied extensively in a book he coauthored,
Cellular Automata Machines (CAM).
For another example of a partitioning scheme, see the
Q*Bert neighbourhood.
Other partitioning schemes
There are probably as many two dimensional partitioning models
as there are ways of tesselating objects to cover planes. Not
all of these exhibit desirable symmetries and regularities, though.
One way of building partitioning schems involves superimposing
two geometricallyrelated grids, and treating these alternately.
The Margolus neighbourhood
can be
seen as the result of doing this with two rectangular grids. The "Q*Bert" neighbourhood
is the result of
doing this with two hexagonal grids. A superposition of hexagons and
appropriatelysized triangles produces the
Hexagonal Triangualar neighbourhood.
For many of the the same reasons that bees build hexagonal honeycombs, and
partly because a hexagonal net represents the best way of packing circles in two
dimensional space, hexagonal neighbourhoods are of particular interest to those
modelling twodimensional physics, or those interested in building computing
structures efficiently in hardware.
There are other hexagonal partitioning neighbourhoods of some interest:
References
 Margulis, Norman H., and Toffoli, Tomasso:
"Cellular Automata Machines: A New Environment for Modelling", 1987;
 Margulis, Norman H., and Toffoli, Tomasso:
"Invertible Cellular Automata", Physics D 45, 1990.
