Euclidean geometry describes very accurately the physical space of the world in which we live, but this is not a logical necessity - rather it is a (nearly exact) observed feature of the physical world. Other geometries are possible; for example, Lobachevskian or hyperbolic geometry which is similar to Euclidean geometry with certain important differences. It is conceivable that hyperbolic geometry describes the world when considered on a cosmological scale; however, for this to be the case, the proportionality constant between the angle deficit for a triangle and its area would have to be exceedingly small (Figure 1). Einstein's general theory of relativity demonstrates that the geometry of our world is, in fact, non-Euclidean, but in an "irregular", more complicated way than is suggested by hyperbolic geometry. As such, Euclidean geometry provides an excellent approximation to hyperbolic geometry on any ordinary observational scale and is the standard geometry for modelling naturalistic systems.
Cellular automata (CA) are a class of mathematical and
geometrical formalisms which are widely used in modelling the
behaviour of naturalistic systems. Examples include lattice gas CA
used to model the behaviour of gases and ecological CA used in
macroevolutionary modelling. In order to describe natural systems
accurately on an ordinary scale, CA models need to approximate
Euclidean geometry as closely as possible. The tessellation (lattice
structure) produced by a number of tiles (cells) is dependent on the
geometry of the individual tiles: standard two-dimensional CA adopt a
homogeneous tile (cell) geometry which is usually regular square,
although hexagonal tile structures have been investigated in the
literature. There are two standard cell neighbourhood templates for
the standard CA with regular square cell geometry: (i) Von Neumann
neighbourhood: 4 cells edge-connected to the central cell (Figure 2(a)),
(ii) Moore neighbourhood: (Von Neumann neighbourhood 
4 cells
vertex-connected to the central cell) (Figure 2(b)). The standard
neighbourhood template in lattices composed of hexagonal cells
comprises 6 cells edge-connected to the central hexagonal cell
(Figure 2(c)). The hexagonal lattice has been shown to have a geometry
which is suitable for modelling the behaviour of a large class of
natural systems including the diffusion of gases and crystal growth
processes in which packing density is assumed to be uniform and
optimal (hexagonal close packing). However, the regular square
geometry is usually adopted in CA models because it simplifies the
computer implementation of the automaton.
Conventional neighbourhood templates ignore an important property connected with the geometry of a lattice; that is, intercellular distance. Intercellular distance is determined according to lattice geometry and the distance metric adopted, but the property itself is geometry- and metric-independent: a cell occupies space in a lattice and since no two cells can ever occupy the same space in the same lattice, they must occupy different spaces separated by a distance equal to the positions of the two cells relative to each other; the magnitude of the separation is calculated according to lattice geometry and distance metric. Intercellular distance has implications for the validity of CA models of natural phenomena in which geometrical considerations such as distance are significant.
Examples of such
phenomena include crystal growth processes in which the strength of a
chemical bond varies with interatomic distance according to an
inverse proportionality relationship which determines the stability
of the crystal structure generated. In CA models of such processes,
incorporating Euclidean geometry with respect to distance metric
becomes significant. However, standard CA assume that cells in a
Moore neighbourhood are equidistant from the central cell; hence,
Euclidean geometry is only coarsely approximated. Other "finer-grain"
distance metrics are available which "correct" the imprecision
associated with CA models. This correction is necessary because CA
generate visual patterns which are interpreted as simulations of
observed natural phenomena; in order for such interpretations to be
valid, the models must approximate reality (including geometric
reality) as closely as possible. The class of CA which incorporate
geometrical properties such as intercellular distance explicitly in
the model are known as masked cellular automata (MCA).