Modelling Spatial Morphologies - GeoComputation - page 26

are transition rules which drive changes of state in each cell as some function of what exists or is
happening in the cell's neighbourhood. There are further assumptions and conditions. It is assumed
that the transition rules must be uniform , that is, they must apply to every cell, state and neighbour-
hood at all times and that every change in state must be local , which in turn implies that there is no
action at a distance. There are conditions which specify the start and end points of the simulation
in space and time which are called initial and boundary conditions , respectively. Initial conditions
apply to the spatial configuration of cells and their states which start the process, as well as the time
at which the process begins. Boundary conditions refer to limits on the space or time over which the
CA is allowed to operate.
To illustrate these principles, we require an elementary example. The most usual configura-
tion of cells comprising a CA is based on a regular 2D tessellation such as a grid where the array
of square cells is contiguous to one another. The simplest categorization of states is that each cell
can be either alive or dead, active or inactive, occupied or empty, on or off, true or false, while the
neighbourhood within which any action changes the state of a cell is composed of the eight adjacent
cells in the band around the cell in question, at the eight points of the compass. This is the so-called
Moore neighbourhood. A very basic rule for changes from cells which are off to on might be as
follows: if any cell in the neighbourhood of any other cell in question is on , then that cell becomes
on . In this way, cells which are off are turned on , and those that are on remain on . To show how
these automata might change the state of an array of cells, we need an initial condition - a starting
point for the configuration of cells and also a stopping rule which in spatial terms is the boundary
condition. We will assume that the temporal is dictated by the spatial conditions in that once the
process begins from time zero, it finishes when the spatial boundary is reached. Our earlier example
manifests these conditions exactly: the starting point is the seed cell in Figure 2.1a. The way the rule
is operated in the Moore neighbourhood is shown in Figure 2.1b where a cell is switched on if it is
to the northwest or northeast of the already active cell. And the boundary condition is the extent of
the space as shown in Figure 2.1c.
If we assume a square grid which is a 100 × 100 square cellular array, we can fix the initial con-
figuration as one active or live on cell in its centre and start the process. Let us now assume that
a cell is switched on if there are one or more cells in its neighbourhood which are on. This differs
from our process in Figure 2.1 where the cell is only switched on if the cell to its bottom left or bot-
tom right is on. The new process is particularly simple. At every time period, each cell in the array
is examined, and if there is a live cell in its neighbourhood, then that cell is made live or switched
on. Here, on the first time cycle, the cells in the band around the centre cell each have a live cell in
their Moore neighbourhoods, and thus they are switched on. In the next iteration, the bands around
this first band all have live cells in their neighbourhoods and the same occurs. A process of growth
begins in regular bands around the initial seed site, with the spatial diffusion that this growth
implies, clearly originating from the operation of the rules on a system with a single seed site. You
can visualise this as a square band of cells spreading around a seed site, and it is clear that the pro-
cess could continue indefinitely if the array of cells were infinite.
This kind of growth and diffusion is an analogue to many systems. For example, consider the cel-
lular array as a grid of light bulbs all wired to those in their immediate (Moore) neighbourhood. The
rule is that we switch one on when one of those to which it has been wired has been switched on. If
we begin by switching the central bulb on, the process whereby all the bulbs are lit follows a regular
and complete diffusion. If the central seed were a population which grew in direct response to the
immediate space around it, like a city, then the process might mirror urban development. These
kinds of example can be multiplied indefinitely for any simple growth process from crystals to
cancers. The morphology produced is very simple in that it is one based on an entirely filled cluster
whose form is dictated by the underlying grid and by the influence of the neighbourhood. We should
also look at the model's dynamics. The number of cells occupied in this model can be predicted as
a function of either time or space. Calling the number of cells at the horizontal or vertical distance
r from the seed N ( r ), the cells occupied can be predicted as N ( r ) = (2 r + 1) 2 . As distance r and time t