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are completely synchronised in this automata, then N ( t ) = (2 t + 1) 2 as well. If we plot the number of
cells which are live at each distance and time, that is, when r = t = 0, 1, 2, 3, …, then the sequence
follows the progression 1, 9, 25, 49, …, and so on.
It is also useful at this point to examine the dynamics of an equivalent 1D CA where the neigh-
bourhood is now a 3 × 1 set of cells, that is, each cell east and west of the cell in question. Using the
same rules as in the 2D CA, the resulting pattern is a horizontal line of cells diffusing from the cen-
tral cell in exactly the same manner as the 2D automata. It is now easy to guess the dynamics which
give the number of cells produced at distance r from the central cell and at time t as N ( r ) = (2 r + 1)
and N ( t ) = (2 t + 1). For now, simply note that the exponent on the general equations for N ( r ) and N ( t ) is
the dimension of the system, 2 for the 2D automata and 1 for the 1D. We will return to this, but now,
all the elements for our study of CA as a basis for spatial simulation have been assembled. Although
we will explore how these features can be adapted to generate many different types of spatial system
in the rest of this chapter, we can anticipate some of this before we digress back into the history and
origins of this powerful approach.
With these definitions and principles in mind, it is worth demonstrating just how flexible the
CA framework is at representing and simulating very diverse types of system. Readers who require
a thorough discussion with many examples, especially from physics, are referred to Toffoli and
Margolus (1987). Although CA give equal emphasis to objects and their relations in space and time,
the focus on cells means that the framework is organised from the spatial rather than the temporal
viewpoint. However, 2D arrays of cells, although the most usual, are simply one case, for CA can
exist in any number of dimensions, and all the principles generalise accordingly. One-dimensional
models can be used to model relations on a line, but space need not be real; it might simply be
used as an ordering principle; in this context, the cells might be, say, time cells. Three-dimensional
automata might be used to represent the explicit 3D world of terrain and built form, but in our expo-
sition, we consider the 2D world the most appropriate for spatial simulation (O'Sullivan and Perry,
2013). States can take on any value in a range of discrete values, while in geographical applications,
there is usually some argument as to whether the concept of neighbourhood should be relaxed. What
we refer to as strict CA are those automata where there is no action at a distance, that is, where the
neighbourhood of interest is entirely local, being composed of those cells which are topologically
nearest neighbours to each cell in question. Geographical systems, however, are often character-
ised by action at a distance, and if this is to be represented, then neighbourhoods must be defined
to reflect it. Such variants are better called CS models (after Albin, 1975), but for the moment, we
will restrict our focus to CA, despite the fact that many realisations of CA models for practical
applications are in fact CS models. In many instances, action at a distance is in fact the product of
the system's dynamics - it is a consequence of local actions through time - and thus it is eminently
feasible to generate morphologies and their dynamics which display such properties using strict CA.
The major representational issue in CA modelling involves the extent to which the discrete-
ness which the framework demands matches the system's elements, relations and behaviour. In
principle, any continuous system can be made discrete, and thus assuming that local action and
interaction characterise the system, CA is applicable. However, in practice, it is often difficult or
even impossible to associate cells and states of the model to those of the real system. For example,
consider a town whose basic elements are buildings. Within each building, there may be several
distinct activities, and thus cells cannot be buildings; they must be parts of buildings disaggre-
gated to the point where each distinct activity - state - is associated with a single cell. Often,
this is impossible from the available data with this problem usually being compounded at higher
levels of aggregation such as the census tract. The same problem may exist in defining states. No
matter how small a cell, there may always be more than one state associated with it in that the
elemental level may not be a state per se but some object that can take on more than one state
simultaneously. Sometimes, redefinition of the system can resolve such ambiguities, but often to
use CA at all, certain approximations have to be assumed; for example, a cell which may be a
geographical location, say, can have only one land use or state, and thus this may have to be the
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