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in many variants of CA on not where the structure grown by limiting the diffusion to enable a con-
nected structure to grow from a seed site which generates a tree-like pattern that mirrors the way a
city might develop around a market or central place (Batty, 2005, 2013).
2.7 SIMULATING FRACTAL GROWTH USING CA
Let us begin our illustrative examples by returning to our first simple diffusion in its 1D and 2D
forms. In one sense, these automata are almost trivial in that they grow under the most unrestric-
tive neighbourhood rule to fill the entire space available. Their rate of growth is regular, with space
and time synchronised through the general equation
N
(
r
) = (2
r +
1)
D
or
N
(
t
) = (2
t +
1)
D
where
D
is the
dimension of the system. Most CA generated from a single seed will utilise transitions which do not
lead to their entire space being filled in such a regular way, but it would seem intuitively attractive
to be able to measure their fractal space filling by the same equations. Interesting patterns will fill
more than a 1D line across the space and less than the entire 2D space, and thus we might expect the
dimension
D
to lie between 1 and 2 for such automata. Such dimensions imply that such automata
are fractals and
D
as noted earlier is the fractal dimension.
Our first examples deal with transition rules that generate a change in cell state (from off to on)
when only a fixed number of cells are active/on in the neighbourhood. In Figure 2.4, we show the
possibilities. The first in Figure 2.4a shows the pattern generated when a cell changes state from off
to on when
one
and
only one
cell is active in the Moore neighbourhood, in Figure 2.4b when
one
or
two
cells are active and in Figure 2.4c when
one
and
only one
cell is active in the von Neumann
neighbourhood. If there are more than two cells active in the Moore and more than one active in the
von Neumann neighbourhoods, the automata generate space that is filled entirely. It is quite clear
that the patterns generated in Figure 2.4 are fractal or fractal-like; they are self-similar in that a basic
motif, which in turn is a function of the neighbourhood rules, is repeated at different scales, and as
the associated space-filling graphs show, they fill more than the 1D space and less than the 2D. In
fact, because of the way they are generated, the dimension
D
in
N
(
r
) = (2
r +
1)
D
and
N
(
t
) = (2
t +
1)
D
must be determined statistically, but in each case, these dimensions are close to but less than 2, that
is,
D
~ 1.947 for Figure 2.4a,
D
~ 1.945 for Figure 2.4b and
D
~ 1.997 for Figure 2.4c.
In this chapter, we will illustrate all our examples using the analogy between CA and the way
cities are developed, although readers are encouraged to think of other examples pertaining to their
own fields of interest and expertise. The patterns generated in Figure 2.4 are highly structured in
that the assumptions embodied within their transition rules are very restrictive. However, these are
reminiscent of a wide array of idealised city forms, such as in the various writings of Renaissance
scholars concerning the optimum layout and size of cities. In fact, these kinds of CA provide excel-
lent analogues for generating those highly stylized residential districts in cities where rules concern-
ing vehicular-pedestrian segregation are used as in the neighbourhoods of English and American
new towns which originated from the pioneering housing layout in Radburn, New Jersey. They
are also suitable for showing how the rules of town layout are exercised in imperial towns such
as Roman castra and colonial towns of the New World such as Savannah, Georgia (Batty, 1997).
Examples can be found in a series of topics that deal with fractal forms from Batty and Longley
(1994) to Salat (2011).
Most cities do not grow with such restrictive conditions on development: completely deter-
ministic, regular patterns are the exception rather than the rule. To make such automata proba-
bilistic, it is necessary to specify that a change in state will occur with a certain probability if a
particular condition(s) in the neighbourhood is (are) met. Thus, the transition rule only operates
with a certain probability. For example, in the case of the complete space-filling automata, a
cell, which has within its Moore neighbourhood an already developed cell, is developed but only
with a given probability ρ which effectively means that it is developed only ρ × 100% of the time
the transition rule is met. This is usually achieved by considering the probability to be a thresh-
old, above which the cell is not developed, below which it is. For example, if the probability of
