Geoscience Reference
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(a)
(b)
(c)
FIGURE 2.7
Various types of DLA. (a) Classic DLA. (b) Compact DLA. (c) DLA growth from an edge.
to demonstrate that anything really useful comes from simply adding patterns together in this
manner, although there can be endless forms from which evocative and fascinating overlap-
ping pattern waves emerge, and who is to say that some of these might not contain insightful
inquiries into real systems? But unlike single-seed automata where global patterns do emerge
from local transitions, there is nothing other than overlap to give coherence and integrity to the
many-seed patterns that are produced by such automata. However, the many-seed approach is
useful for generating automata where the location of the multiple seeds is possible to predict
independently and even for cases where CA is used for design in the manner of shape grammars
(Batty, 1997). Finally, we need to note the space-filling properties of such automata. If we have
regularly spaced seeds and we operate the simplest transition rule which generates the entirely
filled form of
N
(
r
) = (2
r +
1)
2
, then the space is filled as
Ñ
(
r
) ~
nN
(
r
) where
n
is the number of
seeds. Of course, when the overlaps occur, the space becomes entirely filled and this marks the
boundary condition. All the other automata we have introduced can be generalised in the same
way but with caveats imposed by boundary conditions, regular spacing and space-time synchro-
nisation as we have already noted. We can thus generalise
N
(
r
) = (2
r +
1)
D
to
Ñ
(
r
) ~ ϕ
N(r
) where
ϕ is a constant varying with the number of seeds but also accounting for various kinds of noise
introduced where complex overlaps occur and where space and time fall out of sync.
We can now move to a somewhat different and possibly more general variety of CA which begins
with an already developed system. In such cases, there need to be two distinct states other than the
trivial developed/non-developed cases of the previous automata. Consider a completely developed
situation in which each cell is populated by two exclusive types of household - with highbrow or
lowbrow tastes in music, let us say. Imagine that each resident group
prefers
to have at least the
same
number
or
more
of their
own kind
in their Moore neighbourhood. This preference is not unusual, it
is not segregation
per se
and it might even be relaxed in situations where the preference is for say at
least 30%, not 50%, of their own kind to be in their neighbourhood. In terms of the number of cells
in the 3 × 3 neighbourhood, this means that the resident in the central cell would prefer to have 4,
5, 6, 7 or 8 neighbours of the same kind. The transition rule embodying this preference is thus as
follows: if there are less than four cells of type
i
in the Moore neighbourhood around a cell of type
i
, then that cell changes to state
j
, where
i
and
j
are the two types of cell.
This is a very different rule from any we have used so far. It involves a change in state which
depends on the
nature
as well as the
number
of cells in the neighbourhood, not simply the
number
which was the case in the previous growth models. It thus introduces
competition
into the automata.
Imagine that the cellular space is arranged so that every other cell in the grid is of a different type
or state. Highbrow residents are evenly mixed with lowbrow in regular checkerboard fashion so
that each resident has exactly four neighbours of his or her own kind. The system is balanced as
no changes of state will take place - everyone is satisfied - but the balance is precarious. If we
shake the system and shuffle things a bit, the overall proportions are the same - 50% highbrow,
50% lowbrow - but what then happens is that some residents are dissatisfied with the tastes of their
