Geoscience Reference
In-Depth Information
(a)
(b)
FIGURE 2.5
Statistically self-similar fractal growth using probability thresholds. (a) Probability threshold
ρ = 0.2 with development in the Moore neighbourhood using the same rules as in Figure 2.4a. (b) Probability
threshold ρ = 0.5 with development in the Moore neighbourhood using the same rules as in Figure 2.4b.
threshold might become ρ × ρ
= 0.2 × 0.2 = 0.04 the second time and so on. In these randomised
automata,
N
(
r
) and
N
(
t
) are no longer synchronised or even lagged. In Figure 2.5b, we show the
pattern generated to
r
= 40, for a probability threshold of 0.5 where the neighbourhood rule is the
same as that used to generate the fractal pattern shown in Figure 2.4b.
So far, we have shown how very different patterns might be simulated by altering transition rules
based on the two standard Moore and von Neumann neighbourhoods and by introducing probabi-
listic thresholds into the exercise of transition rules. However, we can also change the nature of the
neighbourhood by making certain cells in the Moore neighbourhood illegal for development. This
implies a more general principle of placing a mask over the cellular space to restrict certain areas,
thus artificially changing the nature of the entire space. Here, however, we will use such masks
solely in the 3 × 3 Moore neighbourhood to show how different configurations of cells can lead to
different patterns. This does not destroy the uniformity assumption in that all the neighbourhoods
and transitions are still the same. Moreover, note that the von Neumann neighbourhood is a subset
of the Moore neighbourhood (as shown previously in Figure 2.4a) in that it is formed by making
the diagonal cells to the centre cell illegal to development. In general, this making of cells illegal
is equivalent to altering the transition rules so that certain cells are made ineligible for activating
a change in state, but this is also a more graphic way of illustrating how CA can produce very
different forms.
In Figure 2.6, we show four typically different neighbourhood masks which influence the growth
from a central seed site. If you refer to Figure 2.4c which shows the diagonal grid of growth from
the von Neumann neighbourhood, it is tempting to ask how a similar horizontal grid of alternating
on-off cells might be generated. This is possible by simply displacing the displaced von Neumann
neighbourhood where the mask in Figure 2.6a shows how this is accomplished. Note that the black
cells in the mask show those that are legal, the white those that are illegal: the stippled square is the
origin. If this mask is reduced to only the diagonal half of the 3 × 3 grid as in Figure 2.6b, the resul-
tant pattern which is grown to the edge of the screen is a class of fractal known as the Sierpinski
gasket (Batty and Longley, 1994). It has a fractal dimension
D
~ 1.585 which is also confirmed from
estimation of
D
in
N
(
r
) = (2
r +
1)
D
.
In Figure 2.6c, we have configured the legal neighbourhood as an -shaped block which, if
viewed as a superblock of housing, might be assumed to be a good compromise between access
and daylighting in the position shown. This is the kind of geometry which architects used to
