Modelling Spatial Morphologies - GeoComputation - page 25

The structure which emerges is called a fractal which essentially is a form that is grown from the
bottom up and which manifests strict self-similarity on any scale with respect to the lower-order
scales from which it is generated. In this case, it is a set of trees that overlap one another (as we
cannot see the individual trees due to the limits of the 2D representation), but the way they are
generated is clear from Figure 2.1.
Fractals are in fact discrete forms or patterns that repeat themselves at different scales. In cities,
locations which deliver services to a local population often enshrined in the town centre or central
business district repeat themselves over different spatial scales, there being less numbers of bigger
centres as the spatial scale increases. These centres vary systematically in their size and shape with
their shape being based on a nested hierarchy of hinterlands around each centre serving ever larger
populations with ever more specialised goods. The size of a centre increases exponentially accord-
ing to a power law - the so-called rank size rule - as the space or hinterland around each centre
increases in size with scale. City size distributions usually follow a power law, while the networks
which determine how energy is distributed to each of these centres also follow some sort of fractal
pattern which is usually tree-like. In the last 20 years, these principles have been elaborated for
cities (Batty and Longley, 1994; Batty, 2013), although no completely coherent theory exists as yet.
Essentially in this chapter, we will present methods whereby fractal structures in cities can be
generated using CA. In this sense, cities are essentially fractal structures and thus CA models rep-
resent an obvious way in which they can be generated. In fact, generating fractal structures requires
algorithms that are essentially automata, and the term cell which is often attached to these models
simply implies that the representations are often spatial and that regular cells on a grid or at least
regular tessellations such as those found in hexagonal landscapes are the forms used to generate
fractal morphologies. Fractals come in two varieties - those that are strictly self-similar where the
pattern is highly regular at each spatial/temporal scale such as the trees that we show in Figure 2.1
and those that are statistically self-similar where the pattern at each scale is statistically similar
but not formally identical in terms of shape at other scales. Real-world examples are invariably
statistically self-similar because there is considerable noise that is associated with how these struc-
tures evolve in reality, whereas spatial designs are often produced with no noise whatsoever. Thus,
regular deterministic fractal patterns are much more likely to be generated by designers working to
produce idealised forms, whereas statistical fractals represent the geometry of nature (Mandelbrot,
1983). We will introduce both in the presentation below, as for example, in Figures 2.4 and 2.5.
Here, we will first outline the elements of CA in its strictest form. We then sketch the origins of
CA as a way of generating fractal structure, and we define its basic rudiments which are based on
neighbourhoods, transition rules and initial conditions. We will digress a little and generalise the
approach showing how CA relate to wider spatial models based on the generic reaction-diffusion
approach. We then provide some ideas about the origins of fractal geometry and morphology, start-
ing with basic CA models based on growth from single seeds which we then elaborate as growth
from many seeds, showing how these models pertain to self-organisation. We conclude with a brief
survey of applications, guiding the reader to practical examples and commenting more critically on
the limitations of these approaches as urban simulation models. We then outline further reading and
introduce key references.
2.2 ELEMENTS OF STRICT CA
Formally, we can state the principles of CA in terms of four elements. First, there are cells , objects
in any dimensional space but manifesting some adjacency or proximity to one another if they are
to relate in the local manner prescribed by such a model. Second, each cell can take on only one
state at any one time from a set of states which define the attributes of the system. Third, the state
of any cell depends on the states and configurations of other cells in the neighbourhood of that cell,
the neighbourhood being the immediately adjacent set of cells which are next to the cell in question
where next is defined in some precise and, in terms of strict CA, some local manner. Finally, there