Geoscience Reference
In-Depth Information
2
Modelling Spatial
Morphologies
Fractal Patterns from
Cellular Automata
Michael Batty and Paul A. Longley
CONTENTS
Abstract ............................................................................................................................................ 23
2.1 Cellular Automata, GeoComputation and Fractal Morphologies ..........................................24
2.2 Elements of Strict CA ............................................................................................................. 25
2.3 Origins of CA ......................................................................................................................... 28
2.4 Neighbourhoods, Transitions and Conditions ........................................................................ 29
2.5 The Generic Model: Reaction and Diffusion ......................................................................... 31
2.6 Origins of Fractal Geometry and Morphology....................................................................... 34
2.7 Simulating Fractal Growth Using CA .................................................................................... 35
2.8 More Complicated Growth Regimes ...................................................................................... 39
2.9 Applications to Cities and Related Ecologies ......................................................................... 41
2.10 Conclusions ............................................................................................................................. 44
2.11 Further Reading ......................................................................................................................44
References ........................................................................................................................................ 45
ABSTRACT
In this chapter, we introduce a technique of simulation that articulates the system of interest as
a set of cells, in our case, a 2D tessellation of cells that pertain to different land uses or socio-
economic activities that define cities. Cellular automata (CA) are defined in such a way that these
cells change state, from one land use to another, for example, dependent on a series of rules
that define the way these land uses influence one another, usually in very local neighbourhoods
around each cell in question. This is the way local action translates into global patterns, and
CA tend to be the essential mechanism that determines how global patterns emerge from local
order which can often be interpreted as geometries and spatial morphologies that are fractals.
Having introduced CA, we then outline the idea of fractals which have structures across many
spatial and/or temporal scales that are similar to one another at each scale. The classic example
is a tree-like structure or any hierarchical object such as the set of nested road systems from
freeways to local streets or the set of markets and retail centres which define the hierarchy of
central places. CA, of course, can be used to represent other local processes such as forest fires
and a variety of percolation phenomena that translate into ordered patterns at higher levels and
are not restricted to cities. Having introduced CA, we then develop a generic equation for spatial
processes based on reaction-diffusion and introduce ideas about fractals. We consider different
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