Modelling Spatial Morphologies - GeoComputation - page 34

2.6 ORIGINS OF FRACTAL GEOMETRY AND MORPHOLOGY
There is now a very wide range of basic introductions to fractal geometry, and it is not the intention
here to revisit in detail the basic concepts of scale dependency and self-similarity and their applica-
tion to real-world objects: for a good introduction to this field, see, for example, Peitgen et al. (1992).
Here, we will briefly review the focus upon the application of fractal concepts to the modelling and
simulation of spatial systems (Frankhauser, 1993; Batty and Longley, 1994). In fact, the first appli-
cations of fractal ideas to cities were to render in more realistic detail patterns of land use generated
from cruder models which enable predictions at a higher level of resolution (Batty and Longley,
1986). The spirit of this work chimes with the views of the arch-popularist of fractals Mandelbrot
(1982) who said that '… the basic proof of a stochastic model of nature is in the seeing: numerical
comparisons must come second' and aired such a view in the geographical sciences. The outcome
of these early experiments was an illustration of what is now developing into virtual city representa-
tions of urban structures that were purportedly implicit in many urban models but which had never
been articulated through any explicit land use geometry.
The first serious representations of spatial morphologies at the city level in terms of fractals
go back to the work of the mathematical and theoretical geographers in the 1960s (Tobler, 1970,
1979). These focused on developing the idea of self-similarity for lines defining boundaries where
a series of relations between the perimeter length of a fractal line and its relationship to the scale of
measurement showed that successive self-similar detail is picked up for fractal objects as the scale
gets finer. This builds on the classic coastline conundrum developed by Mandelbrot (1967) which
shows that the length of the line increases indefinitely as the scale gets finer but the area enclosed
by the line tends to a fixed areal limit. There is a range of measures of the ways in which fractal
phenomenon fills space (Batty and Longley, 1994), but the following measure sets out to do so by
relating the number of parts into which a line can be divided, and its length, to some measure of
its scale. Scaling relations may be derived with respect to an irregular line of unspecified length
between two fixed points.
We begin by defining a scale of resolution r 0 such that when this line is approximated by a
sequence of contiguous segments or chords, each of length r 0 , this yields N 0 such chords. Next, we
determine a new scale of resolution r 1 which is less than r 0 , that is, r 1 < r 0 . Applying this scale r 1 to
the line yields N 1 chords. If the line is fractal, then it is clear that reducing the interval always picks
up more than proportionate detail (Mark and Aronson, 1984). Formally, this means that the length
of the line L n at any level of scale n is given by the equation L n = N n r n with L n > L n −1 with the rela-
tionship between successive scales given as L n = ( L n −1 ) D where D > 1. If D is stable at every change
in scale, then the object is considered to be a fractal because the same level of additional detail is
being picked up at every scale, and the only way this can happen is that if this detail is similar to
that at the previous scale. If no additional detail is picked up by the change in scale, then this implies
that the line is straight and that the dimension D = 1. If the detail increases according to the square
of the length of the previous line, then this implies that the dimension D = 2. In fact, this parameter
is called a fractal dimension , and to cut a long story short, it specifies the degree of additional detail
picked up as the line changes its form from the straight line with Euclidean dimension 1 to the plane
with Euclidean dimension 2. All of this can be generalised to any number of dimensions, and many
of the fractals that are well known tend to exist in mathematical rather than in the physical space of
our world of three dimensions.
In fact, the essence of fractal geometry is the generation of successive levels of detail using what
essentially is a recursive algorithm that continually subdivides space. The same idea of growing a
fractal using the algorithm in reverse so to speak immediately introduces the idea of CA, and there
are many definitions that show that the cellular algorithm is a specification in terms of an initiator -
the starting configuration - and a generator - showing how the configuration grows or subdivides
at the next iteration of generation. This is the way we generated the intertwined trees in Figure 2.1
which is the example par excellence of fractals. In fact, the best examples of fractal growth depend,