From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 28

which is surrounded by smaller ones. In general, the interval becomes larger for
sites lying close to sites where spatial organization changes. This type of analysis
seems rather promising for exploring the spatial organization of urban patterns.

2.5.4 Including Boundary Analysis

We now discuss results where urban borders have been extracted and where their
morphology has been compared to that of built-up areas. As pointed out, in
Sierpinski carpets, the built-up space and the entire boundary should have the same
dimension. However, there may be subsets, e.g., outer boundaries (Fig. 2.4 a) or
subclusters, with different fractal dimensions.
Moreover, we recall the peculiar feature of teragons with compact areas but
fractal boundaries. Hence, we may seek out the kind of fractal that urban patterns
resemble most. To this end, we dilated stepwise urban patterns. Buildings were
seen to merge increasingly and form clusters. Investigations show that large clusters
emerge after just a few dilation steps, when courtyards and small streets are filled
in, which usually occurs at distances of about 8-16 m (De Keersmaecker et al.
2003 ; Thomas et al. 2008a ). Here again, we find the distance range for which we
often observed non-fractal behavior when looking at the curves of scaling behavior
(Fig. 2.7 b). Hence, we could say that, beyond this threshold, urban fabrics may be
considered to be prefractal structures.
We show here how boundaries can be extracted. In Tannier et al. ( 2011 ), a
systematic method is discussed based on dilation. Here, the number of clusters
remaining after stepwise dilation is counted. For fractals, the number of clusters
remaining after stepwise dilation again obeys a power-law distribution. When the
number of clusters no longer changes, it can be expected that the morphologic
envelope of the settlement has been identified, since neighboring clusters are far
enough away. This is the case, for example, of the neighboring settlements in
Fig. 2.9 .
Let us now look at the dimension values of this example. The surface dimension
estimated for the zone corresponding to the extracted main cluster corresponds to
that of a rather contrasted pattern. However, we see that the difference between the
entire boundary dimension and that of the main cluster is not very great. This shows
that no significant hierarchy of inner lacunae remains after the smoothing proce-
dure. But the boundary dimensions are rather high. Obviously, no strict planning
constraints were applied for rounding up the urban border. Hence, comparing the
surface area and boundary dimension turns out to provide rather interesting results.
In Thomas et al. ( 2008a , b ), 262 communes of the Walloon region of Belgium
were analyzed. For each commune, the surface dimension was determined. Then,
each pattern was dilated up to three steps, what corresponds to 12 m. Then, the
dimensions were estimated for the surface and the corresponding outer boundaries
of the extracted clusters. The goal was to find a reliable classification based on the
set of dimensions available for each commune. To this end, a method was used based