From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 17

Fig. 2.4 Three different types of fractals. ( a ) A Sierpinski carpet where the outer border is a
fractal, but the dimension of which differs from that of the carpet. ( b ) A fractal, the features of
which remind urban patterns. Iteration has been developed up to the third step only in the upper
square of the cross and up to the second step in the square below .( c ) A fractal consisting of a
hierarchical system of clusters reminding the logic of a central place system (Source: ( b ) Tannier
et al. 2006 )
Figure 2.4 shows other examples of similar fractals. For the Sierpinski carpet of
Fig. 2.4 a, the outer boundary is connected, and its complex shape is reminiscent
of that of the urban pattern on coarse-grained maps as in Fig. 2.1 a. In Sierpinski
carpets, all the elements are connected and so the structure consists of a single
cluster. Fournier dusts follow the same iterative logic, but the elements are
disconnected. By combining both these logics, fractals like those shown in Fig. 2.4 b,
c can be connected. In Fig. 2.4 b, the positions of the elements in the generator
are less regular. In two squares, the second and third iterations are illustrated by
changing the position of the elements each time respecting the hierarchy of lacunae.
Hence, a more city-like shape is obtained at the microscale, with blocks of houses
along the streets and inner courtyards. Finally, in Fig. 2.4 c, two reduction factors,
r D 1/3 and r D 1/6, are combined and thus a hierarchy of clusters occurs. Hence,
a generalized version of Eq. ( 2.1 ) allows the dimension to be computed. This
multifractal structure could be an illustration of the distribution of central places in
an urban system like that of Fig. 2.1 a.
Let us emphasize that the definition of fractal dimensions is entirely consistent
with standard Euclidean geometry; for uniform distributions like that of Fig. 2.2 b,
the dimension is D D 2 and for lines D D 1. But furthermore the fractal dimension
has a clear meaning. It measures the degree of concentration of the occupied sites
across scales or, more specifically, the relative decrease in mass with increasing
distance from any site where mass is concentrated. Accordingly, the more uniformly
mass is distributed within a fractal structure, the closer the dimension will be to
D D 2, and vice versa, if the mass is concentrated in one point, D is zero . Indeed,
since mass is more uniformly distributed in Fig. 2.4 a, the dimension value is higher
for pattern 2.4 a than for patterns 2.4 band 2.4 c which are more contrasted. As
pointed out, boundary and surface area have the same dimensions in Sierpinski
carpets. However, in Fig. 2.4 a, the outer boundary is itself a fractal consisting of
N D 7 elements, and thus, the dimension of this fractal subset is D b (bord)