From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 15

2.3 Fractal Models for Urban Patterns

We now illustrate by means of examples that fractals exhibit properties reminiscent
of those discussed for urban patterns. As pointed out, urban patterns consist
of elements, i.e., buildings forming what are often irregular clusters where the
distances between the buildings vary over a large range. But even when boundaries
are defined (e.g., by defining criteria for simplified maps), these boundaries are not
smooth but display outgrowths and indents of various sizes.
Unlike Euclidian geometrical objects such as circles or squares, fractal geom-
etry allows us to construct geometrical reference models consisting of elements
distributed in a completely nonuniform way, forming clusters at different scales.
It is then possible to illustrate several types of spatial pattern, which resemble
specific aspects of urban patterns like fragmentation or the complex morphology of
boundaries. These structures may look irregular; even so, the spatial distribution of
the constituent parts obeys a powerful distribution law, which may be characterized
by a single value. Thus, if urban patterns really do exhibit the particular features
of fractal objects, it may be concluded that, despite their highly irregular aspect,
they comply with a well-defined principle of spatial organization, which can be
characterized quantitatively. The usual notion of “regularity” or “irregularity” then
becomes meaningless.
Since we focus here on the question of how buildings are distributed within areas
of settlement, we make use of particular types of fractals consisting of “black”
elements distributed in space. We associate these elements with zones containing
buildings, whereas empty, i.e., “white,” zones are essentially unbuilt. Two kinds of
fractal objects prove of interest, Sierpinski carpets and Fournier dust. These two
approaches can be combined. In Fig. 2.3 a-c, we show how a Sierpinski carpet is
generated by iteration. Starting from the square-like initiator, a mapping procedure
known as the generator is applied. Here the procedure reduces the initial figure by
the factor r D 1/3, and N D 5 of these “elements” are assembled to form a cross.
The same operation is repeated for each of the smaller squares. Hence, as this
iteration procedure is repeated, an ever more filigree object appears consisting of

Fig. 2.3 ( a - c ) Generating a Sierpinski carpet and ( d ) the emerging hierarchy of lacunae

(Frankhauser 2005 )