Multifractal models and their formal properties in urban geography
Résumé
Fractal analysis for exploring the spatial organization of settlement patterns is used since a couple
of years. On the one hand, scaling behavior turned out to be a suitable approach for characterizing
such patterns, but town sections, issued from different periods of urban history or corresponding to
particular planning concepts show different types of scaling behavior what aided classifying urban
patterns. However, on the scale of agglomerations these different scaling behaviors are mixed. That
incites asking whether multifractal approaches could be of interest when considering urban patterns
or settlement systems, as local properties like different degrees of concentration of build-up sites
could be better taken into account and several researches focused on using multifractal analysis in
these contexts.
However, the use of fractal geometry is not restricted to measuring scaling behavior. Their intrinsic
geometric properties are of interest, too, when considering urbanization. Indeed, the particular
multiscale feature of fractals brings another perspective on spatial organization. Hence, fractal urban
models have been explored in the context of urban economics, and, more recently, proposed for
planning purposes.
However, we must be aware, that multifractality is a rather polysemic notion. We can use it while
considering the two-dimensional distribution of build-up space or network branches, but also when
looking at the weights, like the heights of buildings or traffic loads, on a two-dimensional fractal
or uniform support, which is soil occupation. In the first case we can still refer to self-similarity,
what is no longer true in the second case where self-affinity comes into play. We distinguish these
different topics and focus on their formal properties, which are nevertheless linked to the common
background of multifractal theory. More explicitly, we consider the properties of Sierpinski carpets
combining different scaling factors, a model which is of interest for describing central place hierarchy
as well as for planning concepts. We show, too, that there exist analytically ambiguities while using
the Minkowski and the Hausdorff approach for measuring fractality for those objects. We consider,
too, by referring the so-called Poincarré-maps, how different degrees of concentration of build-up
space across scales can be treated formally in a coherent way.