From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 22

2.4.4 The Curve of Scaling Behavior

In previous work, it turned out that a particular way of representing the empirical
results helps to provide more detailed information about the spatial organization of
urban patterns (Frankhauser 1998 ; Thomas et al. 2010 ). This type of representation
is known as the “curve of scaling behavior” (Frankhauser 1998 ; Thomas et al. 2010 ).
Palmer ( 1988 ) called this a “fractogram,” but that terminology is confined to ecology
(Leduc et al. 1994 ). We start by taking the logarithm of relation Eq. ( 2.6 ):

log N."/ D log a C D log " (2.7)

which is a linear relation between log N (")andlog" where the dimension is simply
the slope value of the straight line.
We assume now that the prefactor a and the fractal dimension D may both depend
on the distance parameter " in accordance with the idea that real-world textures do
not necessarily obey a strictly fractal law:

log N."/ D log.a/" C D."/log " (2.8)

and thus the variation of log N (") with respect to log " becomes (Frankhauser 1998 ):

d log N."/
d log "

d log a."/
d log "

dD ."/
d log " C D."/

˛."/ D C (2.9)

This shows that if the dimension and the prefactor both depend on the scale
parameter ",the˛-values are no longer constant and may even exceed the upper
limit value of D D 2. Hence, representing the ˛-value as a function of distance "
informs us about deviations from fractal laws due to a variation in D or a . It turns
out that the shape of these curves provides insight into the spatial organization of
urban patterns and can distinguish different types of urban district (for details, see
Thomas et al. 2010 ).

2.5 The Fractality of Urban Patterns

2.5.1 Toward a Typology of Urban Patterns

As pointed out above, a number of papers have been published about fractal analysis
of urban patterns. We focus here on some striking results obtained at the Thema
institute of Besançon (France) in cooperation with the CORE institute at Louvain-la-
Neuve (Belgium). In order to realize fractal analyses of urban patterns, a computer
program was developed which estimates the fractal dimension and the prefactor
from the observed data and makes it possible to verify to what extent the empirical