Geoscience Reference
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can be used to generate a hierarchical city system in accordance with the central
places approach (Frankhauser 2012 ).
Starting from a square-like area, we introduce a generator as illustrated in
Fig. 2.13 . Of course, the reduction factors can be chosen arbitrarily allowing the
biggest square to be adjusted to the size of the largest center offering the whole
range of facilities. The smaller squares correspond to second-order cities which
we assume do not provide the highest level of facilities. Depending on the real-
world situation, more or fewer than four centers may be introduced. Moreover, the
position of all the squares can be chosen freely, the only restriction is that the squares
are not allowed to intersect and they must lie within the initially given square.
Hence, the squares tend to be centered on already existing cities. Moreover, natural
and environmental constraints can be respected which generally condition urban
development (Mohajeri et al. 2013 ).
We see that, in this concept, the logic is not to cover zones containing built-up
space but to define from the outset the areas we wish to develop. By this logic, we
accept that settlements lie in the residual zones, i.e., the “lacunae.” These zones
are interpreted as rural zones. The iteration proceeds by replacing each square by a
smaller replication of the generator.
Hence, areas for urbanization are even more concentrated in zones which we
assume are served by public transportation networks. This prompts us to admit
that, in the zones cut off at this step, a low level of development is possible, thus
weakening the strong fractal model.
We see that by iteration, the reduction factors r 1 and r 0 are now combined
according to all possible permutations, which yields, for example, for the second
step:
r 1 r 1 ;
1 r 0 ;
1 r 0 ;
0 r 0
(2.10)
Since permutations are allowed, we have a degree of “degeneration,” since
r 0 r 1 D r 1 r 0
(2.11)
This is why the areas assigned to the second-order centers are the same as those
of the third-order centers belonging to the highest-ranked center (Fig. 2.14 ). This
corresponds to a particularity of multifractal structures. For the same reason, the
areas belonging to the third-order centers are no longer of the same size. We have
small squares of base length r 0 r 0 and larger ones with base length r 0 r 1 . In our
approach, this is the expression that third-order centers in the direct vicinity of
important centers are usually larger than those belonging to the hinterland. This
assumption differs from Christallers' model where all centers belonging to a certain
level are the same size.
In order to identify the hierarchical level and thus the facilities which are assigned
to the cities, we have introduced a specific coding system. Hence, for the first
iteration, we distinguish the large central square which we denote by the digit 1
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