From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 39

Fig. 2.14 The coding system of the central place system. ( a ) Shows the system at first iteration
( gray squares ) and the second one ( white squares )( b ) corresponds to the third step and illustrates
the underlying hierarchy of service levels (for details, cf. text) (Source: Frankhauser 2012 )
and the four smaller peripheral squares denoted by 0. In each following step, we
add another digit to the right of each digit, by the same logic. Hence, the hierarchy
is created just by combining two factors. In the next step, the highest-order central
square is now called 11 and the four smaller adjacent ones 10. The four peripheral
squares generated in the previous step are replaced, too, by the generator. The central
place is called 01 and the four peripheral ones 00 (Fig. 2.14 a). This procedure is
reiterated at the third step (Fig. 2.14 b). We then obtain a set of eight different codes,
each consisting of three digits. The first level center with the highest facility level
m D 1 has the code 111. The four directly adjacent squares of level m D 2havethe
codes 110. They correspond to suburban areas of the main center. The four centers
011 correspond to the four centers of level m D 2 generated at the second iteration
step and correspond to centers of the facility level m D 3. According to the logic
of iteration, we assume that higher level facilities are provided by the center 111
for the 101 centers, whereas the second level centers 011 provide the same type of
facilities for the centers 001. The small elements 100 and 000, adjacent to these
third level centers, are all low level centers m D 4(Fig. 2.14 b). By introducing
these codes, we have given up the previously discussed commutativity. Indeed,
in the system introduced, the codes 101 and 110 or 011 are not equivalent, even
if the areas are of the same size. Hence, the code introduces a noncommutative
operation. Consequently, the system displays certain properties corresponding rather
to properties of unifractals than to multifractals, which corresponds rather to the
hierarchical city system assumed.
Hence, making abstraction of their size, we confirm that the total number of
centers belonging to the different levels obeys a geometrical series, except for the
transition from the highest to the next level (Table 2.5 ).