From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 20

nor with the confidence interval. Hence they provide, in principle, supplementary
information about the object. Previous investigations (Thomas et al. 2012 )showed
that the more homogeneous the built-up area of a city, the lower the a and the
higher the D .

2.4.2 Covering Methods

The most commonly used methods are based on the idea of covering the elements
of the object analyzed by a minimum of elements like squares or circles of size
". Hence, information is obtained about the number of elements lying within a
distance less than or equal to ". The most widely used method is “box counting,”
although we prefer to speak of it as “grid analysis.” It consists of covering a chosen
zone with a grid of a given mesh (grid square) size ". The number of grid squares
containing buildings is counted. At the next step, the mesh size is reduced and the
procedure repeated. By repeating the procedure, we obtain an empirical relation
between the number of grid squares containing buildings and the grid square size.
Figure 2.6 illustrates the method for a Sierpinski carpet by choosing grid square
sizes in accordance with the iteration procedure. The number N (")aswellassize"
of the grid squares corresponds here to the construction rule of the Sierpinski carpet,
i.e., both follow the series of the mapping procedure and so the fractal law holds. For
empirical structures, it is hence possible to verify whether the empirical data obey
the fractal law. Deviations from this law, e.g., occurring for certain scales ", can
hence be identified. However, it is difficult to find a position of the grid which fulfills
the requirement of minimal covering. Even for the constructed Sierpinski carpet, an
inadequate position of the grid would falsify the results since we would obtain more
grid squares containing elements of the fractal. In order to reduce this artifact, the

Fig. 2.6 Grid analysis applied to a Sierpinski carpet ( a and b ). In ( c ), poor grid positioning leads
to an overestimate of the number of grid squares occupied