1A fundamental issue in all the reflections on the social and economic mechanisms of a territory is the distribution of a population in a given space. Many disciplines have tried to study this phenomenon in different contexts and scales : demographers, geographers, but also town planners and economists (Frankhauser 1994, 1998; Batty and Longley, 1994 ; in Italy : Rabino and Carzaniga, 2003).
2Whatever the scale of observation, the results show that human activities are not homogeneously distributed in space. Various reasons can be given for this : on one hand, some places are more suitable than others for human activity, thereby influencing the spatial distribution of settlement ; on the other hand, the urbanistic processes have generated a hierarchical organization within cities. All these elements have always been the object of formalization.
3Yet if the existence of a heterogeneous distribution seems omnipresent, and features in theoretical analyses of functioning of settlement systems, the measures used are always based on the paradigm of a uniform space : the geometrical reference remains homogeneity. In fact, the most commonly used measure is density, which indicates a mean occupation of the territory, supposing proportionality between the population and the occupied surface. The results highlight the ambiguity of this approach.
4Urban planners cannot refer to the circular or linear city, and the cities are amorphous and irregular, without showing an inner organization. Studies about complex phenomena, such as nonlinear phenomena, have shown the insufficiency of these traditional concepts. As an alternative, fractal geometry is the only approach with a pure geometric character.
5Such an approach is based on binary logic that distinguishes between the urbanized and nonurbanized area. It is thus necessary to introduce fractal geometry (Mandelbrot,1987).
Figure 1.1: the first iterations to build two fractal structures
6In order to obtain a fractal (Fig. 1.1) it is necessary to choose an initial figure, in our case a square with side L. Subsequently a geometric operation that transforms the initial figure must be chosen, this is called a generating phase.
7In our example, a factor of r = 1/3 is applied to the initial figure, creating therefore N = 5 squares (blue arrow) similar to the initial one, with each side l_{1 }= r * L = 1/3L.
8This operation is then repeated for each of the 5 squares. The figure that we obtained is formed by N_{2 }= N^{2 }= 25 squares (red arrow), with each side l_{2 }= r^{2 }* L = 1/9L.
9In the successive passage there are 125 squares (green arrow), and we observe that the chessboard aspect has disappeared, while there is now a hierarchy that manifests itself through a succession of occupied spaces and free space.
10Continuing this iterative application for n times, we will obtain :
11N_{n} = N^{n} squares
12l_{n} = r^{n} * L length of every side
13a_{n} = l_{n}^{2} area of every square
14Calculating the total area and perimeter of the figure we obtain these results :
15A_{n} = N_{n} * a_{n} = (N * r^{2})^{n} * L^{2} = (5/9)^{ n} * L^{2}
16P_{n} = N_{n} * 4 * l_{n} = (N * r)^{n} * 4 * L = (5/3)^{ n} * 4 * L
17Observing the numbers between parenthesis (5/9<1; 5/3>1) we observe that if the number n of iterations were infinite we would obtain a null total area and an infinite total perimeter.
18This behavior is not intuitive, and the fact that the length of a curve grows toward the infinite seems to indicate that it has a dimension greater than one, although it topologically remains a line. These findings do not agree with traditional geometry, and a fractional dimension is introduced (Fig. 1.2).
Figure 1.2: Comparaison between the dimensions of traditional geometry and those of fractional geometry
19The measure of such dimensions is not found in a traditional way, but through the following relation :
20D = log N / log (1/r) 1 < D < 2
21This value is valid for the entire figure, comprising both the area and the perimeter, and disagreeing remarkably with traditional geometry, in which the dimension of a curve is one and two for a surface.
22Measurement methods based on the iteration have been elaborated : they introduce a sequence of measurements of a variable size εand, for every value ε the number of elements N(ε) is determined in order to cover the structure. Many methods have been developed in order to generate concrete algorithms with this theory. More specifically, with regards to urban system analysis, four methods in particular have been used : grid analysis, dilatation analysis, correlation analysis and radial analysis.
23The zone to study is covered with one grid and the size ε of the meshes is varied ; then, for every value ε the number N(ε of meshes is available which contain the occupied points.
24The obvious ambiguity of the result obtained depends on the size of the mesh and the chosen zone (window analysis) in order to apply the grid.
25Each occupied point is surrounded by a square of size ε, the surface of which is considered to be completely occupied. The size of these squares is then gradually enlarged, and the total surface covered at each step is measured. As the squares are enlarged, any details smaller than ε are overlooked and an approximation of the original figure is gradually obtained. By dividing this total surface by the surface of a square, the number of elements N(ε) necessary to cover the whole image is obtained.
26This analysis involves counting the number of occupied points, which lie at a certain distance from each occupied point, and thus the number of correlation. This gives more detailed results about the distribution of occupied points than grid analysis.
27This method refers to a specific point known as the counting center, and gives the law of distribution of the occupied sites around this point. A circle is drawn around this point, and radius is gradually increased. At each step, the total number of occupied points N() inside the circle is counted.
28If the aim is to compare the radial loss of density at different distances from the center, and above all, to identify the changes in fractal behaviour, then bilogarithmic distortion can also be a problem. To avoid this effect, we have calculated for each distance the local value of the slope in the bilogarithmic representation produced by radial analysis. These slope values are represented for the range of distance .
29This mode of representation is especially useful for identifying and measuring changes in the spatial organization of urban structures.
30Founded around 400 B.C. by the Gauls, Milan (Gambi and Gozzoli, 1982) was occupied in 222 B.C. by the Romans, who called it Mediolanum. Milan acquired importance as one of the four capital cities of the tetrarchy of Diocletian's empire. Then Maximian, Diocletian's successor, built up the inner walls of the city, which remained a guide for the urban development of Milan for long time. At the end of the XII century the medieval walls (bastions of earth and poles beside a ditch) were built concentric to the preceding ones. They were then gradually replaced, in the latter half of the XIV century, by embattled walls (in brick). Due to numerous interventions, today, we have lost the traces of the Roman plan. Unlike the Roman plan, the medieval walls with their doors remain visible even today.
Figure 2.1: Roman walls (yellow), Medieval walls (light blue), Spanish walls (blue)
31Still in the XIVth century, the Visconti family built a castle with a square plan, which was then fortified in a northwest zone with regards to the walls. In the following century, the Sforza family reconstructed the castle, thus the origin of the name Castello Sforzesco. In 1546, under Spanish domination, the construction of a new military belt of bastions was undertaken, these walls being external and concentric to the preceding ones, and connected to the castle (figure 2.1).
32Until the end of XIXth century, the population of Milan lived inside the Spanish walls. All the landuse decisions of this period were directed at making Milan a more suitable city to live in : interventions took place on the streets by widening and paving the roads ; on the buildings of the city center by rebuilding and improving the façades ; and finally various areas inside the walls were redesigned.
33Due to the demographic expansion at the beginning of the XX century, it was necessary to formulate a master plan for landuse : in these years the city spread outside of the Spanish walls.
34In the fascist age, the predominant tendency was a radical redesigning of both roads and building systems of the center, in order to have better economic exploitation of the land, in addition to the realization of a complex grid of roads and building structures outside of the walls. The new master plan, designed between 1931 and 1934, covered the town's administrative boundaries almost completely, except for the most southern section occupied by flourishing agricultural firms. Later on, the Agricultural Park "Milan South" will be instituted, in order to restrict urban development in the southern part of the Province of Milan.
35In the period between the two World Wars, the tendency in urban development was to expand toward the outside, until connected to the other villages in the proximity of Milan along the main roads : in this way, the well known conurbation that characterizes Milan appears (figure 2.2).
Figure 2.2 : a summary of the urban expansions since Napoleon epoch a) town in 1800; b) town in 1860; c) town in 1900; d) town in 1940
Figure 2.3 : town of Milan in 1997
36In the last years, urban sprawl has partially stopped, even if development continues following the same patterns of settlement (figure 2.3). This is due to the fact that even if the individual municipalities regulate urban planning, the master plans are not being fully coordinated with a larger plan, which should be the plan of the metropolitan area.
37All the analyses concerning the landuse patterns have been executed by means of the software "Fractalyse", developed under the management of Prof. P. Frankhauser (2003).
38These studies refer to three different aspects :

The global analysis of the territory of the town of Milan, which doesn't concern the municipality alone, but also part of the province around Milan.

The local analysis, that concerns the municipality in its entirety, but also some limited subareas characterizing the evolution of the landuse in the Milanese territory.

The analysis of the perimeter of the city, using the method of dilatation in order to define the boundary and convolution analysis in order to compute the fractal dimension of this boundary.
39The aim of all these analyses is the calculation, using the Fractalyse software, of the fractal dimension characterizing a given area.
40The better techniques for a global analysis are the dilatation and correlation methods ; in fact, different from the radial method, these do not refer to a mobile window of analysis, but they examine the whole urbanized surface. However, these methods have also been used for the study of small areas.
41Use of radial analysis is preferable when it refers to particular zones of the city, for instance the center, the outskirts or distinctive districts. Moreover, it allows us to look at the state of the landuse along a radius departing from the center of the window of analysis. Finally, using the largest admissible window, the value of the fractal dimension for the whole map could be computed.
42Concerning the study of the perimeter, we proceeded with its descent from the given map, through dilatation analysis using the minimum number of expansions necessary to define a reasonable cluster. Then gaussian convolution analysis was used in order to find the fractal dimension. Other methodological aspects concerning the analyses are :

The number of dilatations used has been chosen in order to have only one big cluster, comparing the results obtained with an increasing number of steps.

The shape of the window used in radial analysis depends on the zone considered ; with the window best fitted to the shape of the analyzed area being selected.

The choice of the center of the window of analysis, whether using the baricenter of the figure or a point selected with a pointer, this depends on the structure of the area itself and which part of town we want to analyze.

The size of this window is instead determined by studying the scaling behaviour of the fractal dimension over the whole town map : we look at abrupt variations in the value of α, choosing the value of the window ray corresponding to them. In fact, these variations highlight a breakup in the landuse characteristics, as shown in figure 3.1.
43Scaling behavior
Figure 3.1 : Example of fictitious city

To state the reliability of the calculated D value, a test is made : if the adjustment coefficient (an output of Fractalyse) is less than 0.999, the computed fractal dimension is considered to have a poor adjustment to the theoretical curve, if it is inferior to 0.9999 we have a good adjustment, if it is in the range between 0.9999 and 1.000000 we have an excellent adjustment.
44The local analyses are done for some sections of the original map. For this study we selected the center of Milan, its outskirts, districts that are examples of the rigid planning style of the period between the two World Wars, and finally other zones that have different characteristics from the districts cited above.
45Finally, it is worthwhile to say that all the analyses (the numerical results), are saved and organized in an excel file (Milano.xls). This way they are easily accessible and this file facilitates the task of interpreting the results.
46We recall that the Fractalyse software requires images in tiff format, not compressed, in two colors (white and black). The map, from which we obtained the map of the urbanized surface of Milan, is the socalled Technical Regional Map (CTR) of Lombardy (scale 1:10,000) in raster format. Due to the small scale, the regional CTR is divided in different sheets ; and in order to consider the large metropolitan area of Milan we rejoined 12 sheets together. The CTR includes not only the urbanized areas, but it also displays contours, roads, railroads, and writing and symbols are included too. We therefore removed, by hand, all the details that did not represent the built areas.
47The resulting map measures 30,000 x 30,000 pixels, and is in two colors (2 bit), occupying more than 110 Mb of RAM memory, thus too large for an elaboration with the Fractalyse software. In order to reduce the computation time and to complete the analysis, we reduced the map scale to 1:80,000. A preliminary investigation has shown that the differences between this map and the original one create a difference on the results to the order of 10^{5}, therefore an acceptable difference.
48The analyses of Milan presented in this report refer to a more limited surface with regards to the map just described : in fact, all the local analyses concern the municipality's boundaries, neglecting the towns around Milan.

The first "local section", that is also the biggest, concerns the center of Milan and its outskirts. Note that the radial analysis studying the scaling behavior allows us to divide the surface in circular belts, with a good understanding of the structure of the urban patterns.

Then we analyzed the other three sections, two east of the center and one to the south, showing the characteristics of planning from the beginning of the 1900s (regular networks of streets and geometric districts, very different from the irregular urbanized fabric inside the Spanish walls).

Another area under analysis is MilanoBovisa, localized north of the center, an old industrial zone which has been renewed and serves as an important railway node.

The last zone analyzed is the "Barona" district located southwest of the Milanese municipality, a longtime agricultural area, but now an industrial and residential area (figure 4.1).
Figure 4.1 : Local sections in the Milan metropolitan area
49In this section the results obtained with fractal analysis are introduced and discussed : the global study of the map used, the local study of the sections of the city, and the one on the perimeter of Milan, while focusing our attention on the relationship between the value of the fractal dimension and the city context to which it refers.
50In the tables only the most significant results of the analyses are reported.
51Looking at the map, we immediately notice the presence of a big urbanized nucleus in the center the city of Milan, and one strong dilution in its surroundings. This is due to the phenomenon of the conurbation that has allowed the fusion of Milan with the city centers around it, and it is due to the presence of the south agricultural park, as it has prevented the development in that direction. In fact, the city expansion has happened along main traffic roads and towards the north of the province.
52The fractal study describes the various aspects of the buildings of Milan very well ; next, we will introduce the results obtained with the several analyses carried out (table 5.1).
Table 5.1 : Radial analysis, with a fixed center on Milan
Figure 5.1 : Scaling behavior
53As seen in Table 5.1, the value of the fractal dimension, calculated on the whole map, is very low and equal to 1.075, this implies a strong dilution of the urbanized surface along the radius of the window analysis ; however, the result is not acceptable because of the low value of the adjustment coefficient : the theoretical fractal curve doesn’t fit the real curve of the urbanized surface very well, and this is due to the presence of towns far from the city of Milan and due to big non builtup spaces, which influence the scaling behavior.
54A more indepth study on scale behavior highlights the presence of a constant value of the parameter α, in the initial section, and of increasing or decreasing trends of the represented curve. This allows the characterization of the ranges of the radius in which the fractal dimension is constant and, as it will be shown later, with a good adjustment coefficient.
55Inside Milan, it is homogenous and compact, as shown by the value of D_{r} equal to 1.804. Towards the periphery, the value of the fractal dimension decreases (1.601374) and this means that the radial dilution of the urbanized surface increases : the periphery of Milan is less compact than the center, but its conformation is more dispersed, assuming a tentacular form.
56The external province shows a circular rim around Milan, which is scarcely builtup ; a very low value (D_{r} is under_{ }1) corresponds to this area and this expresses the remarkable effects of the strong suburban radial dilution (François, Frankhauser and Pumain, 1995).
57Then D_{r} returns to high values (1.769): this fact is explained by the presence of other densely populated towns at a distance from Milan, which increase the urbanization level of the province.
58The dilatation analysis of the whole map supplies, with the exception of the radial one, a very reliable result. In fact, the adjustment coefficient is equal to 1 and the corresponding value of the fractal dimension is 1.6628 ( it is obtained with 40 and 60 dilatation steps). As expected, it isn’t a very high number, because the province of Milan has big, nonurbanized spaces ; however, the central urbanized nucleus and the north of the province have an effect on the calculation of D_{d}.
59The correlation analysis shows the presence of homogenous urbanized areas on the provincial territory, but they are not close to one another. The fractal dimension is 1.853897 with an optimal adjustment coefficient (0.999998). Its high value is due to the existing correlations inside the city of Milan, but it is not near 2 because of the distances that separate the cities.
60The first area studied is exclusively the municipality of Milan. Using radial analysis, it is possible to notice that the variations of the scaling behavior perfectly delimit the medieval and Spanish walls in the center of the city, and in the peripheral area outside them.
Table 5.2 : Radial analysis inside the ciyt of Milan
61Inside the medieval walls, the fractal dimension assumes an almost constant value equal to 1.86 with an optimal coefficient of adaptation (table 5.2). The high value of D_{r }indicates that the center of Milan is a densely builtup and homogenous nucleus. Towards the area comprised between the Spanish and medieval walls, the fractal dimension diminishes a little, though always maintaining high values.
62The periphery is less compact than the center and this explains a decrease in the value of D_{r,}however, this value is much too low and it does not adequately explain the variation of the urbanized surface.
63High values are also shown by the other two analyses, the analyses of expansion and correlation, because they are strongly influenced by the presence of an almost homogenous builtup mass.
64The next three sections refer to a precise historical period : in fact, the rigid planning of the early 1900s characterized the evolution of the urbanization process of Milan (figure 5.2).
Figure 5.2 : First example of rigid townplanning
65The radial analysis of this zone shows a poor adjustment coefficient. This is due to big oscillations of the dimension. These oscillations, as observed by the scaling behavior, reduce the number found to a simple average on that area (figure 5.3).
Figure 5.3 : Scaling behavior
66Instead, correlation analysis introduces a value of D_{c} near 2 , this is due to the limited distance between the small masses, constituted by the blocks on the map.
67Dilatation analysis also supplies a high value for the fractal dimension, always testifying a high degree of urbanization.
Figure 5.4 : Second example of rigid townplanning
Figure 5.5 : Third example of rigid townplanning
68Looking at the tables, we observe that the fractal behavior of the several zones analyzed is identical : this indicates that the fractal analyses show the similar characteristics of these zones (figures 5.4 and 5.5).
69The area of MilanBovisa (figure 5.6)shows a very untidy course of the urbanized surface, just like the center of Milan, it has a high degree of aggregation, and even if there are non builtup zones that diminish such an index. Today, on this former industrial area, there are denselyinhabited dwellings, industrial buildings, schools, university centers, and the Bovisa FN railway node.
Figure 5.6 : MilanBovisa area
70The radial analysis of this zone doesn’t supply an univocal result. In fact, the value of the fractal dimension varies between 1.86 and 1.91 depending on which portion of the map is studied. Moreover, the scaling behavior does not follow a linear course, thus testifying the effective disorder of the urbanized surface.
71With correlation analysis we obtained a value of D_{c }that is inferior to the one found for the center of Milan, this is due to the nonbuild up spaces. However, both the correlation and dilatation analyses supply enough elevated values for D, thus confirming a high level of urbanization.
72The section representing the Barona area (figure 5.7) shows a strong concentration of buildings in a small zone. This area is separated from those adjacent to it by large streets that delimit it.
Figure 5.7 : Barona area
73The radial analysis shows the presence of a densely urbanized nucleus at the center of the figure, while extending the study outside of this one, we observe a decrease in the value of D_{r} due to the presence of roads and a different course of urbanization.
74The other two analyses also supply enough high values of D, however they are not near 2, due to the presence of empty spaces around this block. The value of D_{c} is probably higher because it is mainly affected by the existing correlations in the center of this zone.
75The perimeter required for the analyses has been extracted by the expansions from two maps with different scales. With the map at a scale of 1:80 000, 6 dilatation steps were needed in order to obtain a reasonable cluster, while only 3 steps were necessary with the map at a scale of 1:160,000.
76To study the perimeter we used gaussian convolution analysis, with a value of variance equal to 0.02 and 20 steps of convolution.
Table 5.3
77Both calculated values of the fractal dimension are very low, this implies that the perimeter of Milan does not have great ramifications : its variations are contained in a small circular ring around a compact nucleus. It shows neither the external extensions towards the province, nor the inner extensions that can reach the center of Milan.
Figure 5.8 : Perimeter of Milan
78Fractal analysis is a useful instrument to study the spatial organization of urban patterns. However, the results obtained must be regarded in a comparative perspective : the single value of the fractal dimension does not supply sufficient information in order to describe the urban development of a city. This is possible only with a set of fractal indices, which can be studied and confronted together, in order to have a general view of urbanization. An indispensable instrument is scaling behavior (used with radial analysis), which supplies information on the actual variation of the fractal dimension along the radius, showing the variations of such dimensions characterizing zones with different builtup areas, and the limits of homogenous areas on the map.
79Regarding the analysis on the metropolitan area of Milan, the values of fractal dimension have not disregarded the reasonable forecasts made on the course of the city : the city seems to be very compact and homogenous, and the values found respect this development, in fact, they are almost always over 1.8. The only problem found is imputable in the presence of particular areas, in which scaling behavior shows frequent and high oscillations, thereby reducing the fractal dimension calculated to an average in that area, with a poor adjustment coefficient.