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Fractal analysis of the urbanization at the outskirts of the city: Models, measurement and explanation

Analyse fractale de la périphérie urbaine : modèles, mesures et explication
Apostolos Lagarias


Cet article vise à analyser l'urbanisation à la périphérie des villes et des aires métropolitaines  considérées comme des exemples par excellence de structure fractales complexes révélant des processus dynamiques de croissance et de transformation.  Les méthodes d'estimation de la dimension fractale sont présentées.  Une attention particulière est portée à l'interprétation.  Les dimensions fractales sont utilisées comme des indicateurs de l'étalement urbain, révélant le degré de fragmentation du paysage urbain.  Le champ d'application est la périphérie de Thessalonique (Grèce).  Des changements importants  y ont été observés entre 1960 et 1990; ils sont discutés ici à l'aide de l'analyse fractale.

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Notes de l’auteur

I wish to thank my PH.D. Supervisor, Tsoulouvis Lefteris professor of the Aristotle University of Thessaloniki, for the interest he has shown in my work. The discussion and the ideas presented here belong to the author and I apparently accept the responsibility for any deficiencies.

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1The issue of treating cities as complex and emergent structures has been raised during the past two decades, posing serious questions and casting further light upon the understanding of the urban phenomena. Cities are considered to be systems of organized complexity with a distinct hierarchical order. The application of fractal geometry at the analysis of the urban development patterns has been widely investigated during the last two decades, together with the cellular automata and the agent-based models (Batty 2005, Benguigui, Chamanski, Marinov 2001).

2Based on the recognition that urban patterns are highly complex, heterogeneous and hierarchically ordered revealing self-similarity across scales, numerous models of fractal analysis have been applied at the study of the urbanization processes. Fractals are by definition complex, hierarchically ordered structures revealing self-similarity across scales presenting remarkable similarities to urban built-up patterns, urban boundaries, land use distribution etc. Fractals have been widely used in order to distinguish between different urban typologies (Frankhauser 1998), to measure the degree of urban sprawl and examine the way cities expand in space and time (Batty & Longley 1994). Although researches on the field are recent and certainly much remains to be done, the results coming from various application all over the world confirm the fact that urban systems present a fractal structure: It is therefore through this new geometrical perspective that the structural and functional properties of these systems should be described, and new properties which until now where considered too complex to be analyzed should be examined.

3The box-counting method and the radial analysis are two commonly applied models calculating the fractal dimension of an urbanized area (Frankhauser 1998, Benguigui et al 2000). Important attention has also been given to the analysis of urban boundaries (Batty & Longley 1994) which have been considered as fractal-like structures and are related with urban sprawl at the edges of the city (Frankhauser 2004). The fractal dimension is thus considered an extremely useful indicator of the urban spatial structure (Anas, Arnott, Small 1998), used both at the study of the perimeter of the city, and at the study of whole urbanized surfaces, while it’s transformation through time reveals important aspects of urban transformation.

4The objective of the present study is to explore the basic relations between fractal analysis and the urbanization at the outskirts of the city. It is widely accepted that the urban structures at the edges of the metropolitan areas are characterized by dynamic processes of growth, revealing a high degree of complexity and heterogeneity. Those areas can be considered the examples par excellence of complex, fractal urban structures constituting a promising field of inquiry for the new geometry.

5Recently, the accentuated growth of the outskirts of many metropolitan areas has led many to recognize their importance, referring to the emergence of edge cities (Garreau 1991) and polynucleated urban landscapes (Batty 2001, Soja 2000). In European cities the main tendency is the reinforcing of existing peri-urban settlement structures leading gradually to more decentralized patterns (Batty 2001). The application of fractal geometry to the study of these rapidly growing areas can provide a valuable perspective and lead to an explanation of the urban transformations.

6In the present paper the two basic models (box-counting method and radial analysis) and their relation to the analysis of urban development patterns are presented. Ways of evaluating the relationship between the fractal dimension and the degree of fragmentation, the distribution of the land uses and generally the structure of the urban area under investigation, are the main focus. The exploration of these notions is carried out through an application of the box-counting method at the outskirts of Thessaloniki, Greece. The area is located at the north-west sector of the city’s outskirts, characterized by rapid growth. The transformation of the area between 1960 and 1990 is analyzed, while some further current trends are discussed. Finally the evaluation of the fractal analysis is presented and the results are compared with other data concerning the specific area.

Fractal models and the measurement of urban structures

7The most important characteristic of a fractal object is that it possesses a motif that repeats itself on an ever reduced scale. Fractals are by definition complex, hierarchically ordered structures revealing self-similarity across scales (Batty & Longley 1994). The fractal dimension D is a basic parameter of a fractal, revealing important aspects of its structure, as it quantifies the degree of irregularity or fragmentation. It also indicates the level of complexity or the amount of details through scales and describes the distribution of the mass around a center (Sobreira & Filho 2002).

8Many ways for relaxing the strictly self-similar, deterministic fractals have been suggested, in order to relate the fractal concept with urban forms and urban systems. First it has been recognized that urban patterns reveal the property of self-affinity rather than self-similarity (Batty 1994) and thus the fractal dimension is estimated statistically. It has also been shown that sometimes the urban built-up patterns can better be described as multi-fractals or as the overlay of different fractal patterns (Frankhauser 1998). In this case the value of D can change across space and scale and also through time (Lagarias 2005).

  • 1 Μany other methods have been devised such as the dilation, correlation and structured walk algorith (...)

9There are two basic ways for examining a fractal structure, either by examining changes in form by varying the scale, or by keeping the scale fixed and varying the size (Batty & Longley 1994). The basic models used to calculate the fractal dimension are referred to as the box-counting method and the radial analysis1. The approach is based on a binary logic that distinguishes between the urbanized and non-urbanized area, and describes the distribution of the built-up area on the surface that contains it.

10The box-counting method is based on a grid analysis of the structure. The object is covered by a grid made of squares of size l, and the number N of squares in which a part of the object appears is counted. The size of the grid is varied progressively, usually through a sequence of grid lengths equal to li= L/2i and the scatter diagram of l against N is presented. The relation that is expected between the two variables is expressed as N=(L/l)D (1) where D is the fractal dimension and L the total size of the grid. The estimation of D is based on a logarithmic transformation of (1) which takes the form LogN = a – Dlogl (2), where a is a constant and is used as the basis for linear regression.

11The method can be applied both at the study of whole urbanized surfaces and at the study of inner and outer urban boundaries. The value of the dimensions is estimated in the range 1<D<2 (Batty 2005). The relation between the fractal dimension of a surface (areal fractal dimension, Anas, Arnott, Small, 1998) and the corresponding boundary can be understood if we consider a perfect homogeneous circle: The value of Da (dimension of the surface) is equal to 2 while the value of Db (dimension of the boundary) is equal to 1. The more fragmented a pattern appears, the smaller is the value of Da and the greater of Db. The fractal dimension therefore describes how the built up area is distributed on the surface, and how dense or fragmented is the spatial patterning of the city. It has been shown through various studies that the box-counting dimension of the scattered, irregular urban patterns at the edges of the city present a lower value, especially on the first stages of development. Progressively the fragmented structure is transformed through processes of coalescence, while the dendritic outline border is smoothed. The quantification of those processes provided by the box-counting fractal dimension offers important insights on the analysis and explanation of those transformations.

12Radial analysis refers to a specific point known as the counting center, and gives the law of distribution of the occupied sites around this point. The scale of the analysis is fixed and we vary the radius R of the distance from the center. At each step, the total number of occupied points N inside the circle is counted and it is expected that Ν = a RD (3).

13In the case of a homogeneous, uniform distribution D is calculated close to 2, while the more concentrated is the mass around the center the closest to 1 is the value of D estimated. The radial analysis offers therefore a convenient way to examine the hierarchical organization of an urban area and to determine whether the structure is monocentric or reveals a greater degree of decentralization.

14The two methods do not give identical results when applied over the same area, as a cause to the fact that the urban patterns are not equivalent to the theoretical fractals and often display properties of multi-fractals. Each one however measures a different attribute of the area under investigation and therefore they should be applied in combination.

  • 2 As the case study does not treat the city of Thessaloniki as a whole but is limited to the northwes (...)
  • 3 Outskirt areas which have been developed by an organized plan and with a great control in their bui (...)

15The case study presented here concentrates on the box-counting dimension of the built-up area and the corresponding boundary of the northwest sector of Thessaloniki2. Tables 1 and 2 therefore present the basic theoretical relationships between those fractal dimensions and the urban context to which they refer. In order to explore the divergence of values in space we can state that central, densely built-up zones of a city display a greater value of Da, (Frankhauser 1998) while the peripheral a lower value, especially on the first stages of their development3 (De Keersmaeker, Frankhauser, Thomas 2003), as a cause to their irregularity and fragmentation. An increase in the value of Da in an outskirt area indicates a denser, homogeneous pattern of development with a fill-in of the existing vacant land. On the contrary a decrease in the value of Da indicates a process of urban growth characterized by fragmented, leap-frog development. The inverse is expected for the dimension of the urban boundary.

Table 1: Change of values in space

Table 1: Change of values in space

Table 2: Change of values through time

An application of fractal analysis at the outskirts of Thessaloniki

16Thessaloniki is at present a large metropolis, the second in the hierarchy of Greek urban areas with a total population that exceeds a million if we include the residents of its wider hinterland. The form of the city has been extremely compact until approximately the end of the 19th century, while during the 20th century the city went through rapid urbanization expanding almost in every direction and forming a complex and dynamic urban system.

17The particularity of the geographical setting, with high wooded hills on the north and the sea on the south excluding urban development, limited the expansion towards the west on the large plains and the smooth hills on the east. An important particularity was the creation of numerous spontaneous settlements on the eastern and western areas forming a disconnected network around the central area. The gradual incorporation of these areas in the city plan and the rapid urbanization of the whole area led to a complete transformation and created a dense, unified urban area with expanding boundaries.

Figure 1: Satellite image of Thessaloniki showing the area under investigation located at the North West sector.

18The area chosen for the analysis is located at the North West sector of the city containing the municipalities of Stauroupolis and Polichni, and also a part of Euosmos and Efkarpia (figures 1, 2). The area is close enough to the center and connected to it by the major transportation axis of Lagada Street, while its furthest part is located at the edge of the city’s administrative boundary, close to the inner ring road. A densely populated area during the last decades, it has gone through major transformations since the middle of the century when it was a low density, suburban area characterized by the growth of spontaneous settlements in areas not included in the city plan.

Figure 2: The location of the municipalities and communities of the area in relation to the CBD.

  • 4 The analysis was carried out by a combination of the software programmes, Fractalyse 2.3.2, Photosh (...)

19The analysis was carried out by comparing the patterns of development of the years 1960 and 1990. As a basis for the estimation of the fractal dimension4 two maps of the built-up area were used (figures 3, 4). The complete absence of maps of the urban development of Thessaloniki has made necessary the extraction of the urban built up area directly from a combination of aerial photos which where digitized, transformed to the same scale and cropped to display exactly the same area. The resolution of the images was set to 300 pixels/inch, while the total size of the image is 1270 X 1131 pixels.

20A threshold filter was applied to the photo which was transformed into a binary image. Considerable attention is given to the parameters of the filter so that all buildings are included as black pixels in the final image. However the filter distinguishes the buildings only to certain degree and some surfaces such as roads and other impervious surfaces are also included. The classification was then checked and the basic errors are corrected manually, with the exception of roads inside the residential blocks which can be considered as ‘urban land use’. This does not seem to alter in any way the basic results of the analysis (Sobreira & Filho 2002). Maps of the corresponding urban boundary were also used which were derived through a dilation of the image and extraction of its outline (Frankhauser 2004).

Figure 3: Aerial photo and the extracted built-up area in 1960.

Figure 3: Aerial photo and the extracted built-up area in 1960.

Figure 4: Aerial photo and the extracted built-up area in 1990.

  • 5 The values of N for grid size over 64 were excluded from the estimation of D. This is a common tech (...)

21The box-counting method was first applied at the map of 1960 using a sequence of grid sizes 1-2-4-8-16-32-645. The software ‘Fractalyse’ was used and the method ‘box-testing’ was chosen which consists in finding the least number of square of size ε needed to cover all black pixels. The number of iterations wad fixed at 10.

22A linear regression was performed on the logarithmic plot of N against l (where N is number of grid cells in which a part of the object appears is counted and l the corresponding size of the grid cell) and the fractal dimension was estimated equal to D=1.466 with a constant a=11.635. Therefore the law of distribution of the developed urban area across scales is given by the function:

LogN= 11.635 – 1.466logl

23It must be noted that the reliability of the calculated D value is examined by the correlation coefficient r, which in the present case is 0,9989. The value of the coefficient can be considered good if we take into account the suggestions of M.Batty (2005) and Benguigui et al. (2000) giving. We can therefore state that the urban pattern under investigation displays clearly a fractal structure.

24A further examination of the structure was carried out through the extraction of the urban boundary and the calculation of its fractal dimension. The map was dilated incrementally 2 times and the outline was extracted, as presented in figures 5, 6. Attention was given so that through the dilation procedure one united cluster was obtained in the compact area of the city. The inner boundaries created by the significant empty areas between the settlements were removed before the estimation. The fractal dimension was estimated using the box-testing method equal to Db = 1.328 with a correlation coefficient equal to 0.988.

Figure 5: The urban boundary in 1960.

25The same calculations were applied to the map of 1990 giving a fractal dimension equal to 1.741 and the constant factor a=13.054. The law of the distribution across scales is LogN=13054 – 1.741logl, while r equals to 0,9936. The increase in the value of Da implies an important change of the form of the urban area, which will be discussed later. The same technique was applied for the extraction of the boundary of 1990 and Da gave a lower value equal to 1.199 with a correlation coefficient r = 0.981. Therefore, the change of the dimension of the urbanized surface is followed by a change of the dimension of the boundary.

Figure 6: The urban boundary in 1990.

  • 6 The issue of the minimum value of correlation coefficient necessary in order to rcognize a steuctur (...)

26In a discussion of the results of the models it must first be stated that the urban patterns examined display a clear relationship with the fractal structures. The application of the fractal models at the case of Thessaloniki gave thus results that conform well to existing theory. The scatter diagrams show a strong linear association between LogN and Logl and therefore we can consider both the structure of the built-up area and of the boundary, fractal6. The change of Da and Db follows an inverse logic and the increase of Da through time is consistent with the results of various applications all over the world (Benguigui, Chamanski, Marinov 2000, Shen 2002).

Towards explanation: Fractal analysis and the transformation of the area under investigation

27The focus of the final part is in the interpretation of the results and the explanation of what has happened in the area under investigation. The sector of the city analyzed here is part of the west suburban area. A rapid urbanization process took place during the middle of the century mainly due to the proximity of the industrial zone, the harbor, and the main transportation axes of the city. The largest part of the area has been built illegally through a process of land colonization by rural migrants settling at the outskirts of the city (Tsoulouvis 1985). There was always a belated expansion of the city boundary, and large areas where urban growth had already taken place, were gradually incorporated in the city plan (Triantafillidis 1968).

28Taking into account the results of fractal analysis, we can conclude that the distribution of the built-up area was highly heterogeneous and fragmented especially in the year 1960. During the period 1960-1990 significant changes led to a remarkable increase of density on every scale leading into an increase in the value of Da from 1.466 to 1.741. What can further be stated is that the fragmentation and heterogeneity of the urban development were reduced, while the scattered settlements of 1960 coalesced into larger clusters. The form of the urban boundary confirms the above conclusion as its fractal dimension reduced from 1.328 to 1.199. The urban boundary has been extended to areas that in 1960 were empty especially in the northern part of the area around the intersection of Lagada Street and the Ring Road. We can mark two basic processes of urban growth:

  • The further sprawl of the built-up area at the areas that surround the development of 1960.

  • The filling of many empty spaces inside the urban fabric, leading to a nearly saturation of the more central parts of the area, and a considerable increase in density.

Figure 7: Population change for the municipalities of the area from 1940 to 2001.

29The examination of the population size of the municipalities in the course of the 20th century (1940 – 2001), accords with the above conclusion. As we can see in figure 7 the population of the municipalities of the area has risen rapidly according to the census data of ESYE (National Statistical Service of Greece). Important has also been the rise in the number of buildings which is presented in figure 8. The area was transformed from a spontaneous low-density outskirt into a more central, denser sector of the city. The fractal analysis provides important insights on how this urbanization process took place in a spatial sense and quantifies the distribution of the new development.

Figure 8: Change in the number of buildings for the municipalities of the area from 1919 to 1990.

30Another issue, which has not been analyzed by the models presented here, was the significant change in the heights of the buildings during the period of 45 years. The small, one or two floors height houses, where replaced by buildings up to 4 and 5 stories height, especially in the more central parts of Stauroupolis and Polichni. This change explains the enormous shift in the residential density over the whole area.

31The rates of population growth decreased during the last decades, and a move of population from the inner suburbs towards the newly incorporated areas near the urban boundary is observed. However, the growth of the built-up area is continued leading to a considerable rise in density and the almost complete fill-in of every available empty land. In the same time the urban sprawl leads to a further expansion of the developed area beyond the inner ring road and the growth of settlements in the peripheral zone of Thessaloniki. Therefore the conclusion that the urbanized area became denser and more homogeneous holds only for the area that was analyzed. If a larger field is considered, further sprawl towards the outer ring road and the rapid urbanization of the peripheral settlements can be observed (figure 9).

Figure 9: Recent photo of the area under investigation and the zones that surround it.

32There is another way we can extend our analysis and relate the fractal dimension with other, more general data concerning urban growth. As we have already noted, the fractal dimension varies in space, and different parts of an urbanized area are characterized by different dimensions. We can therefore classify the different parts of the area by grouping them according to their box-counting dimension. It is obvious that the area under investigation contains sub-areas of different type and it is important to make a distinction between the more homogeneous, central parts and the fragmented development near the urban boundary.

Figure 10: Estimation of fractal dimensions at different parts of the area. The results are displayed by an appropriate color range from blue (lower values) to yellow (greater values).

33By dividing the area into sub-areas of 150 x 150 pixels of the maps –the box-counting was applied estimating local fractal dimensions. The results are displayed at figure 11 where the range of values corresponds to a color range.

34As the maps in figure 10 illustrate, the differentiation of the values of D is more important in the first period where the parts closer to the center of the city (upper part of the map) are more dense and homogeneous, while the areas with proximity to the external boundary (lower part) are extremely fragmented and heterogeneous. On the contrary, in 1990 the different sub-areas are more similar and the homogeneity of the whole built-up area is expressed by a fractal dimension which varies less across space.

  • 7 Technical Chambers of Greece

35The different character of those areas can also be explored through census data concerning population density, building heights, ratio of incorporated - unincorporated areas and permitted plot ratios. By referring to data of the year 1979, when a systematic research was carried out by a working group of TEE7 (Tsoulouvis 1981), a distinction between the different parts of the area can be observed. The more distant areas at the edge of the boundary remain completely unincorporated, even though they present considerable residential development. There is a considerable higher density (number of residences per ha) in the central parts of the municipalities with a maximum of 178.26, while near the boundary the value varies between 16 and 50 people per acre. Moreover the saturation degree of the area (calculated by dividing the constructed by the permitted plot ratios) ranges from 0.4 to 0.85. We can therefore understand that the urbanization process was still on the way in 1979, and continued until a higher level of saturation was reached. All the areas were gradually incorporated in the city plan, and the high plot ratios that were established launched a wave of new development.


36In the present paper, fractal analysis was considered as a method of examining the transformations taking place in the urbanized areas located in the outskirts of the city. What was basically examined was the built-up pattern of the city and the main objective was the determination of whether its structure can be described as dense or scattered, compact of dispersed. By presenting a case study for the city of Thessaloniki and comparing two different periods, we have been able to observe a process of urban intensification, which was reported by an increase in the value of the fractal dimension of the urbanized area and a corresponding decrease in the value of the urban boundary. By estimating these changes we succeeded in measuring in an accurate way the transformations that took place in the area under investigation and its transformation from a low density, fragmented, system of peripheral settlements to a compact unified urban cluster. Moreover the box-counting analysis revealed a fractal character of the urban patterns where similar structure is observed at many scales, verifying what has already been reported in other researches (Batty & Longley 1994, Frankhauser 1998, Bengugui et al. 2000).

37The importance of the method proposed here is that it provides a concrete way of measuring the urban structure and its transformation through time. Many recent researches have also explored similar ways of measuring the dynamic urban morphology. Shen (2002) for example compared the morphology of 20 urban areas in USA obtaining a wide range of results due to the different size and character of each case study. P.Frankhauser (1998) has also used the fractal dimension in the examination of outskirt areas in European cities trying to obtain a typology of urban agglomerations. Finally Benguigui et al. (2000) by examining the built-up patent of Tel Aviv conclude that the fractal dimension tends to increase through time. In the present analysis we have confirmed this trend despite the fact that the field of analysis was limited to an outskirt area and not the whole metropolitan area.

38The case study presented here offers some new evidence on the above issues due to the particularity of the area under investigation: Being at first a low density system of spontaneous settlements, it was gradually incorporated in the city plan almost reaching a saturation level. This example contrasts with the sprawling outskirts observed in many metropolises of the USA, where a low density development is reported. In the present analysis evidence was also provided on the relation between fractal dimension and population data. We have seen that the rise in D is followed by an impressive increase in population numbers and in the density of development. This fact further confirms the observed process of intensification, leading to a highly compact urban structure.

39The rapidly changing outskirts of metropolitan areas can therefore be examined through the estimation of fractal dimension and the change of its value through time. However the study is preliminary and highly suggestive and its main objective was to present some ways of applying the fractal models to the analysis of urban areas at the outskirts of cities. Many extensions are possible: The need for the presentation of more time periods and a comparative analysis between many outskirt areas is obvious, while the application of radial analysis and the focusing on the variation of fractal dimension across space and through time constitutes a major path of further investigation.

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1 Μany other methods have been devised such as the dilation, correlation and structured walk algorithms

2 As the case study does not treat the city of Thessaloniki as a whole but is limited to the northwest suburban sector, the radial analysis was not applied.

3 Outskirt areas which have been developed by an organized plan and with a great control in their building process, display high values just like central areas. We can state therefore that D is not equivalent to density but takes also into account the distribution and the structure of the patterns of development

4 The analysis was carried out by a combination of the software programmes, Fractalyse 2.3.2, Photoshop 7.0, Image J and the plug-in, Fraclac 2.2.

5 The values of N for grid size over 64 were excluded from the estimation of D. This is a common technique (Batty 2005) for at the grid sizes over a certain limit we obtain a rather poor approximation of the object.

6 The issue of the minimum value of correlation coefficient necessary in order to rcognize a steucture as fractal appears to be rather controversial. L.Benguigui et al ( Environment and Planning B: Planning and Design 2000, volume 27, pages 507 - 519) proposed a smaller minimum for the determination of a fractal structure by taking r2 = 0,996 (and therefore r = 0,998), while according to M.Batty (2005) even smaller values can be accepted.

7 Technical Chambers of Greece

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Apostolos Lagarias, « Fractal analysis of the urbanization at the outskirts of the city: Models, measurement and explanation », Cybergeo: European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 391, mis en ligne le 16 juillet 2007, consulté le 28 janvier 2023. URL : ; DOI :

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Apostolos Lagarias

Ph.D. Candidate
Division of Urban and Regional Planning, School of Architecture, Faculty of Engineering, Aristotle University of Thessaloniki, Greece

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