The application of fractal analysis and spatial technologies for urban analysis

Michael A. McAdams[1]

Abstract

Urban areas and their built form are the result of complex cultural, social, economic, and technological agents/processes and their physical environment. Traditional forms of analysis (i.e., linear modeling, statistics) have wrestled with the chaotic nature of urban systems and the urban built form with an increasing amount of apparent sophistication (i.e. additional algorythms, new models) often with no significant increase in accuracy or understanding. Fractal analysis, which is firmly grounded in chaos theory, allied with spatial technologies (i.e., Geographic Information Systems, Remote Sensing, spatial modeling) is proving to an innovative technique to study urban form/structure. This paper discusses how fractal analysis can be used to evaluate urban form. The findings of those that have applied fractal anlysis to study urban form, including the author, have concluded that the fractal dimenions of urban form are not infinite, but settle in a defined fuzzy range of values based on the distinct characteristics of urban processes. The amalgamation of fractal analysis and other chaotic analytical tools (i.e., nueral networks, agent based modeling) with spatial technologies to study urban form and dynamic forces represents a major paradigm shift that is just begining to be observed in urban analysis and urban planning.

Keywords: chaos theory, fractal analsyis, built environment, spatial analysis, urban geography, urban planning, spatial technologies

[1] Geography Department, Fatih University, 34500 Büyükçekmece / İstanbul, Türkiye. E-mail: mcadams@fatih.edu.tr

Introduction

To a growing number in the scientific community, chaos and complexity theories represent a major paradigm shift in understanding numerous phenomena [8] [25] [13]. In the field of urban geography, these developing areas could have a major impact on its direction. Michael Batty in his seminal book, Fractal Cities [3] revealed the promise of using fractal analysis in studying cities. Recently, he explored agent based modeling and cellular automata in Cities and Complexity [1] . The combination of these methods are ‘unraveling the urban fabric’ and giving urban geographers, urban planners and others interested in urban development greater insights. Fractal analysis has revealed that urban areas have patterns which can be distinquished, measured and dissected quantitatively, revealing the simplicity and complexity of their geometric formation. Spatial metrics is also contributing to our knowledge of the geometry of urban areas and intergrating with cellular automata and fractal analysis [1] [9]. The analysis of urban areas combining chaos related tools and spatial analysis software is proliferating such that there is a growing number of urban geographers which are utilizing them to study different aspects of urban development.

Urban processes are natuarally chaotic. Urban analysis have traditionally used linear and standard statistical techniques. These techniques have been found to be significantly flawed for the study of different aspects of urbanization such as growth, transportation, economic development. Chaos theory and other ‘new’ methods of analysis such as cellular automata, agent based modeling, spatial metrics, artificial intelligence, nueral networks, non-linear simulation and fractal generation represent means to study urban phenomen from the ‘bottom up’. These new developing methods of analysis are challanging the tenents of logical-positivism and associated methods of analysis such as statistics and aggregate linear modeling techniques. The ability to analyze the processes of urbanization has been greatly inhanced by the increasing abililty to collect and process geographic information that is now found in the spatial technologies such as GIS and Remote Sensing.

Fractal Analysis and Urban Analysis

How did fractal analysis become a part of urban analysis? Fractal analysis, grounded in mathematical theory, developed independently of traditional urban analytical techniques. Urban geographers, being involved in a discipline which is inherently inter-disciplinary, began to realise that the attribues of abstract fractals and fractal generation were complementary in the ınvestigation of urban form. Michael Batty [3] was the first to comprehensively examine the use of fractal analysis to investigate urban structure. Batty further developed these ideas in his book Complexity and Cities [1] , incorporating complexity theory and related fields such as cellular automata and agent-based modeling. These areas have been facilitated by Geographic Information Systems, Remote Sensing, the development of advanced spatial analysis tools and increased computer processing capacities. The areas of complexity and chaos theory along with related spatial analysis techniques such as agent-based modeling, simulation, and cellular automata have a rapidly developing literature in urban geography and spatial analysis, carrying with them the potential to transform the traditional areas of examining cities such as applications of regression analysis, econometric models etc. Although some may disagree, this author perceives that this is beginning of a major paradigm shift that is not only apparent in Urban Geography, but in other disciplines. To introduce the basis for fractal analysis, the next paragraphs will discuss its development and some of the major tenents. However, it should not be considered a comprehensive analysis of this subject, but an overview.

Some of the elements of fractal analysis were first introduced by D’Arcy Wentworth Thompson in his book On Growth and Form [22], indicating that although Mandebrot [12] is credited with developing fractal mathematics, the roots are obviously streaching back much further. Mandelbrot [12] introduced the basic concepts of fractals and fractal analysis such as self-simularity, multiple iterations of simple formulas, and scaling and dimenisions. Now, fractals and their analysis are robust areas of study in mathematics, but a still developing sub-field.

The basic elements related to fractal creation/generation and analyis are: 1) an object; 2) a generator (initiator); 3)iteration and 4) the emerged form. Fractal analysis is concerned with the study of the emerged form. For example, the object may be a line. This line may be subject to different rules or a generator (i.e, divide the line by a third) and iterated for an established amount of times. At the conclusion of those iterations, which theoretically could be infinite, an emergent form would appear. The resulting form/object will not be the same as the original object as it is not a replication but a mutation. However, the paradox is that the abstract fractal actually has self-similarity and scalelessness so in essence it is a replication, but, is mysteriously different. The best example in nature would be a tree whose structure was initiated by the bifurcation (dividing by 2) of a single twig and repeated numberous times. Cell development or mutation is similar. Another definition of a fractal is that it is an object which is less than a plane and more than a line possessing self similar elements. It would follow that fractal analysis is the study of the characteristics of fractals. Theories of simplicity, complexity, chaos theory and analysis techniques such as agent-based modeling and celluar automata are intrisically coupled with fractal analysis. The ‘butterfly effect’ is the most simple explanation of the chaos theory. Thie is the concept which states that the single action of a butterfly flapping its wings in South America might result in a hurricane developing in the Atlantic Ocean. This concept of one action repeating and mutating is also one of the basic tenents of fractal generation and chaos theory. James Glieck, himself a mathamatician, gives a more detailed explanation of the history of the development of chaos theory and also fractals in his entertaining book, Chaos: Making a New Science [8].

Fractals can be analyzed in a number of manners. One of the most common is to examine its dimension, lucunarity, and scaling [6]. Dimension refers to the fractal variation or how the fractal fills the space. The dimension for a fractal is always between 1 and 2 with 1 being a line and 2 being a plane. Dimension for a point would be 0 and for a cube, 3.

The forumla for calculating a fractal dimension is as follows:

Dimension (Dim) = log (number of self-similar pieces)/log (magnification factor) [5] Or Dim = - (log Nk/log rk) = log Nk/log (1/ rk)[3]

Whereas the negative of the logarithm of the number (N) of self similar objects (vector or raster) is divided by the log of the scaling ratio ( rk). Therefore, for a square which is divided into 4 equal parts and a scaling dimenison of 2, the dimension would be 2.

Fractal dimenison can be better analyzed when combinded with lacunaity. Lacunarity refers to the texture of a fractal. A fractal with more gaps, bays or tears has a higher lacunarity. Tolle et al. [22] state, ‘Fractal dimension only measures how much space is filled. Lacunarity complements fractal dimension by measuring how the data fills the space.’ Tolle et al. [23] further state that muliple fractals may have the same dimension, but will have different lacunarity.

While there are multiple types of fractals that can be generated, only a few can be used as models for the city (i.e., Seripenski Carpet, diffusion.) This demonstrates that although the city is complex, there are limited forms that are being manifested in the evolultion of urban areas. ‘Real life’ urban fractal dimensions can be compared against abstract fractal objects (see following figure) as a standard to determine the simililarity or difference of different urban forms. Abstract fractal objects which could be seen as the basis for examining cities are the Dendrictic Pattern (a), Sierpenski Carpet (b) and Sierpenski Triangle Variation (c) [21]. The Sierpenski Carpet is similar to a regular gridded city. Its fractal dimension is 1.77 and luncanrity is 292.08 This indicates a pattern that is highly regular, but containing a large amount gaps or ‘bays’. The Dendrictic Pattern, similar to the growth of urban development along transportation lines, has a fractal demenison of 1.60 and a lucanarity of 29.91. The Sierpenski Triangle Variation has a fractal dimension of 1.65 and lacunarity of 17.35. This also indicates a dendrictic type pattern and small bays. The closer to a the dimenision to linear demenision indicates more hierarchy and the small lucanity means smaller gaps or ‘bays’.These dimension and lacunarity of abstract fractal can useful as a means to compare the different fractal of cities.

Abstract Fractals—a- Dendritic Pattern b- Sierpenski Carpetand c-Sierpenski Triangle Variation

In the urban environment, as opposed to a theoretical one, ‘real’ urban fractals are also subject to the influence of physical topography in limited and shaping its direction and sometimes form. The fractal formation in an urbanized area is not equal in the way it developes over time. The economic and technological agents effect the manner in which the fractal grows or diffuses. In addition, there are entropy factors based on economic functioning of the city. In a polycentric city, the distance from the core is related to the ability of commercial and industrial concentrations to form on the perifery of the urban agglomeration [16]. Some of these issues are being taken up in the application of agent based and cellular automata modeling of cities.

Comparative Fractal Analysis of Cities

There are two objectives when analyzing the fractal dimension and lacunarity of urbanized areas. One is to examine the characteristics of a particular city and the other is to compare it with that of other cities. There have been several studies which examined the fractal aspects of urbanized areas. Batty [3] conjectured that all cities have fractal dimensions between 1 and 2. In this spectrum, there are high amounts of variability depending on the time period of the analysis. Batty [3] stated that most of the values were greater that 1.4 and most between 1.6 and 1.8 with a mean of 1.7. However, this was for a global analysis of the entire urban area. Frankhauser [7] took a different approach and compared cities with idealized geometric structures. He stated that, ‘the value of the fractal dimension of the occupied sites is directly linked to the parameters of the generator, in particular to the number of N of elements and the reduction factor..’ [7].

In examining French and other European cities, Frankhauser [7] found that city centers were between 1.8 to 1.95, regular estates without public space were between 1.8 and 1.99, new town between 1.6 and 1.77 and irregular or less controlled growth between 1.64 and 1.85. It should be noted that these measurements are not mutually exclusive, having a significant amount of overlap in some cases. In a study of Milan [4], a global analysis indicated a low fractal dimension of 1.075. Near the periphera of the urban area, the fractal value was 1.601, but near the center it was close to 1.804. This is consistent will other values found by Frankhauser [7]. In a study by Lagarias [10] of Thessaloniki, Greece the dimension of a selected suburban area was 1.741 McAdams [14] in his analyis of Istanbul found that its fractal dimensions when compared to European cities were similar but slightly higher at 1.83. In North America, the fractal dimensions of city were investigated by Shen [19] . He stated, concerning past studies using fractal analysis of cities, that ‘While these studies have provided some interesting theoretical formulations and empirical results revealing the fractal nature of urban form and growth, they are not systematic in the sense that cities were not selected according to a spatial scheme (e.g., city or population size hierarchy) and a common set of parameters (i.e. map coverage, resolution, scale.) Thus, the results are incomplete and less useful for purpose of inter-city comparison from the urban system perspective.’ Shen [19] selected 40 cities ranked by 1992 population and examined the relationship between population and fractal dimension. In this study, it is concluded that overall population size when regressed against dimension does reveal a good fit. In the study, if one inspects the highest populated city, New York City and the lowest populated city, Omaha, Nebraska it was found that their fractal dimensions are 1.701 and 1.277 respectively. Shen [19] did not inspect the fractal dimensions of sub-areas such as central city versus suburban areas in these urbanized areas, nor did he analyze the lacunarity of these cities. Shen [19] emphasizes that fractal dimensions do not appear to be related to density and that other factors are influencing the fractal nature of a city.

Despite some irregularity in fractal dimensions, there appears to be some quidelines as to the fractal dimensions of cities. In the center, if a city is occupied with buildings and little open space, the fractal dimenison would be approximately 1.8 or above. This would appear to be particulary true of large metropolitan areas There could be some indication from some of the author’s preliminary studies using remote sensing images that this could be lower for smaller cities, perhaps in the range of 1.75. As the city diffuses outward, there is a tendency for less concentration and more dendricity due to uneven development and also more space devoted toward highways, which occupy a large amount of space in modern cities. The tendency is for the dimension of approximately to be near to 1.8 in the center to rapidly change to one of about 1.75. This seems to indicate that cities are being fragmented by the forces of modern urbanization processes.

Linking fractal analyis with spatial technologies/modeling

Fracatal analysis for urban analysis and linking it to spatial technologies is still in the developmental stage. There are still major issues that need to be addressed in this area. The sampling of cities at various levels under controlled circumstances and the relaionship to other spatial metrics measurements need to be better defined to improve the reliability and interpretation. At the base of understanding fractal analysis is the rules that are forming the fractals. Although the generation of fractals to approxiate different urban forms has been done for sample cities [3], more research is necessary to understand why certain rules in fractal generation produce certain urban forms. These rules and fractal generation linkages to spatial technolgies and spatial modling would greatly increase the knowledge of urban systems.

Another unexplored area is the application of fractal analysis in the measurement of modeling urban growth using agent-based modeling. The fractal dimensions and lacunarity of the actual and the modeled urban areas could serve as calibration measurements. This possibilty occured to the author during a special course that he offered during Fall of 2007 on the use of agent-based modeling and GIS [15]. One of the group projects used the Dynamic Urban Evolutionary Model (DUEM) a cellular automata model as a basis for their project [26]. The students delinated the area of of Istanbul, placed ‘seeds’ at growth points and then let the model proceed on its own. The model allows for growth to follow roads. The pattern that developed was very similar to the actual growth patterns of Istanbul. The fractal pattern dimensions would be similar, but natuarally not identical to that of Istanbul. The use of fractal measurements has not been used, to the author’s knowldege for calibration of agent based urban models.

Spatial technologies, particularly GIS and Remote Sensing are able presently able to collect and analyze static spatial data The problems lie in ‘hard coupling’ time series and dynamic processes such as those that are modeled in fractal generation and agent-based modeling. There area programs such as Fractalyse [24] which can take images that are generated from geographic data and examine their fractal dimensions and lacunarity. However, the program would be much more useful if it was linked directly with a GIS or Remote Sensing program. Agent-based and fractal generation models have inherent problems with display due the graphic limitations of GIS programs because they were primarily constructed to display static images. The improved abilty of coupling of these different programs would make fractal analysis and GIS more robust tools for urban spatial analysis.

Implication of Fractal Analysis and Related Methods of Analysis for Urban Planning

At this period, fractal analysis, space syntax, agent based modeling and other other forms of urban analysis are in their infancy for use as tools for urban planning. While the use spatial technologies by most urban and regional planning agencies is widesrpead in high and middle income economies, the use of fractal analysis for urban analysis appears to be concentrated in the academic sphere. Salingaros [18], a mathematics professor at University of Texas at San Antonio, is one of a few academicians who is attempting to bridge the gap between fractal analyis therory and urban planning practice. He considers chaos theory and fractal analysis as overall philosophies which are challanging the concepts of urban planning which were based on rationalism and logical-positism, elitism and top-down decision-making. However, just as logical-positivism, linear modeling and Newtonian physics is an outcome of a philosophy connected with industrialization, modernism and Fordism, chaos theory and fractal analysis is a product of post-modernism [20]. As such, chaos theory represents a philosophical break from the ideas that dominated much of the 19th and 20th Century. While the proponents’ of chaos theory claim that it represents a scientific paradigm shift, the root may more have its basis in Post-Modern concepts with science backing it up [20].

Consider these two references related to urban planning:

Urban planning involves forecasting future population growth and planning

for possible changes. Planners consider: rate of growth, rates of natural

increases and migration, age profile of the forecasted population and housing

types, employment services required [17].

.

Planning under the modernist approach uses grand plans and the "big broom" or "clean sweep" approach to development. This implies that planning serves the interests capitalists and develpers as opposed to those with less power as only a select few can participate in the "major" schemes involved. Large-scale plans, development, and redevelopment activities can hardly be construed as being condusive to promoting the interests and wellbeing of those with less power as they are not in a postions to participate in the planning process and gain the benefits of the outcomes of modernist planning. The accomodation of a small-scale approach of post-modern planning, with its sensitivity to local interests and context human scale demands that planning encompse a wider range of interests [11].

The statement by the City of Prince Albert (Canada) implies that planning is a linear rational process which is based on making the best projections and then accommodating growth. The latter is a criticism of modernist urban planning depicting it as a process that is controlled by a small group of individuals for their own benefit. Implied is that the methods of justification are linear models and linear long range planning. In reality, most planners are still operating with planning methods based on logical-positism and regard any discussion concerning Post-Modern planning theory and concepts as purely academic and philisophical. Some of these criticisms of Post-Modern planning theory are well-grounded as it is still developing [11]. Also, Post-Modernistic planning is the antithesis of the concepts that most planners were taught in urban planning schools which are firmly situated within the elitist logical positism philosophy from the latter part of the 19th and early 20th Century.

Fractal analysis and other related chaos based tools such as agent-based modeling and cellular automata are associated with the developing area of Post-Modern scientific thought. To encorporate chaotic methods into urban planning requires an overall change in the concept of urban planning-being one of not one future but numerous futures involving not just a few people and forces but a unknowable amount within and outside of an urban area with an indefinite amount of goals and objectives. As such, this concept leads an urban planning to enter a new realm—one of uncertainity and non-conclusiveness where future outcomes are unknown. The illusion of long-range rational planning was that urban areas could plan 20 years in advance and arrive at some future state. Examinng fractal analysis and cellular automata, one realises that there is not one outcome, but an infinite amount based on different forces at a variety of scales. In fractal analyis and agent-based modeling, the outcomes-however varied- are determined by the rules and environment. If one changes the rules (sometimes only slightly) and/or the environment, the outcomes can be significantly different. This very premise changes the way one views urban development and planning.

Conclusion

Fractal analysis is a diverse and promising method to examine the built form of cities. While examining fractals in a theoretical manner is considerably advanced, its methods when examining actual urbanization is complex and sometime results in conflicting measurements. The interpretation of fractal analysis is not an easy task as there appears to be differences in measurments based on resolution and other factors. There is a developing literature concerning measurements from different cities. The application of fractal analysis of Istanbul by the author [14] futher demonstrates that fractal analyis can prove to be a worthly technique to study urban structure but raised a multitude of questions that can not be adequately addressed within the scope of this article.

What is evident is that there needs to be more research into this area as to standardizing the methods and interpretation. While there is ample evidence that fractal analysis and related analysis methods such as cellular automata/agent based modeling are promising, they are remaining as theoretical tools and have not entered the mainstream of urban analysis and planning. The developing area of urban syntax hints at additional new tools that can further examine the city with unique tools. At this time, research into fractal analysis, agent based modeling and other analytical tools that seek to probe further into the composition of the urban enivironment are being conducted in a limited number of locations around the world. It is anticipated that this field will become even more diverse yielding a whole set of tools that those who are working in the field of urban geography and planning will utitize to better understand cities and discover new ways of managing them.

The use of fractal analysis for urban analysis opens up more questions than it answers. This is somewhat disconcerting to those attached to the general paradigm of urban planning which is based on using linear models and encorporating them within a process of rational decision making. However, it should be self-evident that the current methods of urban planning and likewise the role of planners are ineffective and do not mesh with the present realities of the urban enviroment. The ability of urban planning theory and methods of analysis to address Post-Modern scientific thought which includes chaos theory and their methods could bring new life into urban planning making it a powerful player in transforming urban environments. How those developing urban theory and methods of analysis address these issues could greatly effect the health of the profession and its effectiveness and role in shaping the urban environment.

References

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[3] Batty, M. and Longley, P., (1996), Fractal Cities: a Geometry of Form and Function, London and San Diego, Academic Press.

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[8] Gleick, J., ( 1987), Chaos: Making a New Science, London. Penguin Books.

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[10] Langarias, A., ( 2007), ‘‘ Fractal analysis of the Urbanization at the Outskirts of the City, Measurement and Explantion, Cybergeo, article 391, 16 July 2007; http://www.cybergeo.eu/index8902.html. Accessed on8 May 2008.

[11] MacLeod, D., (2008), ‘‘ Post Modernism and Urban Planning’’; http://www3.sympatico.ca/david.macleod/POMO.HTM#POMO7 . Accessed on 8 May 2008.

[12] Mandelbrot, B. , (1983), The Fractal Geometry of Nature, 3rd Edition, San Francisco. W.H. Freeman.

[13] Merali, Y. and McKelvey, B. , ( 2006), ‘‘ Using Complexity Science to Effect a Paradigm Shift in Information Systems for the 21st Century ’’, Journal of Information Technology, 21, 4, 211–215; http://www.palgrave-journals.com/jit/journal/v22/n3/index.html. Accessed on 8 May 2008.

[14] McAdams, M., (2007), ‘’Fractal Analysis and the Urban Morphology of a City in a Developing Country: A Case Study of Istanbul’’, Marmara Coğrafya Dercisi (Marmara Geographical Review), 15, 147-170.

[15] McAdams, M., ( 2007). ‘’Course webpage for agent based modeling and GIS’’; http://www.fatih.edu.tr/~mcadams/geo391/. Accessed on 8 May 2008.

[16] McAdams, M., (1995), The Land Use Impact of an Airport and Urban Structure : a Case Study in Milwaukee, Wisconsin (dissertation), Milwaukee: University of Wisconsin-Milwaukee.

[17] City of Prince Albert ( Saskatchewan, Canada), (2007), ‘ ‘ Urban Growth and Urban Form

Managing Urban Growth.’’; http://www.citypa.ca/TheCity/Departments/EconomicDevelopmentandPlanning/ThePrinceAlbertDevelopmentPlanProcess/tabid/347/Default. Accesssed on 8 May 2008

[18] Salingaros, N., ( 2003). ‘‘Connecting the Fractal City (keynote speech)’’, 5th Biennial of towns and town planners in Europe, Barcelona, April 2003.; http://www.math.utsa.edu/sphere/salingar/connecting.html . Accessed on 8 May 2008.

[19] Shen, G. (2002), Fractal Dimension and Fractal Growth of Urbanized Areas, International Journal of Geographical Information Science, 16, 5, 419-437. Accessed on 7 May 2008.

[20] Smith, W. , and Higgins, M., (2003), ‘‘Postmodernism and Popularisation: The Cultural Life of Chaos Theory ’’, Culture and Organization 9, 2 , 93-104.

[21] Tannier C. and Pumain D., ( 2005), ‘‘ Fractals in Urban Geography: a Theoretical Outline and an Empirical Example, Cybergeo , 307, 20 ; http://193.55.107.45/articles/307res.htm. Accessed on 8 May 2008.

[22] Thompson, D.W., (1992 [c1917]), On Growth and Form, Cambridge. Cambridge University Press.

[23] Tolle, C., McJunkin, T., Rohrbaugh, D. , and LaViolette, R., (2004), ‘‘Lacunarity definition for ramified data sets based on optimal cover’’, Physica D ,179, 129–152; http://www.inl.gov/physics/d/lacunarity_physicad2003.pdf. Accessed on 8 May 2008

[24] University of Franche-Comté, France, ThéMA Research Group, ( 2007) , ‘Documentation for Fractalyse’’; http://www.fractalyse.org/. Accessed on 8 May 2008.

[25] Wolfram, S., (2002), A New Kind of Science, Champaign, Illinois. Wolfram Media.

[26] Xie, Y and Batty, M., (2003), ‘‘Integrated Urban Evolutionary Modeling, CASA Working Paper 75, Center for Advanced Spatial Analysis (CASA); http://www.casa.ucl.ac.uk/publications/workingPaperDetail.asp?ID=68 . Accessed 8 May 2008.

## 07 May 2008

### The application of fractal analysis and spatial technologies for urban analysis

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