0.1 Abstract i

0.2 Acknowledgements ii

0.3 Table of contents iii

0.4 List of figures xi

0.5 List of tables xvii

1 CHAPTER ONE: INTRODUCTION ………………………….. 1

1.1 Introduction and the rationale for the research ………………………….. 2

1.1.1 The city of organized complexities ………………………….. 2

1.1.2 The research background and the theoretical motivation 4

1.1.3 The fractal analysis of urban morphology and the research focus 5

1.1.4 The fractal analysis of planned and unplanned/organic urban patterns 7

1.2 Aims, objectives and the research questions ………………………….. 10

1.2.1 Aims and objectives ………………………….. 10

1.2.2 The research questions ………………………….. 11

1.3 The research methodology ………………………….. 12

1.3.1 The case study ………………………………………………………. 12

1.3.2 The thesis structure and the stages of the investigation . 13

1.3.2.1 Stage I: the literature review ………………………….. 13

1.3.2.2 Stage II: the data collection and examination ………………………….. 14

1.3.2.3 Stage III: the data analysis ………………………….. 16

1.4 Chapter summary ………………………………………………………. 17

2 CHAPTER TWO: THE CONVENTIONAL GEOMETRY OF STRAIGHT LINES ………………………….. 18

2.1 Part one: the implications and interpretations of Euclidean geometry as

applied to architecture and urban form ………………………….. 20

2.1.1 The origins of geometry and its role in shaping built environment . 20

2.1.2 A brief history of geometric implications for architecture and urban form 22

2.1.2.1 Geometry of antiquity ……………………………. 23

2.1.2.2 Divine geometry of dark ages …………………………… 28

2.1.2.3 Rational geometry of the Renaissance ………………………….. 33

2.1.2.4 Pure geometry of modernism ………………………….. 40

2.1.3 Pure geometry and the quality ………………………….. 42

2.1.3.1 Purism and critics ………………………………………………………. 42

2.1.3.2 Scaling relationships and fractal concepts ………………………….. 44

2.2 Part two: the failures of the conventional geometry 47

2.2.1 The strengths and weaknesses of Euclidean geometry in interpreting planned and organic city form ………………………….. 48

2.2.2 The association of mistakes with the application of Euclidean geometry 52

2.3 Chapter summary ………………………………………………………. 56

3 CHAPTER THREE: COMPLEXITY, CHAOS, AND FRACTAL GEOMETRY (THEORETICAL BACKGROUND) 58

3.1 Part one: chaos theory, literature review ………………………….. 60

3.1.1 Understanding of chaos and complex systems ………………………….. 60

3.1.1.1 Step I: order out of chaos (linear dynamics) ………………………….. 63

3.1.1.2 Step II: chaos out of order (thermodynamics) ………………………….. 65

3.1.1.3 Step III: at the edge of chaos (nonlinear dynamics) . 68

3.1.2 Interpretations of chaos theory ………………………….. 71

3.2 Part two: complexity, characteristics of chaotic complex systems, and their analogies to urban systems 79

3.2.1 Complexity theory and its relationship with chaos theory 79

3.2.2 Characteristics of complex systems, and their analogies to urban systems 83

3.2.2.1 Variety (large number of components with dynamical interactions) . 85

3.2.2.2 Irreducibility ……………………………………………………….. 86

3.2.2.3 Deterministic chaos (duality of determination and randomness) … 87

3.2.2.4 Positive and negative feedback …………………………….. 88

3.2.2.5 Sensitivity to initial conditions (the butterfly effect) ….. 91

3.2.2.6 Limited predictability ……………………………. 92

3.2.2.7 Emergence ………………………………………………………. 93

3.2.2.8 Self-organization ………………………………………………………. 95

3.2.2.9 Adaptability ………………………………………………………. 97

3.2.2.10 Interconnectedness (synergy) ……………………………. 98

3.2.2.11 Hierarchy and levels of scale ……………………………. 99

3.2.2.12 Self-similarity and fractal pattern (the image of complexity) ……………………………… 102

3.3 Part three: fractals; the geometry of complexity …………………………………………………………………………………… 104

3.3.1 Fractal; terminology, meaning and definition ………………………………………………………………………………………………………………… 105

3.3.2 Fractal dimension ……………………………………………………………………………………………………………….. 106

3.3.3 Types of fractals ……………………………………………………………………………………………………………………………………………. 111

3.3.3.1 Linear fractals (self-similar fractals) ………………………………………………………………………………………………………………… 112

3.3.3.2 Nonlinear fractals (self-affine fractals)…………………………………………………………………………………………………………………………………………….113

3.3.4 Fractal generators………………………………………………………………………………………………………………………………………………………………………117

3.3.5 Methods for measuring fractal dimension………………………………………………………………………………………………………………………………………….119

3.3.5.1 Box counting method (grid dimension)…………………………………………………………………………………………………………………………………………..120

3.3.5.2 Structured walk method (ruler dimension)……………………………………………………………………………………………………………………………………….123

3.3.6 Fractals in everyday life………………………………………………………………………………………………………………………………………………………………..125

3.4 Chapter summary………………………………………………………………………………………………………………………………………………………………………….127

4 CHAPTER FOUR: THE APPLICATION OF COMPLEXITY THEORY AND FRACTALS IN ARCHITECTURE, URBAN PLANNING AND DESIGN .129

4.1 Part one: fractal architecture; the applications of complexity and fractals in architecture ……………………………… 134

4.1.1 The applications of fractal concept in architectural design .. 134

4.1.1.1 Fractal concept as a critical tool ……………………………. 135

4.1.1.2 Fractal concept as a design tool ……………………………. 138

– Group one, the idea of self-similarity …… 139

– Group two, the idea of fractal rhythm (unity with variety) ….. 148

4.1.2 Fractal architecture …………………………….. 152

4.1.2.1 Non-fractal architecture ……………………………. 153

4.1.2.2 The key features of fractal architecture ……………………………. 156

– Integrity and multiplicity ……………………………. 157

– Self-similarity/self affinity ………………………………… 157

– Fractality (the degree of physical complexity) .. 158

– Hierarchies of connections ………………………….. 158

– Change over time/space ………………………….. 160

4.2 Part two: fractal city; the applications of fractals and complexity theory in urban planning and design ………………………….. 162

4.2.1 Conceptualizing city complexity ………………………….. 163

4.2.1.1 Transitions in urban planning and design ………………………….. 166

– City as machine (planning transition I) ………………………….. 166

– City as system (planning transition II) ………………………….. 169

– Advantages and limitations of systems planning 171

– Planning in the conventional top down approach . 173

4.2.1.2 City as complex system (planning transition III) 175

– Planning in complex bottom up approach ………………………….. 176

– Conceptual complex planning models ………………………….. 180

4.2.2 Simulating city complexity ………………………….. 185

4.2.3 Measuring city complexity ………………………….. 194

4.3 Chapter summary ………………………….. 200

5 CHAPTER FIVE: THE IMPLICATIONS OF THE CONVENTIONAL MASTER PLANS OF TEHRAN FOR URBAN GROWTH AND FORM; CASE STUDY SELECTION AND INTERPRETATION ………………………….. 203

5.1 Part one: the impacts of the conventional master plans (comprehensive plans) in controlling urban change in Tehran and other Iranian cities 205

5.1.1 Thee history of the city growth, large scale planning, and urban interventions in Tehran …………………………… 205

5.1.1.1 The initial period (pre-1786) ………………………….. 206

5.1.1.2 The second period (1785-1920) ………………………….. 207

5.1.1.3 The third period (1925-1965) ………………………….. 209

5.1.1.4 The fourth period (after 1965) ………………………….. 212

5.1.2 Tehran’s current condition ………………………….. 217

5.1.3 The overall comments on the former master plans of Tehran 220

5.1.4 The failures of the top-down master planning in managing change in other Iranian cities ………………………….. 223

5.1.4.1 Economic studies and predictions ………………………….. 223

5.1.4.2 Population ………………………………………………………. 224

5.1.4.3 Prediction in settlement land-use ………………………….. 224

5.1.4.4 Assessment of the directions and zones the city has expanded 225

5.1.4.5 Assessment of network expansion ………………………….. 225

5.1.4.6 The correspondence of proposed densities with the actual occurred densities 226

5.1.4.7 The changes in the city structure ………………………….. 226

5.1.5 The overall comments on the conventional urban planning in Iran . 227

5.2 Part two: the case study of Shemiran and sample selection .. 228

5.2.1 The factors of the case study and sample selection 229

5.2.2 Historical and geographical interpretation of the urban patterns in Shemiran 230

5.2.3 Fractal interpretation of the organic urban pattern of Tajrish 234

5.2.4 The case study and sample selection by measuring fractal dimension 236

5.3 Chapter summary ………………………….. 239

6 CHAPTER SIX: FRACTAL MAPPING; THE PILOT STUDY AND THE FRACTAL ASSESSMENT OF THE SELECTED CASE STUDIES 240

6.1 Part one: the research target and the method refinement .. 242

6.1.1 The source of data for measuring urban morphological evolution 244

6.1.2 Fractal mapping …………………………………………………………. 246

6.1.3 An introduction to the employed method and data processing steps 247

6.1.4 Advantages and limitations of using Benoit 1.3 and ArcMap 9.2 248

6.1.4.1 Fractal analysis software, Benoit 1.3 ………………………….. 248

6.1.4.2 GIS software, ArcMap 9.2 ………………………….. 250

6.2 Part two: the pilot study ………………………….. 251

6.2.1 Software calibration � the validity test ………………………….. 251

6.2.2 Image control � the sensitivity test ………………………….. 256

6.2.2.1 Contrast/Brightness test ………………………….. 256

6.2.2.2 Resolution test ………………………….. 257

6.2.2.3 Contents test ………………………….. 258

6.3 Part three: the case study examination 260

6.3.1 Case studies’ preparation and image processing 260

6.3.2 Fractal dimension measurement ………………………….. 261

6.3.2.1 Experiment one (fractal assessment at the neighbourhood scale) 261

6.3.2.2 Experiment two (fractal assessment at the local scale) 262

6.3.3 Mapping fractal dimensions; data processing to fractal maps . 263

6.4 Chapter summary ………………………….. 267

7 CHAPTER SEVEN: FRACTAL ANALYSIS OF THE CASE STUDY AND INTERPRETATIONS 269

7.1 Part one: fractal identification of urban patterns .. 271

7.1.1 fractal map as an urban fingerprint ………………………….. 271

7.1.2 Fractal Neighbourhood Identification code (FNID) …. 273

7.2 Part two: fractal classification of urban patterns 278

7.2.1 Advantages of fractal classification ………………………….. 278

7.2.1.1 Urban patterns with high complexity ………………………….. 281

7.2.1.2 Urban patterns with medium complexity ………………………….. 284

7.2.1.3 Urban patterns with low complexity ………………………….. 285

7.2.2 Discussions of Fractal classification of urban patterns 287

7.3 Part three: measuring the change in urban patterns .. 288

7.3.1 Change analysis of the urban patterns of Tajrish from 1956 to 2002 288

7.3.2 Change analysis through the �add and remove� technique (A-R-Technique) 294

7.3.2.1 Proposal testing …………………………… 294

7.3.2.2 Policy testing ………………………….. 297

7.3.3 Vegetation, physical complexity and environmental sustainability 300

7.4 Chapter summary …………………………… 304

8 CHAPTER EIGHT: CONCLUSIONS 307

8.1 The main findings from the research questions 309

8.1.1 Euclidean geometry and morphological complexity 309

8.1.2 the relationships between chaos, fractals and complexity 311

8.1.3 Fractal and non-fractal architecture ………………………….. 313

8.1.4 The bottom up nature of urban morphological evolution 314

8.1.5 Complexity theory and fractals as applied to urban form and function 318

8.1.6 Mapping and measuring complexity ………………………….. 319

8.1.7 Fractal dimension as a mathematical criterion for urban pattern analysis … 320

8.2 The original contribution to new knowledge and the significance of the proposed method ………………………….. 323

8.2.1 Fractal maps ………………………….. 323

8.2.2 FNIDs ………………………………………………………. 324

8.2.3 The change analysis of urban patterns ………………………….. 325

8.3 The research limitations ………………………….. 327

8.3.1 The limitations of the research scope and the methodology 327

8.3.2 The limitations of the proposed method ………………………….. 328

8.3.3 The limitations of the case study selection and examination 329

8.4 The recommendations for future research …………………………….. 330

8.4.1 Opportunities for further research based on the research literature review 330

8.4.2 Opportunities for further research based on the advantages and limitations of the proposed fractal assessment method ………………………….. 332

9 BIBLIOGRAPHY ………………………….. 336

10 APPENDICES ………………………….. 363

Appendix A: A glossary of urban complexity 363

Appendix B: Fractal measurement methods 377

Appendix C: The logarithmic graphs of the 22 districts of Tehran 384

Appendix D: Images of Shemiran ………………………….. 394

Appendix E: The aerial photos of Tajrish from 1956 to 2002 401

Appendix F: The fractal analysis charts of 24 neighbourhoods in Tajrish 407

Appendix G: The published paper in 2006: Controlling future urban developments by fractal dimensions 412

Appendix H: The published paper in 2004: The nature of urban morphological evolution based on chaos theory and fractal geometry 425

List of Figures

Figure 1.1 The interdependence between local interactions and the emerging

global structure in a complex system. p.3

Figure 1.2 The plan of Bewdley. p.8

Figure 1.3 The first stage of the methodology; the literature review. p.14

Figure 1.4 The second stage of the methodology; the data collection and examination.

p.15 Figure 1.5 The third stage of methodology, the data analysis. p.16

Figure 2.1 The geometrical perception of nature in cave paintings. p.21

Figure 2.2 Geometry of early settlements. p.22

Figure 2.3 The geometrical analysis of the Parthenon. p.24

Figure 2.4 Athens’ rough plan and a view of its agora. p.25

Figure 2.5 The imposition of the Hippodamian grid in Miletus and Priene. p.25

Figure 2.6 The Roman geometry. p.26

Figure 2.7 Timgad, North Africa; a typical example of a Roman castra. p.27

Figure 2.8 The geometrical evolution from the Roman forms to Byzantine and Romanesque churches. p.29

Figure 2.9 The geometry of the rib vault as compared to the domical vault. p.29

Figure 2.10 The geometrical emphasis on height in the Gothic architecture. p.30

Figure 2.11 Chartres Cathedral’s interiors. p.30

Figure 2.12 The geometrical dominance of the medieval cathedrals at city scales. p.31

Figure 2.13 The Centre of Damascus. p.32

Figure 2.14 Three stages in the development of the city of Regensburg. p.33

Figure 2.15 The geometrical translation of Pythagoras’s theory. p.34

Figure 2.16 The arcade of Brunelleschi’s Foundling Hospital. p.35

Figure 2.17 The plan and aerial view of St Peter’s Square. p.35

Figure 2.18 Reconstruction of Brunelleschi’s experiment (perspective). p.36

Figure 2.19 Examples of the ideal cities of the Renaissance. p.37

Figure 2.20 The conformity of Baroque streets to the Renaissance perspective. p.38

Figure 2.21 The Baroque street layout. p.39

Figure 2.22 Traditional symmetry in architecture. p.40

Figure 2.23 Steiner Haus, designed by Adolf Loos. p.41

Figure 2.24 The modulor man and the modulor lattices by Le Corbusier. p.42

Figure 2.25 Levels of scale in a mosque in Meshed, Iran, as compared to Le Corbusier’s Marseilles block of apartments. p.44

Figure 2.26 Constitution Arch in Hyde Park, London. p.45

Figure 2.27 Three levels of scale at a residential project. p.46

Figure 2.28 Geometrical evolution in ground plan of Baghdad. p.50

Figure 2.29 The gradual transformation of a gridded Roman colony into an Islamic city. p.51

Figure 2.30 Generated organic and fabricated planned structures. p.54

Figure 3.1 Post-Modern Sciences of Complexity. p.62

Figure 3.2 Time reversibility in the Newtonian laws of motion. p.64

Figure 3.3 An open system model shifting from an equilibrium state to a non-equilibrium one. p.65

Figure 3.4 from Lorenz’s 1961 printout. p.69

Figure 3.5 Three-dimensional model of Lorenz’s attractor. p.71

Figure 3.6 The self-similar pattern in the Feigenbaum diagram. p.75

Figure 3.7 Three main transition phases in the Feigenbaum diagram. p.75

Figure 3.8 Self-similar branching seen in the Feigenbaum diagram. p.78

Figure 3.9 Transfer function with and without feedback. p.89

Figure 3.10 Emergence of extreme segregation from local cellular automata rules. p.95

Figure 3.11 Adaptive redevelopment in central Newcastle upon Tyne. p.98

Figure 3.12 Interconnectedness and levels of hierarchy in complex systems p.100

Figure 3.13 The semi-lattice and the tree-like hierarchies. p.101

Figure 3.14 Euclidean integer dimensions. p.106

Figure 3.15 The Koch curve. p.108

Figure 3.16 The Cantor set. p.109

Figure 3.17 The Sierpinski carpet and the Menger sponge. p.110

Figure 3.18 The Sierpinski triangle and its three-dimensional analogue. p.111

Figure 3.19 The Koch snowflake. p.113

Figure 3.20 Bifurcation in a bush as an example of natural fractals. p.114

Figure 3.21 The Mandelbrot set and the Julia sets. p.115

Figure 3.22 Random Koch curves. p.117

Figure 3.23 Random coastlines. p.118

Figure 3.24 The fractal dimension of the coastline of Britain using the Box-Counting method. p.121

Figure 3.25 The log-log graphs of the assessed fractal dimensions. p.122

Figure 3.26 The fractal dimension of the coastline of Britain using the Structured Walk method. p.123

Figure 4.1 A pie chart comparing quantitatively the cited papers. p.133

Figure 4.2 Fractal analysis of the elevations of Frank Lloyd Wright’s Robie house and Le Corbusier’s Villa Savoye. p.135

Figure 4.3 Self-similarity in Hindu Temples. p.140

Figure 4.4 Self-similar patterns in Gothic elements (pointed arch, gable, etc). p.140

Figure 4.5 Self-similarity of Gothic style at varied scales. p.142

Figure 4.6 Koch island and Gothic column compared. p.143

Figure 4.7 A denticular Doric entablature and the Devil’s staircase compared. p.143

Figure 4.8 Ruskin’s fractal tree. p.144

Figure 4.9 Thiepval memorial and the Amsterdam housing. p.145

Figure 4.10 Bruce Goff, Price house. p.146

Figure 4.11 Daniel Libeskind, proposed addition to the Victoria, Albert Museum. p.147

Figure 4.12 Storey Hall, Melbourne. p.148

Figure 4.13 A diverse residential designs by Lucien Kroll. p.149

Figure 4.14 An elevation of Aalto’s home and office. p.150

Figure 4.15 Fractal distribution of height and width for a row of townhouses. p.151

Figure 4.16 Bruce Goff, the interior of the Bavinger house. p.152

Figure 4.17 Jewish Museum, Berlin. p.153

Figure 4.18 Cinema Center, Dresden, Designed by Coop Himmelblau. p.154

Figure 4.19 Federation Square, Melbourne, designed by LAB with Bates Smart. p.154

Figure 4.20 Some avant-garde examples of architectural language in the twentieth-century creating fabricated structure. p.155

Figure 4.21 Community Center, New Caledonia, Designed by Renzo Piano. p.156

Figure 4.23 Chartres Cathedral’s spires, France, and Eurhythmics Center, Spain. p.161

Figure 4.24 The sketch for the centre of Paris proposed by Le Corbusier. p.168

Figure 4.25 The conventional planning and the action planning compared. p.181

Figure 4.26 Structural (Sierpinski) order in one-dimensional cellular automaton. p.188

Figure 4.27 Four types of dynamics for a one-dimensional CA model. p.190

Figure 4.28 The mechanism of diffusion-limited aggregation model (DLA). p.190

Figure 4.29 The diagram of flows in CAST. p.193

Figure 4.30 Fractal dimension as an indication of visual variety. p.195

Figure 4.31 The city boundary of Cardiff in 1886, 1901, 1992, and 1949. p.197

Figure 4.32 The growth of London from 1820 to 1962. p.198

Figure 5.1 The first map of Tehran in 1841 (Berezin Map). p.207

Figure 5.2 Map of Tehran drawn by Abd-ol-Ghafar Khan in 1891. p.208

Figure 5.3 Map of Tehran up to 1953. p.210

Figure 5.4 Tehran’s Master Plans (Comprehensive Plans). p.214

Figure 5.5 The model of Shahestan-e Pahlavi; High rise residential complexes in Shahrak-e Gharb. p.215

Figure 5.6 Two examples of structural planning in Tehran. p.216

Figure 5.7 The demographic growth in Tehran city and its metropolitan region. p.217

Figure 5.8 Tehran’s growth up to 2004 and its new network system. p.218

Figure 5.9 Shemiran in 1957 and 2005. p.219

Figure 5.10 A caricature of changes to street vistas of Tehran. p.222

Figure 5.11 Aerial photo of Shemiran. p.230

Figure 5.12 Tehran’s geographical location between the desert and Caspian sea. p.231

Figure 5.13 A traditional inward plot layout in the south of Tehran. p.233

Figure 5.14 The new plot layout and the old layout of Persian gardens compared. p.233

Figure 5.15 Tajrish Bazaar and its surrounding neighbourhoods in 1981. p.234

Figure 5.16 Building and plot layout in the neighbourhood area of Tajrish. p.235

Figure 5.17 Self-similar, semi U-shape, pattern of the neighbourhoods of Tajrish. p.236

Figure 5.18 A graphic presentation of semi U-shaped fractal pattern of Tajrish. p.236

Figure 5.19 Degrees of urban pattern diversity within Three districts of Tehran. p.238

Figure 6.1 The main steps of the examination method. p.247

Figure 6.2 The main steps of the pilot study. p.251

Figure 6.3 The adjustable parameters in Benoit 1.3’s interface. p.252

Figure 6.4 The pilot samples. p.253

Figure 6.5 The steps of adjustments of contents for the pilot sample (T-N10) . p.259

Figure 6.6 The sequence of image/data processing at each steps of examination. p.260

Figure 6.7 The steps of image processing. p.261

Figure 6.8 The histograms of the fractal dimensions of Velenjak and Tajrish. p.263

Figure 6.9 Fractal shapes created for Velenjak and Tajrish. p.264

Figure 6.10 The fractal map of Velenjak and of Tajrish . p.265

Figure 6.11 The fractal map of Shemiran. p.265

Figure 6.12 The summary of the sequential steps to produce the fractal map. p.267

Figure 7.1 The structure of analysing the results. p.270

Figure 7.2 The number of occupied built-up units in Tehran at different scales. p.274

Figure 7.3 The hierarchical logic behind FNIDs. p.274

Figure 7.4 Fb measured for Tehran at different scales. p.275

Figure 7.5 Fractal classification: high physical complexity. p.280

Figure 7.6 Fractal classification: medium physical complexity. p.280

Figure 7.7 Fractal classification: low physical complexity. p.280

Figure 7.8 The areas which their patterns begin to be distorted. p.281

Figure 7.9 Some examples of the old and organically grown urban patterns with high physical complexity. p.282

Figure 7.10 Some linear examples of areas with high physical complexity. p.283

Figure 7.11 Some examples of urban patterns with medium complexity. p.284

Figure 7.12 Some examples of urban patterns with low complexity. p.286

Figure 7.13 Examples of designed areas with low physical complexity . p.287

Figure 7.14 The change in physical complexity of Tajrish at local scale from 1956 to 2002. p.289

Figure 7.15 Fractal Dimensions of 24 neighbourhood units of Tajrish for the years 1956, 1969, 1979, and 2002. p.291

Figure 7.16 The fluctuation of fractal dimensions for individual neighbourhoods of Tajrish between 1956 and 2002. p.292

Figure 7.17 The large-scale change occurring in T-N10 and T-N1. p.293

Figure 7.18 Neighbourhoods in Tajrish that have undergone minimal change in their urban patterns. p.294

Figure 7.19 Fractal assessment of a hypothetical design proposal. p.295

Figure 7.20 The street widening plan around Tajrish Square. p.298

Figure 7.21 the change in physical complexity of the neighbourhoods in Tajrish after the Implementation of the street widening policy. p.299

Figure 7.22 Aerial photos of V-N1 and V-N19 in Velenjak. p.301

Figure 7.23 The aerial photos of Tajrish in winter 1965 and summer 1969. p.302

Figure 7.24 A view of Vali-e Asr Street. p.303

Figure 7.25 The aerial photo, land-use and fractal maps of Vali-e Asr Street. p.303

List of Tables

Table 3.1 The properties of the organic universe and the mechanical universe. p.67

Table 3.2 The fractal dimension measured for the coastline of Britain using the Box Counting method. p.121

Table 3.3 The fractal dimension measured for the coastline of Britain using the Structured Walk method. p.124

Table 4.1 A summary of applications of complexity, chaos and fractal theories. p.132

Table 4.2 Results of box counting for the Robie house and the Villa Savoye at the range of scales from 1/14 to 1/64. p.136

Table 4.3 Results of box counting for the Robie house and the Villa Savoye at the window scale. p.136

Table 4.4 Planning transitions in the 20th and 21st centuries. p.166

Table 4.5 The Sierpinski triangle, the rules behind a one-dimensional cellular automaton. p.189

Table 4.6 The results of the fractal calculation of Cardiff city’s boundary. p.197

Table 4.7 The assessed fractal dimension for some cities around the world. p.199

Table 5.1 The rate of employment for some major cities in Iran. p.224

Table 5.2 Prediction in settlement land-use for some major cities in Iran. p.225

Table 5.3 The built-up area inside and outside the proposed boundary. p.225

Table 5.4 the percentage of the streets constructed based on the proposed plan. p.226

Table 5.5 The rate of annual temperature and rainfall in Tehran. p.231

Table 6.1 Measurable morphological elements. p.244

Table 6.2 Calibration of the fractal analysis tool with constant CD and varied SL. p.254

Table 6.3 Calibration of the fractal analysis tool with constant SL and varied CD. p.254

Table 6.4 The sensitivity to the contrast/brightness of the pilot images. p.257

Table 6.5 The sensitivity to image resolutions. p.258

Table 6.6 The pilot study – contents test. p.259

Table 6.7 Fractal calculation at neighbourhood scale of Tajrish. p.261

Table 6.8 Fractal calculation at neighbourhood scale of Velenjak. p.262

Table 6.9 Fractal dimensions measured at local scales for Tajrish and Velenjak. p.262

Table 7.1 Estimating the fractal dimension of Tehran by fixing the input image- size varying the scale of the city. p.273

Table 7.2 The assessed fractal dimensions for the different scale levels of the city of Tehran. p.276

Table 7.3 FNIDs of the neighbourhoods T-N1, T-N10, T-N11, and T-N15. p.276

Table 7.4 FNIDs of the neighbourhoods V-N1, V-N12, V-N19, and V-N23. p.276

Table 7.5 The change in fractal dimensions of the urban patterns at local scale of Tajrish from 1956 to 2002. p.289

Table 7.6 The fractal dimensions of 24 neighbourhoods in Tajrish for the years 1956, 1969, 1979, and 2002. p.290

Table 7.7 The degree of physical changes imposed by a hypothetical proposal. p.295

Table 7.8 Fractal dimensions of the neighbourhoods in Tajrish after the implementation of the street widening policy. p.299