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