The Mathematics of Urban Morphology (eBook)

Foreword: The Morphology of Cities 6
Contents 11
On Urban Morphology and Mathematics 14
1 Morphology, Urban and Mathematics 14
2 Urban Morphology 16
3 Mathematical Treatment of Urban Forms 17
4 Cities Between Hard and Soft Sciences 20
5 In Search of Universal Laws: From Calvino to Santa Fe 21
6 Complex Self-organizing Systems Urban Approaches 24
7 Laplace’s Demon in Cities 25
8 A Science but not a Science? 26
9 The Language of Mathematics 26
10 The Art and Science of Cities 27
Bibliography 29
Fractals 32
Fractal Dimension Analysis of Urban Morphology Based on Spatial Correlation Functions 33
1 Introduction 33
2 Theoretical Models 35
2.1 Basic Postulates 35
2.2 Spatial Correlation Functions 37
2.3 Fractal Parameter Equations 41
2.4 New Analytical Framework for Urban Morphology 45
3 Case Study 48
3.1 Methodology 48
3.2 Study Area, Datasets, and Results 50
4 Questions and Discussion 55
4.1 Methodological Outline 55
4.2 Model Generalization 58
5 Conclusions 62
References 63
Central Place Theory and the Power Law for Cities 66
1 Introduction 66
2 Central Place Theory 70
2.1 Model and One-Good Equilibrium 70
2.2 Hierarchy Equilibrium 72
3 Power Law for Cities 73
4 A Dynamic Programming Approach to Central Place Theory 76
4.1 The Sequence Problem 77
4.2 The Dynamic Programming Problem 77
4.3 The Central Place Property 82
5 Concluding Remarks 84
References 85
Distribution of City Size: Gibrat, Pareto, Zipf 87
1 Introduction 88
2 Methodology and Data 89
2.1 Testing for a Power-Law Tail 90
2.2 Data Description 92
3 Empirical Results 93
3.1 Test Results 93
3.2 Estimates of the Shape Parameter 94
3.3 Emergence of a Power-Law Upon Aggregation 95
4 Discussion and Conclusion 98
References 99
The Signature of Organic Urban Growth 102
1 Introduction 103
2 Of Street Networks and Degree Centrality Distributions 104
2.1 The Topology of Street Networks 105
2.2 Street Network Representation 105
2.3 Degree Distribution and Power Law Fitting 106
2.4 Related Work 107
3 Datasets and Methods 107
3.1 Objects of Study 108
3.2 Sources of Street Network Data 108
3.3 Street Network Model Considerations 110
3.4 Methodological Process 114
3.5 Software Used in the Process 115
4 Results 115
4.1 Derived Parameters of Degree Distribution 115
4.2 The Evolution of Paris 120
4.3 Comparison of Cities 122
5 Discussion 123
5.1 Organic and Planned Growth Processes 123
5.2 The Classification of Cities 124
6 Conclusions 127
References 128
A Fractal Approach to Explore Australian Urban Form and Its Impacting Factors at Neighbourhood Scale 131
1 Introduction 131
2 Methodology, Study Context and Data 134
2.1 The Fractal Methods 134
2.2 Spatial Correlation Analysis 139
2.3 Study Context: Brisbane, Australia 139
2.4 Data 140
3 Results 142
3.1 D and ? Values of Brisbane Neighbourhoods 142
3.2 Correlation Between Fractal Measures and Classical Spatial Form Metrics 144
4 Discussion 146
5 Conclusion 149
References 150
Cellular Automata 153
Geographic Cellular Automata for Realistic Urban form Simulations: How Far Should the Constraint be Contained? 154
1 Introduction 155
2 CA-Based Discrete Modeling 156
2.1 CA Formal Definition 156
2.2 CA Limits 157
3 Suitable Constraints for Urban Modeling 160
3.1 Temporal Constraint 161
3.2 Spatial Constraint 163
4 Constraint Geographical CA 164
5 Discussion 166
6 Conclusion 167
References 168
Mathematical Foundations of Cellular Automata and Complexity Theory 170
References 175
Spatial Networks and Space Syntax 178
Assessing Complexity of Urban Spatial Networks 179
1 ``We Shape Our Buildings Thereafter They Shape Us''
2 Spatial Graphs of Urban Environments 180
2.1 Locally Anisotropic Random Walks on Graphs 181
2.2 Stationary Distributions of Locally Anisotropic Random Walks 183
2.3 Entropy of Anisotropic Random Walks 184
2.4 The Relative Entropy Rate for Locally Anisotropic Random Walks 185
3 Information Decomposition for Markov Chains 186
3.1 Conditional Information Measure for the Downward Causation Process 187
3.2 Conditional Information Measure for the Upward Causation Process 188
3.3 Ephemeral Information in Markov Chains 188
4 Exploring Graph Structures by Random Walks 189
4.1 Affine Probabilistic Geometry of Graphs 190
4.2 Probabilistic Interpretation of Euclidean Geometry 190
5 How a City Should Look? 192
5.1 Labyrinths 192
5.2 Manhattan's Grid 195
5.3 German Organic Cities 198
5.4 The Diamond Canal Network of Amsterdam 200
5.5 The Canal Network of Venice 201
6 Conclusion 202
References 202
Space Syntax: A Network-Based Configurational Approach to Studying Urban Morphology 204
1 Introduction 205
2 Space Syntax: Standard Concepts, Methods, and Measures 206
2.1 Configurations 207
2.2 Axial and Segment Maps 208
2.3 Measures 210
3 Mathematical Developments in Space Syntax 215
3.1 Normalization of Integration and Choice 215
3.2 Geometric and Topological Measures Versus Metric Measures in Space Syntax 219
3.3 Scaling and Universality in the Axial and Segment Maps 221
3.4 Boundary Effects on Space Syntax Measures 222
3.5 Shape and Space in Space Syntax 224
3.6 3D Descriptions in Space Syntax 226
3.7 Space Syntax and GIS 228
4 Research Applications of Space Syntax 231
4.1 The Syntactic Cores and Types of Spatial Configurations 231
4.2 The Whole and the Parts Relationships of Spatial Configurations 233
4.3 The Re/Production of Society and Culture Using Spatial Configurations 235
4.4 The Generative Functions of Spatial Configurations 238
4.5 Spatial Configurations and Social Capital 240
4.6 Spatial Configurations and Urban Historical Studies 241
5 Summary and Conclusions 243
References 244
Applied Mathematics on Urban Space 257
1 Introduction 257
2 Definition of Urban Space 258
3 The Method of Calculations 259
4 Space Syntax’ Contribution to Theory Building and Understanding on How to Build Environments Works 270
References 272
The Morphology and Circuity of Walkable and Drivable Street Networks 275
1 Introduction 275
2 Analytical Background 277
3 Methods 279
4 Results 282
5 Discussion 282
6 Conclusion 287
References 288
Complexity 292
Emergence of Complexity in Urban Morphology 293
1 Introduction 293
2 Fractal Nature of Urbanized Regions in Seoul 295
3 State Vector Description of Urban Areas 298
4 Modeling Urban Morphology 302
4.1 Model Description 302
4.2 Monte Carlo Simulations and Mean-Field Approximation 305
4.3 Criticality in the Urban Morphology of Seoul 308
5 Discussion 313
References 315
On the Complex Interaction Between Mathematics and Urban Morphology 317
1 Motivations and Plan of the Chapter 317
2 Reasonings on the Metamorphosis by Cornelis Escher 319
3 Mathematical Models for Complex Systems 323
4 Towards a Mathematics for Urban Morphology 326
4.1 Agent-Based Modeling Tools 327
4.2 Active Particles Kinetic Modeling Tools 328
4.3 Towards Research Perspectives 330
References 333
A Topological Representation for Taking Cities as a Coherent Whole 336
1 Introduction 337
2 Wholeness or Living Structure, and Its 15 Fundamental Properties 338
3 The Topological Representation for Taking Things as a Whole 341
4 Living Structures of China and UK Natural Cities 343
4.1 Data and Data Processing 343
4.2 Visualization of Differentiation and Adaptation 344
5 Discussions on the Topological Representation and Analysis 348
6 Conclusion 350
References 351
Other Forms of Quantification 354
A Multiscale Clustering of the Urban Morphology for Use in Quantitative Models 355
1 Introduction and State of the Art 355
2 Data Model 356
2.1 Local and Global Data 356
2.2 An Object Driven Data Model 357
2.3 Methods of Geoprocessing 357
3 Scales and Attributes 358
3.1 Building 358
3.2 Composition 359
3.3 Neighborhood and District 360
3.4 Municipality and Region 362
4 Examples of Location Types 364
4.1 Building 364
4.2 Composition 365
4.3 Neighborhood 365
4.4 Municipality 366
5 Clustering 366
5.1 Classification Versus Clustering 366
5.2 Methodology 367
5.3 Principal Component Analysis 368
5.4 Number of Clusters and Runtime 369
6 Clusters Results 371
7 Summary and Discussion 378
References 380
An Urban Morphogenesis Model Capturing Interactions Between Networks and Territories 383
1 Introduction 384
1.1 Urban Morphology 384
1.2 Urban Morphology and Interactions Between Networks and Territories 385
2 Measuring Morphology: Method 386
2.1 Urban Morphology 386
2.2 Network Measures 386
2.3 Correlations 388
3 Empirical Application 388
3.1 Urban Morphology 389
3.2 Network Topology 389
3.3 Effective Static Correlations and Non-stationarity 392
4 Urban Morphogenesis Model 398
4.1 Model Rationale 398
4.2 Model Description 399
4.3 Simulation Results 400
5 Discussion 404
5.1 Quantifying Urban Form 404
5.2 Modeling Urban Morphogenesis 405
5.3 Implications for Policies 406
References 407
Continuum Percolation and Spatial Point Pattern in Application to Urban Morphology 410
1 Introduction 410
2 Percolation in Mathematical Physics 413
2.1 Description 413
2.2 Phase Transition 413
3 Continuum Percolation and Spatial Arrangement of Points 415
3.1 General Ideas 415
3.2 Characterisation of Spatial Distribution of Points 417
3.3 Comparison with Other Methods 421
4 Interpretation of the Patterns in Urban Morphology 422
4.1 Single-scale Regular Pattern 423
4.2 Multi-scale Regular Pattern 424
4.3 Clustered Pattern 424
4.4 Dispersed Pattern 425
References 426
Urban Compactness: New Geometric Interpretations and Indicators 429
1 Introduction 429
2 Compact Cities 431
2.1 The Compact City Policy Debate 431
2.2 Urban Compactness 431
2.3 Learning from Density 432
3 Geometric Interpretations of Compactness 433
4 A New Interpretation and Indicator of Compactness 437
4.1 Interpreting Compactness 437
4.2 Quantification of Compactness 438
5 Application to Urban Areas 440
5.1 Initial Illustration 440
5.2 Compactness for a Range of Urban Areas 440
5.3 Compactness Distinct from Density 445
6 Compactness Revisited 445
6.1 Wholes and Parts 446
6.2 The Core Meaning of Compactness 448
6.3 Further Kinds of Compactness 449
7 Conclusions 451
References 452
Using Google Street View for Street-Level Urban Form Analysis, a Case Study in Cambridge, Massachusetts 455
1 Introduction 455
2 Data Preparation 456
2.1 Google Street View (GSV) Panorama Collection 456
2.2 Geometric Transform of Google Street View (GSV) Panoramas 458
2.3 Image Classification 458
3 Estimating and Mapping the Sky View Factor in Different Seasons 461
4 Estimating Direct Sunlight Duration in Street Canyons 463
5 Other Potential Applications 464
6 Discussion and Conclusion 465
Appendix A 467
References 467
Examining Spatial Structure Using Gravity Models 469
1 Introduction 469
2 Functional Spatial Structure 470
2.1 Functional Polycentricity 471
2.2 Spatial Interdependencies 473
3 Gravity Models and Spatial Structure 473
4 Estimation of the Gravity Model 474
5 Concluding Remarks 475
References 476
Humanistic and Multidisciplinary Commentaries 478
Urban Morphogenesis: Putting Mathematics in Its Place 479
1 Introduction 479
2 Morphogenesis: Conventional Approaches 480
3 Complexity: From Weak to Strong Emergence 481
References 484
Not Only … But Also: Urban Mathematical Models and Urban Social Theory 486
1 Introduction 486
2 Mathematics Modelling Versus Social Theory 488
3 From Either/or to not Only/but also 490
References 491
Urban Morphology or Townscape? Wholes Made of Many Parts 493
References 496
Extending Urban Morphology: Drawing Together Quantitative and Qualitative Approaches 497
1 The Nature of Urban Morphology 497
2 Contributions to a ‘New Urban Morphology’? 500
3 Conclusion: Unhelpful Dichotomies of Old and New, Quantitative and Qualitative—Moving Forward 504
References 506
Mathematics and Cities: A Long-Standing Relationship Fit for the Future? 510
References 514
Mathematics and/as Humanities–Linking Humanistic Historical to Quantitative Approaches 516
1 Mathematics as Humanities 516
2 Quantitative Approaches in Urban Morphology 517
3 Imminent Changes 520
Urban Forms, Agents, and Processes of Change 522
1 Urban Morphology 522
2 The Historico-Geographical Approach 523
3 The Process-Typological Approach 525
4 Combining Different Approaches in the Study of Urban Form 527
4.1 The Historico-Geographical Approach 527
4.2 The Process-Typological Approach 527
References 528
Future of Streets 529
References 537
Understanding and Quantifying Urban Density Toward more Sustainable City Form 539
1 Introduction 539
2 Defining Urban Density 540
3 Defining Density with Regard to the Compact City 542
4 The Ongoing Density Debate 544
5 A Proposed Framework for “Quality Density” 545
6 Learning to Live in More Compact and Denser Communities 546
References 547
To Not Talk Past Each Other: An Immodest Proposal for Cross-Conceptual Research in Urban Morphology 549
1 Desire for Integration 551
2 Cross-Conceptual Initiatives 552
3 Evidence and Theory 553
4 The Challenge 554
References 555