Our daily universe is rough and infinitely diverse. The fractal approach clarifies and orders these disparities. It helps us to envisage new explanations of geographical phenomena, which are, however, considered as definitely understood.
Written for use by geographers and researchers from similar disciplines, such as ecologists, economists, historians and sociologists, this book presents the algorithms best adapted to the phenomena encountered, and proposes case studies illustrating their applications in concrete situations.
An appendix is also provided that develops programs written in Mathematica.
1. A Fractal World.
2. Auto–similar and Self–affine Fractals.
3. From the Fractal Dimension to Multifractal Spectrums.
4. Calculation and Interpretation of Fractal Dimensions.
5. The Fractal Dimensions of Rank–size Distributions.
6. Calculation and Interpretation of Multifractal Spectrums.
7. Geographical Explanation of Fractal Forms and Dynamics.
8. Using Complexity Theory to Explain a Fractal World.
9. Land–use Planning and Managing a Fractal Environment.
Chapter 1. A Fractal World 1
1.1. Fractals pervade into geography 2
1.2. Forms of fractal processes 10
1.3. First reflections on the link between power laws and fractals 14
1.4. Conclusion 19
Chapter 2. Auto–similar and Self–affine Fractals 21
2.1. The rarity of auto–similar terrestrial forms 22
2.2. Yet more classes of self–affine fractal forms and processes 24
2.3. Conclusion 37
Chapter 3. From the Fractal Dimension to Multifractal Spectrums 39
3.1. Two extensions of the fractal dimension: lacunarity and codimension 40
3.2. Some corrections to the power laws: semifractals, parabolic fractals and log–periodic distributions 43
3.3. A routine technique in medical imaging: fractal scanning 48
3.4. Multifractals used to describe all the irregularities of a set defined by measurement 50
3.5. Conclusion 57
Chapter 4. Calculation and Interpretation of Fractal Dimensions 59
4.1. Test data representing three categories of fractals: black and white maps, grayscale Landsat images and pluviometric chronicle series 60
4.2. A first incontrovertible stage: determination of the fractal class of the geographical phenomenon studied 62
4.3. Some algorithms for the calculation of the fractal dimensions of auto–similar objects 75
4.4. The fractal dimensions of objects and self–affine processes 80
4.5. Conclusion 85
Chapter 5. The Fractal Dimensions of Rank–size Distributions 87
5.1. Three test series: rainfall heights, urban hierarchies and attendance figures for major French museums 88
5.2. The equivalence of the Zipf, Pareto and Power laws 89
5.3. Three strategies for adjusting the rank–size distribution curve 92
5.4. Conclusion 101
Chapter 6. Calculation and Interpretation of Multifractal Spectrums 103
6.1. Three data sets for testing multifractality: a chronicle series, a rank–size distribution and satellite images 104
6.2. Distinguishing multifractal and monofractal phenomena 104
6.3. Various algorithms for calculation of the singularity spectrum 111
6.4. Possible generalizations of the multifractal approach 116
6.5. Conclusion 118
Chapter 7. Geographical Explanation of Fractal Forms and Dynamics 121
7.1. Turbulence generates fractal perturbations and multifractal pluviometric fields 122
7.2. The fractality of natural hazards and catastrophic impacts 126
7.3. Other explanations from fields of physical geography 128
7.4. A new geography of populations 129
7.5. Harmonization of town growth distributions 131
7.6. Development and urban hierarchies 132
7.7. Understanding the formation of communication and social networks 136
7.8. Conclusion 137
Chapter 8. Using Complexity Theory to Explain a Fractal World 139
8.1. A bottomless pit debate140
8.2. General mechanisms for explaining power laws 143
8.3. Four theories on fractal universality 144
8.4. Conclusion 164
Chapter 9. Land–use Planning and Managing a Fractal Environment 167
9.1. Fractals, extreme values and risk 168
9.2. Fractals, segmentation and identification of objects in image processing 173
9.3. Fractals, optimization and land management 177
9.4. Fractal beauty and landscaping 179
9.5. Conclusion 180
C.1. Some tools and methods for quantifying and qualifying multiscale coarseness and irregularity 184
C.2. A recap on geographical irregularities and disparities 186
C.3. A paradigm that gives rise to new land–use management practices 189
A.1. Preliminary thoughts on fractal analysis software 191
A.2. Instructions for the following programs 192
A.3. Software programs for the visual approach of a satellite or cartographic series or image 193
A.4. Software programs for calculating fractal dimensions for a chronicle or frequency series 198
A.5. Software programs for calculating the fractal dimensions of a satellite image or map 208
A.6. Software programs for calculating multifractal spectrums of a series and an image 213