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<p>Introduction xi</p> <p><b>Chapter 1. A Fractal World 1</b></p> <p>1.1. Fractals pervade into geography 2</p> <p>1.2. Forms of fractal processes 10</p> <p>1.3. First reflections on the link between power laws and fractals 14</p> <p>1.4. Conclusion 19</p> <p><b>Chapter 2. Auto-similar and Self-affine Fractals 21</b></p> <p>2.1. The rarity of auto-similar terrestrial forms 22</p> <p>2.2. Yet more classes of self-affine fractal forms and processes 24</p> <p>2.3. Conclusion 37</p> <p><b>Chapter 3. From the Fractal Dimension to Multifractal Spectrums 39</b></p> <p>3.1. Two extensions of the fractal dimension: lacunarity and codimension 40</p> <p>3.2. Some corrections to the power laws: semifractals, parabolic fractals and log-periodic distributions 43</p> <p>3.3. A routine technique in medical imaging: fractal scanning 48</p> <p>3.4. Multifractals used to describe all the irregularities of a set defined by measurement 50</p> <p>3.5. Conclusion 57</p> <p><b>Chapter 4. Calculation and Interpretation of Fractal Dimensions 59</b></p> <p>4.1. Test data representing three categories of fractals: black and white maps, grayscale Landsat images and pluviometric chronicle series 60</p> <p>4.2. A first incontrovertible stage: determination of the fractal class of the geographical phenomenon studied 62</p> <p>4.3. Some algorithms for the calculation of the fractal dimensions of auto-similar objects 75</p> <p>4.4. The fractal dimensions of objects and self-affine processes 80</p> <p>4.5. Conclusion 85</p> <p><b>Chapter 5. The Fractal Dimensions of Rank-size Distributions 87</b></p> <p>5.1. Three test series: rainfall heights, urban hierarchies and attendance figures for major French museums 88</p> <p>5.2. The equivalence of the Zipf, Pareto and Power laws 89</p> <p>5.3. Three strategies for adjusting the rank-size distribution curve 92</p> <p>5.4. Conclusion 101</p> <p><b>Chapter 6. Calculation and Interpretation of Multifractal Spectrums 103</b></p> <p>6.1. Three data sets for testing multifractality: a chronicle series, a rank-size distribution and satellite images 104</p> <p>6.2. Distinguishing multifractal and monofractal phenomena 104</p> <p>6.3. Various algorithms for calculation of the singularity spectrum 111</p> <p>6.4. Possible generalizations of the multifractal approach 116</p> <p>6.5. Conclusion 118</p> <p><b>Chapter 7. Geographical Explanation of Fractal Forms and Dynamics 121</b></p> <p>7.1. Turbulence generates fractal perturbations and multifractal pluviometric fields 122</p> <p>7.2. The fractality of natural hazards and catastrophic impacts 126</p> <p>7.3. Other explanations from fields of physical geography 128</p> <p>7.4. A new geography of populations 129</p> <p>7.5. Harmonization of town growth distributions 131</p> <p>7.6. Development and urban hierarchies 132</p> <p>7.7. Understanding the formation of communication and social networks 136</p> <p>7.8. Conclusion 137</p> <p><b>Chapter 8. Using Complexity Theory to Explain a Fractal World 139</b></p> <p>8.1. A bottomless pit debate140</p> <p>8.2. General mechanisms for explaining power laws 143</p> <p>8.3. Four theories on fractal universality 144</p> <p>8.4. Conclusion 164</p> <p><b>Chapter 9. Land-use Planning and Managing a Fractal Environment 167</b></p> <p>9.1. Fractals, extreme values and risk 168</p> <p>9.2. Fractals, segmentation and identification of objects in image processing 173</p> <p>9.3. Fractals, optimization and land management 177</p> <p>9.4. Fractal beauty and landscaping 179</p> <p>9.5. Conclusion 180</p> <p>Conclusion 183</p> <p>C.1. Some tools and methods for quantifying and qualifying multiscale coarseness and irregularity 184</p> <p>C.2. A recap on geographical irregularities and disparities 186</p> <p>C.3. A paradigm that gives rise to new land-use management practices 189</p> <p>Appendices 191</p> <p>A.1. Preliminary thoughts on fractal analysis software 191</p> <p>A.2. Instructions for the following programs 192</p> <p>A.3. Software programs for the visual approach of a satellite or cartographic series or image 193</p> <p>A.4. Software programs for calculating fractal dimensions for a chronicle or frequency series 198</p> <p>A.5. Software programs for calculating the fractal dimensions of a satellite image or map 208</p> <p>A.6. Software programs for calculating multifractal spectrums of a series and an image 213</p> <p>Bibliography 221</p> <p>Index 239</p>

<p><strong>André Dauphiné</strong> is Emeritus Professor at the?University of Nice Sophia-Antipolis, France.

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