Fractal Geometries Theory and Applications

Part 1 The discovery of fractal geometry: the surveyor's task - rectifiable curves, measuring by arc lengths a journey into pathology - the search for lost rectifiability what is a measure? from line to surface, a simple expression of fractality - self-similarity fractal closed curves (loops) - perimeter, area, density, fractal mass dimension, the concept of co-dimension scaling laws with variable ratios from self-similarity to self-affinness. Part 2 Measures of dimension - time in fractal geometry practical methods - different measures of dimension two methods for measuring the fractal dimension relation between time and measure, parametrization of fractal curves case where the geometry is a transfer function, physical analysis and non-integral derivation an irregularity parameter for continuous non-derivation of non-integral order spectral analysis and non-integral derivation an irregularity parameter for continuous non-derivable functions - the maximum-order of derivation. Part 4 Composition of fractal geometries: statistical aspects - multifractality - the combination of several fractal dimensions hyperfractality - combination of derivations of different orders. Part 5 Applications: measure and uncertainty fractal morphogenesis from fractal geometry to irreversibility complexity.