简介
magine an equilateral triangle. Now, imagine smaller equilateral triangles perched in the center of each side of the original triangle--you have a Star of David. Now, place still smaller equilateral triangles in the center of each of the star's 12 sides. Repeat this process infinitely and you have a Koch snowflake, a mind-bending geometric figure with an infinitely large perimeter, yet with a finite area. This is an example of the kind of mathematical puzzles that this book addresses. The Fractal Geometry of Nature is a mathematics text. But buried in the deltas and lambdas and integrals, even a layperson can pick out and appreciate Mandelbrot's point: that somewhere in mathematics, there is an explanation for nature. It is not a coincidence that fractal math is so good at generating images of cliffs and shorelines and capillary beds.
目录
1. Introduction
Theme
The irregular and fragmented in nature
Dimension, symmetry, divergence
Variations and disclaimers
2. Three classic fractals, tamed
How long is the coasts of Britain?
Snowflakes and other Koch curves
Harnessing the Peano Monster curves
Fractal events and Cantor dusts
3. Galaxies and eddies
Fractal view of galaxy clusters
Geometry of turbulence; Intermittency
Fractal singularities of differential equations
4. Scaling fractals
Length-area-volume relations
Islands, clusters and percolation; Diameter-number relations
Ramification and fractal lattices
5. Nonscaling fractals
Surfaces with positive volume, and flesh
Trees; Scaling residues; nonuniform fractals
Trees and the diameter exponent
6. Self-mapping fractals
Self-inverse fractals, Apollonian nets and soap
Cantor and fatou dusts; Self-squared dragons
Fractal attractors and fractal (Chaotic) evolutions
7. Randomness --
Chance as a tool in model making
Conditional stationarity and cosmographic principles
8. Stratified random fractals
Random curds: contact clusters and fractal percolation
Random chains and squigs
Brownian motion and brown fractals
Random midpoint displacement curves
9. Fractional brown fractals
River discharges; scaling nets and noises
Relief and coastlines
The areas of Islands, lakes and cups
Isothermal surfaces of homogeneous turbulence
10. Random tremas; Texture
Interval tremas; Linear levy dusts
Subordination; Spatial levy dusts; ordered galaxies
Disc and sphere tremas: moon craters and galaxies
Texture: gaps and lacunarity; Cirri and succolarity
General tremas, and the control of texture
11. Miscellany
Logic of fractals in statistical lattice physics
Price change and scaling in economics
Scaling and power laws without geometry
Mathematical backup and addenda
12. Of men and ideas
Biographical sketches --
Historical sketches
The path to fractals.
Theme
The irregular and fragmented in nature
Dimension, symmetry, divergence
Variations and disclaimers
2. Three classic fractals, tamed
How long is the coasts of Britain?
Snowflakes and other Koch curves
Harnessing the Peano Monster curves
Fractal events and Cantor dusts
3. Galaxies and eddies
Fractal view of galaxy clusters
Geometry of turbulence; Intermittency
Fractal singularities of differential equations
4. Scaling fractals
Length-area-volume relations
Islands, clusters and percolation; Diameter-number relations
Ramification and fractal lattices
5. Nonscaling fractals
Surfaces with positive volume, and flesh
Trees; Scaling residues; nonuniform fractals
Trees and the diameter exponent
6. Self-mapping fractals
Self-inverse fractals, Apollonian nets and soap
Cantor and fatou dusts; Self-squared dragons
Fractal attractors and fractal (Chaotic) evolutions
7. Randomness --
Chance as a tool in model making
Conditional stationarity and cosmographic principles
8. Stratified random fractals
Random curds: contact clusters and fractal percolation
Random chains and squigs
Brownian motion and brown fractals
Random midpoint displacement curves
9. Fractional brown fractals
River discharges; scaling nets and noises
Relief and coastlines
The areas of Islands, lakes and cups
Isothermal surfaces of homogeneous turbulence
10. Random tremas; Texture
Interval tremas; Linear levy dusts
Subordination; Spatial levy dusts; ordered galaxies
Disc and sphere tremas: moon craters and galaxies
Texture: gaps and lacunarity; Cirri and succolarity
General tremas, and the control of texture
11. Miscellany
Logic of fractals in statistical lattice physics
Price change and scaling in economics
Scaling and power laws without geometry
Mathematical backup and addenda
12. Of men and ideas
Biographical sketches --
Historical sketches
The path to fractals.
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