The Fractal Geometry of Nature

Benoit Mandelbrot was a graduate of Ecole Polytechnique and the University of Paris, and received a masters degree in aeronautics at the California Institute of Technology. In 1958 he joined the IBM Thomas J. Research Council, where he was an IBM Fellow, taking periodic leaves to teach at Harvard University as Visiting Professor of Economics, and, later, of applied mathematics. He also taught at Yale, the University of Paris-Sud, M.I.T., and the Albert Einstein College of Medicine. He was a Fellow of the Guggenheim Foundation, Trumbull Lecturer at Yale, Samuel Wilks Lecturer at Princeton, and Abraham Wald Lecturer at Columbia. It was as Lecturer at Collge de France that he gave in 1973 and 1974 the lessons which eventually were developed into the present essay; he discovered the Mandelbrot set in 1980 through his use of the computing power available to him at IBM.

i introduction

1 theme

2 the irregular and fragmented in nature

3 dimension, symmetry, divergence

4 variations and disclaimers

ii three classic fractals, tamed

5 how long is the coast of britain?

6 snowflakes and other koch curves

7 harnessing the peano monster curves

8 fractal events and cantor dusts

iii galaxies and eddies

9 fractal view of galaxy clusters

10 geometry of turbulence; intermittency

11 fractal singularities of differential equations

iv scaling fractals

12 length-area-volume relations

13 islands, clusters, and percolation; diameter-number relations

14 ramification and fractal lattices

v nonscaling fractals

15 surfaces with positive volume, and flesh

16 trees; scaling residues; nonuniform fractals

17 trees and the diameter exponent

vi self-mapping fractals

18 self-inverse fractals, apollonian nets, and soap

19 cantor and fatou dusts; self-squared dragons

20 fractal attractors and fractal (' chaotic") evolutions

vii randomness

21 chance as a tool in model making

22 conditional stationarity and cosmographic principles

viii stratified random fractals

23 random curds: contact clusters and fractal percolation

24 random chains and squigs

25 brownian motion and brown fractals

26 random midpoint displacement curves

ix fractional brown fractals

27 river discharges; scaling nets and noises

28 relief and coastlines

29 the areas of islands, lakes, and cups

a book-within-the-book, in color

30 isothermal surfaces of homogeneous turbulence

x random tremas; texture

31 interval tremas; linear levy dusts

32 subordination; spatial levy dusts; ordered galaxies

33 disc and sphere tremas: moon craters and galaxies

34 texture: gaps and lacunarity; cirri and succolarity

35 general tremas, and the control of texture

xi miscellany

36 logic of fractals in statistical lattice physics

37 price change and scaling in economics

38 scaling and power laws without geometry

39 mathematical backup and addenda

xii of men and ideas

40 biographical sketches

41 historical sketches

42 epilog: the path to fractals

list of references

acknowledgments

index of selected dimensions

index of names and subjects