The Fractal Geometry of Nature

976 Pages·1982·10.89 MB·English

by  Benoit B. Mandelbrot

Table of Contents

Title Page
Copyright Page
Dedication
Foreword

I - INTRODUCTION
Chapter 1 - Theme

PRESENTATION OF GOALS
A SCIENTIFIC CASEBOOK
A MANIFESTO: THERE IS A FRACTAL FACE TO THE GEOMETRY
OF NATURE
MATHEMATICS, NATURE, ESTHETICS
“FRACTAL” AND OTHER NEOLOGISMS
RESTATEMENT OF GOALS

Chapter 2 - The Irregular and Fragmented in Nature

FROM THE PEN OF JEAN PERRIN
WHEN A “GALLERY OF MONSTERS” BECOMES A MUSEUM OF
SCIENCE

Chapter 3 - Dimension, Symmetry, Divergence

THE IDEA OF DIMENSION
DEFINITION OF THE TERM FRACTAL
FRACTALS IN HARMONIC ANALYSIS
OF “NOTIONS THAT ARE NEW,... BUT”
A MATHEMATICAL STUDY OF FORM MUST GO BEYOND
TOPOLOGY
EFFECTIVE DIMENSION
DIFFERENT EFFECTIVE DIMENSIONS IMPLICIT IN A BALL OF
THREAD
SPATIAL HOMOGENEITY, SCALING, AND SELF-SIMILARITY
“SYMMETRIES” BEYOND SCALING
DIVERGENCE SYNDROMES

Chapter 4 - Variations and Disclaimers

OBSCURITY IS NOT A VIRTUE
ERUDITION IS GOOD FOR THE SOUL
“TO SEE IS TO BELIEVE”
THE STANDARD FORM, AND THE NEW FRACTAL FORM, OF
GEOMETRIC “ART”
POINTS OF LOGISTICS
BACK TO THE BASIC THEME

II - THREE CLASSIC FRACTALS, TAMED
Chapter 5 - How Long Is the Coast of Britain?

MULTIPLICITY OF ALTERNATIVE METHODS OF MEASUREMENT
ARBITRARINESS OF THE RESULTS OF MEASUREMENT
IS THIS ARBITRARINESS GENERALLY RECOGNIZED, AND DOES
IT MATTER?
THE RICHARDSON EFFECT
A COASTLINE’S FRACTAL DIMENSION (MANDELBROT 1967s)
HAUSDORFF FRACTAL DIMENSION
A CURVE’S FRACTAL DIMENSION MAY EXCEED 1; FRACTAL
CURVES
Chapter 6 - Snowflakes and Other Koch Curves

SELF-SIMILARITY AND CASCADES
COASTLIKE TERAGONS AND THE TRIADIC KOCH CURVE
THE KOCH CURVE AS MONSTER
THE KOCH CURVE, TAMED. THE DIMENSION D=log 4/log 3=1.2618
THE SIMILARITY DIMENSION
CURVES; TOPOLOGICAL DIMENSION
INTUITIVE MEANING OF D IN THE PRESENCE OF CUTOFFS A
AND λ
ALTERNATIVE KOCH GENERATORS AND SELF-AVOIDING KOCH
CURVES
KOCH ARCS AND HALF LINES
DEPENDENCE OF MEASURE ON THE RADIUS, WHEN D IS A
FRACTION
KOCH MOTION
PREVIEW OF RANDOM COASTLINES
COMPLEX, OR SIMPLE AND REGULAR?
CAPTION OF PLATE 55, CONTINUED

Chapter 7 - Harnessing the Peano Monster Curves

PEANO “CURVES,” MOTIONS, SWEEPS
THE PEANO CURVES AS MONSTERS
THE PEANO CURVES’ TRUE NATURE
RIVER AND WATERSHED TREES
MULTIPLE POINTS ARE UNAVOIDABLE IN TREES, HENCE IN
PEANO MOTIONS
PEANO MOTION AND PERTILING
ON MEASURING DISTANCE BY AREA

Chapter 8 - Fractal Events and Cantor Dusts

NOISE
ERRORS IN DATA TRANSMISSION LINES
BURSTS AND GAPS
A ROUGH MODEL OF ERROR BURSTS: THE CANTOR FRACTAL
DUST C
CURDLING, TREMAS, AND WHEY
OUTER CUTOFF AND EXTRAPOLATED CANTOR DUSTS
DIMENSIONS D BETWEEN 0 AND 1
C IS CALLED DUST BECAUSE D=0
GAPS’ LENGTH DISTRIBUTION
AVERAGE NUMBERS OF ERRORS
TREMA ENDPOINTS AND THEIR LIMITS
THE CANTOR DUSTS’ TRUE NATURE

III - GALAXIES AND EDDIES
Chapter 9 - Fractal View of Galaxy Clusters

IS THERE A GLOBAL DENSITY OF MATTER?
ARE STARS IN THE SCALING RANGE?
IS THERE AN UPPER CUTOFF TO THE SCALING RANGE?
THE FOURNIER UNIVERSE
DISTRIBUTION OF MASS; FRACTAL HOMOGENEITY
FOURNIER UNIVERSE VIEWED AS CANTOR DUST. EXTENSION
TO D#1
THE CHARLIER MODEL AND OTHER FRACTAL UNIVERSES
FOURNIER’S REASON TO EXPECT D=1
HOYLE CURDLING; THE JEANS CRITERION ALSO YIELDS D=1
EQUIVALENCE OF THE FOURNIER AND HOYLE DERIVATIONS
OF D=1
WHY D=1.23 AND NOT D=1?
THE SKY’S FRACTAL DIMENSION
ASIDE ON THE BLAZING SKY EFFECT (WRONGLY CALLED
OLBERS PARADOX)
ASIDE ON NEWTONIAN GRAVITATION
ASIDE ON RELATIVITY THEORY
AN AGGLUTINATED FRACTAL UNIVERSE?
FRACTAL TELESCOPE ARRAYS
SURVEY OF RANDOM FRACTAL MODELS OF GALAXY
CLUSTERS
CUT DIAMONDS LOOK LIKE STARS

Chapter 10 - Geometry of Turbulence; Intermittency

CLOUDS, WAKES, JETS, ETC.
ISOTHERMS, DISPERSION ETC.
OTHER GEOMETRIC QUESTIONS
THE INTERMITTENCY OF TURBULENCE
A DEFINITION OF TURBULENCE
ROLE OF SELF-SIMILAR FRACTALS
INNER AND OUTER CUTOFFS
CURDLING AND FRACTALLY HOMOGENEOUS TURBULENCE
DIRECT EXPERIMENTAL EVIDENCE THAT INTERMITTENCY
SATISFIES D>2
GALAXIES & TURBULENCE COMPARED
(IN)EQUALITIES BETWEEN EXPONENTS (MANDELBROT 1967k,
1976o)
THE TOPOLOGY OF TURBULENCE REMAINS AN OPEN ISSUE

Chapter 11 - Fractal Singularities of Differential Equations

A SPLIT IN TURBULENCE THEORY
THE IMPORTANCE OF SINGULARITIES
CONJECTURE: THE SINGULARITIES OF FLUID MOTION ARE
FRACTAL SETS (MANDELBROT 1976c)
NONVISCOUS (EULER) FLUIDS
VISCOUS (NAVIER-STOKES) FLUIDS
SINGULARITIES OF OTHER NONLINEAR EQUATIONS OF
PHYSICS

IV - SCALING FRACTALS
Chapter 12 - Length-Area-Volume Relations

STANDARD DIMENSIONAL ANALYSIS
PARADOXICAL DIMENSIONAL FINDINGS
FRACTAL LENGTH-AREA RELATION
HOW WINDING IS THE MISSOURI RIVER?
GEOMETRY OF RAIN AND OF CLOUDS
THE AREA-VOLUME RELATION. CONDENSATION BY MICRO-
DROPLETS
MAMMALIAN BRAIN FOLDS
ALVEOLAR AND CELL MEMBRANES
MODULAR COMPUTER GEOMETRY

Chapter 13 - Islands, Clusters, and Percolation; Diameter-Number Relations

KORČAK EMPIRICAL LAW, GENERALIZED
KOCH CONTINENT AND ISLANDS, AND THEIR DIVERSE
DIMENSIONS
THE DIAMETER-NUMBER RELATION
GENERALIZATION TO E>2
FRACTAL DIMENSION MAY BE SOLELY A MEASURE OF
FRAGMENTATION
THE INFINITY OF ISLANDS
SEARCH FOR THE INFINITE CONTINENT
ISLAND, LAKE AND TREE COMBINATION
THE NOTION OF CONTACT CLUSTER
MASS-NUMBER AND WEIGHTED DIAMETER-NUMBER
RELATIONS. THE EXPONENTS D-D AND D/D.
THE MASS EXPONENT Q=2D-D
NONLUMPED CURDLING CLUSTERS
CROSS LUMPED CURDLING CLUSTERS
KNOTTED PEANO MONSTERS, TAMED
BERNOULLI PERCOLATION CLUSTERS
THE CLUSTERS’ FRACTAL GEOMETRY
CRITICAL BERNOULLI CLUSTERS’ D
THE CYPRESS TREES OF OKEFENOKEE
Chapter 14 - Ramification and Fractal Lattices

THE SIERPIŃSKI GASKET AS MONSTER
THE EIFFEL TOWER: STRONG AND AIRY
CRITICAL PERCOLATION CLUSTERS
THE TRIADIC SIERPIŃSKI CARPET
NONTRIADIC FRACTAL CARPETS
TRIADIC FRACTAL FOAM
MENGER’S TRIADIC FRACTAL SPONGE
NONTRIADIC SPONGES AND FOAMS
GAPS’ SIZE DISTRIBUTIONS
THE NOTION OF FRACTAL NET, LATTICE
THE SECTIONS’ FRACTAL DIMENSIONS
THE RAMIFIED FRACTALS VIEWED AS CURVES OR SURFACES
A CURVE’S ORDER OF RAMIFICATION
APPLICATIONS OF RAMIFICATION
ALTERNATIVE FORM OF RAMIFICATION
SECRETS OF THE KOCH PYRAMID
SPHERICAL TREMAS AND LATTICES
PREVIEW OF LACUNARITY

V - NON SCALING FRACTALS
Chapter 15 - Surfaces with Positive Volume, and Flesh

CANTOR DUSTS OF POSITIVE MEASURE
SLOWLY DRIFTING log N/log (1/r)
CURVES WITH POSITIVE AREA
GEOMETRY OF ARTERIES AND VEINS
LEBESGUE-OSGOOD MONSTERS ARE THE VERY SUBSTANCE OF
OUR FLESH!
OF INTUITION, OLD AND NEW