Benoit Mandelbrot

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Benoit B. Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

Published Works

• Fractional Brownian Motions, Fractional Noises and Applications
• The Variation of Certain Speculative Prices
• The Fractal Geometry of Nature
• Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
• Fractal character of fracture surfaces of metals
• Noah, Joseph, and Operational Hydrology
• Fractals and Scaling in Finance
• Some long-run properties of geophysical records
• Self-Affine Fractals and Fractal Dimension
• Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence
Random Walk Models for the Spike Activity of a Single Neuron
Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances
New Methods in Statistical Economics
The Pareto-Levy Law and the Distribution of Income
Critical Phenomena on Fractal Lattices
Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation
On the Distribution of Stock Price Differences
When Can Price be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models
The Variation of Some Other Speculative Prices
The Science of Fractal Images
A Fast Fractional Gaussian Noise Generator
Forecasts of Future Prices, Unbiased Markets, and "Martingale" Models
Geometric Implementation of Hypercubic Lattices with Noninteger Dimensionality by Use of Low Lacunarity Fractal Lattices
A Multifractal Walk down Wall Street
On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars
Long-Run Linearity, Locally Gaussian Process, H-Spectra and Infinite Variances
FRACTAL ASPECTS OF THE ITERATION OF z →Λz(1- z) FOR COMPLEX Λ AND z
Multifractals and 1/ƒ Noise
Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence
Limit theorems on the self-normalized range for weakly and strongly dependent processes
Computer Experiments with Fractional Gaussian Noises: Part 2, Rescaled Ranges and Spectra
Negative fractal dimensions and multifractals
Die fraktale Geometrie der Natur
Fractals in physics: Squig clusters, diffusions, fractal measures, and the unicity of fractal dimensionality
Fractals and Chaos
Stable Paretian Random Functions and the Multiplicative Variation of Income
Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix
Physical Properties of a New Fractal Model of Percolation Clusters
The potential distribution around growing fractal clusters
Exactly self-similar left-sided multifractal measures
The Role of Sufficiency and of Estimation in Thermodynamics
Plane DLA is not self-similar; is it a fractal that becomes increasingly compact as it grows?
SELF-AFFINE FRACTAL SETS, I: THE BASIC FRACTAL DIMENSIONS
Intermittent turbulence and fractal dimension: Kurtosis and the spectral exponent 5/3+B
Paretian Distributions and Income Maximization
New “anomalous” multiplicative multifractals: Left sided ƒ(α) and the modelling of DLA
Temperature Fluctuation: A Well‐Defined and Unavoidable Notion
The variation of certain speculative prices
Renewal sets and random cutouts
On the Derivation of Statistical Thermodynamics from Purely Phenomenological Principles
A note on a class of skew distribution functions: Analysis and critique of a paper by H. A. Simon
Comments on: "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," by Peter K. Clark
Fractal Analysis and Synthesis of Fracture Surface Roughness and Related Forms of Complexity and Disorder
Angular Gaps in Radial Diffusion-Limited Aggregation: Two Fractal Dimensions and Nontransient Deviations from Linear Self-Similarity
On the theory of word frequencies and on related Markovian models of discourse
The Stable Paretian Income Distribution when the Apparent Exponent is Near Two
Invariant multifractal measures in chaotic Hamiltonian systems, and related structures
Multifractality of the harmonic measure on fractal aggregates, and extended self-similarity
Lévy dusts, Mittag-Leffler statistics, mass fractal lacunarity, and perceived dimension
On the quadratic mapping z→z2-μ for complex μ and z: The fractal structure of its set, and scaling
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk
Broken line process derived as an approximation to fractional noise
Parallel diffusion-limited aggregation
Convergence of complex multiplicative cascades
Self-similarity of harmonic measure on DLA
On Dvoretzky coverings for the circle
The Canopy and Shortest Path in a Self-Contacting Fractal Tree
Inverse Measures, the Inversion Formula, and Discontinuous Multifractals
“New Methods of Statistical Economics,” revisited: Short versus long tails and Gaussian versus power-law distributions
Local Regularity of Nonsmooth Wavelet Expansions and Application to the Polya Function
A Fractal’s Lacunarity, and how it can be Tuned and Measured
Gap Independence and Lacunarity in Percolation Clusters
Final note on a class of skew distribution functions: Analysis and critique of a model due to H. A. Simon
Exceptions to the multifractal formalism for discontinuous measures
Multifractal Measures, Especially for the Geophysicist
Random Walks, Fire Damage Amount and Other Paretian Risk Phenomena
Inversion Formula for Continuous Multifractals
Measures of Fractal Lacunarity: Minkowski Content and Alternatives
Fractal Geometry in Physics
Renormalization and fixed points in finance, since 1962
Correction of an Error in "The Variation of Certain Speculative Prices" (1963)
3. Mild Vs. Wild Randomness: Focusing on Those Risks that Matter
Mandelbrot on price variation
A Class of Multinomial Multifractal Measures with Negative (Latent) Values for the “Dimension” f(α)
The Fractal Geometry of Trees and Other Natural Phenomena
Squig sheets and some other squig fractal constructions
Note on the definition and the stationarity of fractional Gaussian noise
Multifractals and Fractals
Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures. II)
Scalebound or Scaling Shapes: A Useful Distinction in the Visual Arts and in the Natural Sciences
Fractals and the Rebirth of Iteration Theory
Fractional multiplicative processes
Introduction to infinite products of independent random functions (Random multiplicative multifractal measures. I)
Simpie games of strategy occurring in communication through natural languages
Uniform convergence for complex [0, 1]-martingales
Parallel cartoons of fractal models of finance
The inescapable need for fractal tools in finance
Comment on the equivalence between fracton/spectral dimensionality, and the dimensionality of recurrence
Some “Facts” That Evaporate Upon Examination
SELF-AFFINE FRACTAL SETS, II: LENGTH AND SURFACE DIMENSIONS
Heavy Tails in Finance for Independent or Multifractal Price Increments
Fractals and an Art for the Sake of Science
The variation of the prices of cotton, wheat, and railroad stocks, and of some financial rates
Lecture V Fractals and turbulence: Attractors and dispersion
Variability of the form and of the harmonic measure for small off-off-lattice diffusion-limited aggregates
The Fractal Geometry of Nature
Industrial concentration and scaling
Exactly Self-Similar Left-Sided Multifractals
Fractal aggregates, and the current lines of their electrostatic potentials
Post scriptum to “final note”
Reply [to “Comments on ‘Noah, Joseph, and Operational Hydrology’ by Benoit B. Mandelbrot and James R. Wallis”]
Comment on ‘Stochastic Models in Hydrology’ by Adrian E. Scheidegger
Mathematical Methods in the Social Sciences, 1959, Proceedings of the First Stanford Symposium (Kenneth J. Arrow, Samuel Karlin, and Patrick Suppes, eds.)
Stable Fractal Sums of Pulses: The Cylindrical Case
Introduction to fractal sums of pulses
Linear Regression With Non-Normal Error Terms: A Comment
New methods in statistical economics
A case against the lognormal distribution
Wie lang ist die Küste Britanniens?
Sporadic Turbulence
Fractal analysis and synthesis of fracture surface roughness and related forms of complexity and disorder
Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
The Fractal Geometry of Percolation, Polymers and Almost Everything Else
The Statistics of Natural Resources and the Law of Pareto
Unser Thema
On the Dynamics of Iterated Maps VIII: The Map z→λ(z+1/z), from Linear to Planar Chaos, and the Measurement of Chaos
Peano-pólya motions, when time is intrinsic or binomial (uniform or multifractal)
Fractals
ON THE AGGREGATIVE FRACTALS CALLED “SQUIGS”, WHICH INCLUDE RECURSIVE MODELS OF POLYMERS AND OF PERCOLATION CLUSTERS
SELF-AFFINE FRACTAL SETS, III: HAUSDORFF DIMENSION ANOMALIES AND THEIR IMPLICATIONS
A note on a law of berry and on insistence stress
TESTS OF THE DEGREE OF WORD CLUSTERING IN SAMPLES OF WRITTEN ENGLISH
The Fractal Geometry of Nature.
Les fractales, les monstres et la beauté
Fractals
Hurst Coefficient (Rescaled Range Analysis)
Fractal Sums of Pulses and a Practical Challenge to the Distinction Between Local and Global Dependence
Louis Bachelier
Easy and Natural Generation of Multifractals: Multiplying Harmonics of Periodic Functions
Iterated random multiplications and invariance under randomly weighted averaging
Filtering and specification of self-affinity
Harold Edwin Hurst
Introduction
Scaling distributions and income maximization
Comments by P. H. Cootner, E. Parzen, and W. S. Morris (1960s) and responses
Nonlinear forecasts, rational bubbles, and martingales
Discontinuity and scaling: their scope and likely limitations
Limitations of efficiency and of martingale models
Rank-size plots, Zipf’s law, and scaling
Introduction to papers on quadratic dynamics: a progression from seeing to discovering
Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski
Das Unregelmäßige und Splitterhafte in der Natur
Fraktale Attraktoren und fraktale («chaotische») Evolutionen
Schneeflocken und andere Koch-Kurven
Some Mathematical Questions Arising in Fractal Geometry
Exactly Self-similar Left-sided Multifractals
Editorial
Self-inverse fractals osculated by sigma-discs and the limit sets of inversion groups
Book review
Gibbs and Mandelbrot at Yale
DISCUSSION PAPER: FRACTALS, ATTRACTORS, AND THE FRACTAL DIMENSION
Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling (Ingram Olkin, Sudhist G. Ghurye, Wassily Hoeffding, William G. Madow, and Henry B. Mann, eds.)
La Theorie Physique au Sens de Boltzmann (René Dugas)
Some Aspects of the Random walk Model of Stock Market Prices: Comment
Logique, langage et theorie de l'information
Fractals—I
Fractional Brownian Motions and Fractional Gaussian Noises
Fractional Brownian Motions and Fractional Gaussian Noises
Fractional Integrodifferentiation
Fractional Integrodifferentiation
Gibbs Distributions I
Gibbs Distributions—I
Hurst Coefficient
Panorama of grid-bound self-affine variability
Sporadic random functions and conditional spectra; self-similar examples and limits
Random sets of multiplicity for trigonometric series (Kahane & M 1965)
Sporadic turbulence
Intermittent free turbulence
Lognormal hypothesis and distribution of energy dissipation in intermittent turbulence
“On certain martingales of Benoit Mandelbrot” Guest contribution (Kahane & Peyrière 1976)
Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3 + B
Fractal dimension, dispersion, and singularities of fluid motion
Sketches of prehistory and history
Scaling, invariants and fixed points
Short excerpts (1964–1984)
New model for the clustering of errors on telephone circuits (Berger & M 1963)
Self-similarity and conditional stationarity
1/ƒ noises and the infrared catastrophe
Co-indicator functions and related 1/ƒ noises
L-stable model for the distribution of income
L-stability and multiplicative variation of income
Computation of the L-stable distributions
Sources of inspiration and historical background
States of randomness from mild to wild, and concentration from the short to the long run
Self-similarity and panorama of self-affinity
Proportional growth with or without diffusion, and other explanations of scaling
Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs
Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets
Introduction to papers on chaos in nonquadratic dynamics: rational functions devised from doubling formulas
The map z → λ (z+1/z) and roughening of chaos, from linear to planar (computer-assisted homage to K. Hokusai)
Two nonquadratic rational maps devised from Weierstrass doubling formulas
Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments
Self-inverse fractals, Apollonian nets, and soap
Symmetry with respect to several circles: dilation/reduction, fractals, and roughness
Self-inverse fractals osculated by sigma-discs: the limit sets of (“Kleinian”) inversion groups
Acknowledgments related to quadratic dynamics
Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988)
The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems
Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991)
The inexhaustible function z squared plus c
The Fatou and Julia stories
Mathematical analysis while in the wilderness
Fractal aspects of the iteration of z→λz(1-z) for complex λ and z
Cantor and Fatou dusts; self-squared dragons
The complex quadratic map and its ℳ-set
Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell
The “normalized radical” of the ℳ-set
The boundary of the ℳ-set is of dimension 2
Certain Julia sets include smooth components
A Geometry Able to Include Mountains and Clouds
Fractal Financial Fluctuations
Die Geometrie der Turbulenz. Intermittenz
Fraktale Singularitäten von Differentialgleichungen
Länge-Fläche-Volumen-Relationen
Inseln, Klumpen und Perkolation. Durchmesser-Anzahl-Relationen
Verzweigung und fraktale Gitter
Flächen mit positivem Volumen und Fleisch
Bäume. Skaleninvarianz-Residuen. Ungleichmäßige Fraktale
Bäume und der Durchmesser-Exponent
Selbstinverse Fraktale, apollonische Netze und Seife
Cantor-Staub und Fatou-Staub. Selbstquadrierte Drachen
Der Zufall als ein Mittel zur Modellierung
Bedingte Stationarität und kosmographische Prinzipien
Zufälliger Quark: Kontaktklumpen und fraktale Perkolation
Zufällige Ketten und Schnörkel
Brownsche Bewegung und Brown-Fraktale
Kurven aus zufälligen Mittelpunktsverschiebungen
Wasserstände. Skaleninvariante Netze und skaleninvariantes Rauschen
Relief und Küstenlinien
Die Flächeninhalte von Inseln, Seen und Schüsseln
Dimension, Symmetrie, Divergenz
Isothermalflächen bei homogener Turbulenz
Intervalle als Tremata. Linearer Lévy-Staub
Subordination. Räumlicher Lévy-Staub. Geordnete Galaxien
Kreis- und Kugeltremata: Mondkrater und Galaxien
Textur: Lücken und Lakunarität; Zirruswolken und Sukkolarität
Allgemeine Tremata und die Steuerung der Textur
Die Logik der Fraktale in der statistischen Gitterphysik
Preisänderungen und Skaleninvarianz in der Ökonomie
Skaleninvarianz und Potenzgesetze ohne Geometrie
Mathematische Hintergründe und Ergänzungen
Variationen und Dementis
Biographische Skizzen
Historische Skizzen
Epilog: Der Weg zu den Fraktalen
Nachtrag vom Dezember 1982
Vom Nutzen der Peanoschen Monster-Kurven
Fraktale Ereignisse und Cantor-Staub
Die fraktale Sicht auf Galaxienhaufen
Vorwort
Discussion Panel
Information sans Interpretation dans la Description des Langues Reelles
Limit Theorem on the Self-Normalized Range for Weakly and Strongly Dependent Process
A new model of percolation clusters
Théorie de la pré-correction des erreurs de transmission
Book review
Book review
Book review
Book review
Book review
Book review
Comment on Dr. Herdan's letter
Book review
Book review
Benoit Mandelbrot
Continuous Interpolation of the Complex Discrete Mapz→ λz(1 -z), and Related Topics.On the dynamics of iterated mapsIX
Erratum: Variability of the form and of the harmonic measure for small off-off-lattice diffusion-limited aggregates
Information and Decision Processes (Robert E. Machol, ed.)
Scalebound or Scaling Shapes: A Useful Distinction in the Visual Arts and in the Natural Sciences
Foreword
Fractais
FOREWORD
TOWARDS A REVIVAL OF THE STATISTICAL LAW OF PARETO
Fractals: Form, Chance, and Dimension
Fractals: Form, Chance, and Dimension
Fractals: Form, Chance, and Dimension.
FOREWORD
Algunas cuestiones de teoría de la observación en el contexto de las modernas teorías de la inferencia estadística (Presentación y traducción de A. Javier Izquierdo)
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Schools

California Institute of Technology

Private research university located in Pasadena, California

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Yale University

Private research university in New Haven, Connecticut, United States

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