The Fractal Geometry of Nature

1983/03/01 by Benoit B. Mandelbrot, Benoît B. Mandelbrot, John A. Wheeler +1 · 21,973 citations
Computer Science · Mathematics · #Classical mechanics #Fractal #Geometry #Image Processing and 3D Reconstruction #Mathematical analysis #Mathematics #Morphological variations and asymmetry #Physics #Theoretical physics

paper · doi:10.1119/1.13295

published in American Journal of Physics 51(3), 286-287 (American Institute of Physics)

openalex publication_date 1983/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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• Interoccurrence time statistics in the two-dimensional Burridge-Knopoff earthquake model
• Computer-Aided Diagnosis of Depression Using EEG Signals
• Zipf′s Law, Hierarchical Structure, and Cards‐Shuffling Model for Urban Development
• City size distributions for India and China
• Cellular models for river networks
• Fat Tailed Distributions in Catastrophe Prediction
• Extended electron states and magnetotransport in a 3-simplex fractal
• Small-angle scattering from fat fractals
• Are earthquakes predictable?
• Fractional Schrödinger equation
• Is Earthquake Seismology a Hard, Quantitative Science?
• Unfolding the procedure of characterizing recorded ultra low frequency, kHZ and MHz electromagnetic anomalies prior to the L'Aquila earthquake as pre-seismic ones. Part II
• Self-organized criticality and earthquake predictability: A long-standing question in the light of natural time analysis
• Information storing by biomagnetites
• Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals
• Fractal dimensions of speech sounds: Computation and application to automatic speech recognition
• Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis
• Suboptimal minimum cluster volume cover-based method for measuring fractal dimension
• Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging
• Fractal Dimension of Electrochemical Reactions
• An improved box-counting method for image fractal dimension estimation
• Random matrices, generalized zeta functions and self-similarity of zero distributions
• Texture Analysis of Aggressive and Nonaggressive Lung Tumor CE CT Images
• Contribution of fractal dimension theory into the uniaxial compressive strength prediction of a volcanic welded bimrock
• Static properties of polymer melts in two dimensions
• Self-affine fractality in π+p and K+p collisions at 250 GeV / c
• Random walk on the prime numbers
• Strong-disorder renormalization-group method on fractal lattices: Heisenberg models and magnetoresistive effects in tight-binding models
• Fractal and multifractal analysis: A review
• Multifractal phenomena in physics and chemistry
• ANISOTROPIC MULTIFRACTAL SCALING OF MOUNT LEBANON TOPOGRAPHY: APPROXIMATE CONDITIONING
• Multifractal analysis of crack patterns in reinforced concrete shear walls
• Numerical studies of planar closed random walks
• Polydisperse adsorption: Pattern formation kinetics, fractal properties, and transition to order
• Fractal and nonfractal behavior in Levy flights
• Measuring capital market efficiency: Global and local correlations structure
• Evolution of the human brain: when bigger is better
• Second law of thermodynamics in non-extensive systems
• Anomalous Fluctuations in Autoregressive Models with Long-Term Memory
• Network topology of the desert rose
• Self-similar factor approximants
• GIANT MOLECULAR CLOUD FORMATION IN DISK GALAXIES: CHARACTERIZING SIMULATED VERSUS OBSERVED CLOUD CATALOGS
• Model for anisotropic directed percolation
• Percolation and roughness in ballistic deposition of complex shapes
• Fractal zeta functions and complex dimensions of relative fractal drums
• Evolution and selection of river networks: Statics, dynamics, and complexity
• Conformal invariance and 2𝐷 statistical physics
• Conformal restriction and related questions
• Automatic morphological classification of galaxy images
• The invariances of power law size distributions
• Breakdown of Scale Invariance in the Phase Ordering of Fractal Clusters
• Finite subdivision rules
• An analysis of ridge axis gravity roughness and spreading rate
• Can One Hear the Shape of a Fractal Drum? Partial Resolution of the Weyl-Berry Conjecture
• Multifractal detrended fluctuation analysis of combustion flames in four-burner impinging entrained-flow gasifier
• Fissural volcanism, polygenetic volcanic fields, and crustal thickness in the Payen Volcanic Complex on the central Andes foreland (Mendoza, Argentina)
• Winding number of fractional Brownian motion
• Power Laws for Energy Efficient and Resilient Cities
• Estimating fractal dimension
• The Divider Dimensions of the Irish Coast
• Galactic cold cores
• Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis
• Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background
• Extracting the scaling exponents of a self-affine, non-Gaussian process from a finite-length time series
• Intermittency of three-dimensional perturbations in a point-vortex model
• Is the universe homogeneous? (On large scales)
• Fractal Analysis Based on Hierarchical Scaling in Complex Systems
• Entropy-Variance curves of binary sequences generated by random substitutions of constant length
• Instability and subsequent evolution of electroweak bubbles
• Fractal cosmology in an open universe
• Hidden scale invariance of metals
• Finite-element analysis of contact between elastic self-affine surfaces
• Lévy anomalous diffusion and fractional Fokker–Planck equation
• Multiscale Charge Transport in van der Waals Thin Films: Reduced Graphene Oxide as a Case Study
• Fractional Velocity as a Tool for the Study of Non-Linear Problems
• Tunneling in quantum superlattices with variable lacunarity
• CASA—Practical Aspects
• Tunable power law in the desynchronization events of coupled chaotic electronic circuits
• Multiscale unfolding of real networks by geometric renormalization
• A multifractal detrended fluctuation description of Iranian rial–US dollar exchange rate
• q-exponential, Weibull, and q-Weibull distributions: an empirical analysis
• An observation of a nascent fractal pattern in MD simulation for a fragmentation of an fcc lattice
• Fractal dimension and degree of order in sequential deposition of mixture
• Unique Morphologies of Molybdenum Disulphide: Sheets, Diffusion Limited\n Cluster Aggregates and Fractals, by Langmuir- Blodgett Assembly for Advanced\n Electronics
• On the Complexities of Complex Economic Dynamics
• CHAOTIC MOTION IN MULTI-BLACK HOLE SPACETIMES AND HOLOGRAPHIC SCREENS
• Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems
• Poisson-to-Wigner crossover transition in the nearest-neighbor statistics of random points on fractals
• Complex Networks Renormalization: Flows and Fixed Points
• Anomalous paths in quantum mechanical path-integrals
• WAVE EQUATION FOR FRACTAL SOLID STRING
• Critical behavior in porous media flow
• Critical region in the spatiotemporal dynamics of a turbulent thermoacoustic system and smart passive control
• Fractal entropy of a chain of nonlinear oscillators
• Fractal-based exponential distribution of urban density and self-affine fractal forms of cities
• Scaling of a Slope: The Erosion of Tilted Landscapes
• Galaxies as Rotating Buckets - a Hypothesis on the Gravitational Constant Based on Mach's Principle
• Virial statistical description of non-extensive hierarchical systems
• Spatial Heterogeneity, Scale, Data Character and Sustainable Transport in the Big Data Era
• Time Evolution of Quantum Fractals
• FERMI SURFACES OF CRYSTALS IN A HIGH MAGNETIC FIELD
• What kind of a wave is Hokusai's Great wave off Kanagawa ?
• Fractal scaling of turbulent premixed flame fronts: Application to LES
• Lévy on-off intermittency
• Fast Entropy Estimation for Natural Sequences
• Remarks on quadratic rational maps
• Fractal propagators in QED and QCD and implications for the problem of confinement
• Optimum inhomogeneity of local lattice distortions in La 2 CuO 4+ y
• Multifractal detrended cross-correlation analysis for two nonstationary signals
• Simple model for turbulence intermittencies based on self-avoiding random vortex stretching
• Big Data Is a New Paradigm
• Line Simplification
• Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China
• The relation between the true and observed fractal dimensions of turbulent clouds
• Loading a linear Paul trap to saturation from a magneto-optical trap
• Classical evolution of fractal measures on the lattice
• Structure and Stability of Bovine Casein Micelles
• Design and evolution of modular neural network architectures
• Lacunarity indices as measures of landscape texture
• Lacunarity analysis: A general technique for the analysis of spatial patterns
• Lacunarity definition for ramified data sets based on optimal cover
• Quasirandom Spanning Tree Model for the Early River Network
• Mathematical Intuition Vs. Mathematical Monsters*
• Renormalization of the Hutchinson Operator
• Three-dimensional fingering structures in immiscible flow at the crossover from viscous to capillary fingering
• Box-Counting Dimension Revisited: Presenting an Efficient Method of Minimizing Quantization Error and an Assessment of the Self-Similarity of Structural Root Systems. [europepmc]
• In-Situ X-ray Tomography Observation of Structure Evolution in 1,3,5-Triamino-2,4,6-Trinitrobenzene Based Polymer Bonded Explosive (TATB-PBX) under Thermo-Mechanical Loading. [europepmc]
• Bio-Signal Complexity Analysis in Epileptic Seizure Monitoring: A Topic Review. [europepmc]
• Similarity analysis between species of the genus Quercus L. (Fagaceae) in southern Italy based on the fractal dimension. [europepmc]
• Analysis and classification of droplet characteristics from atomizers using multifractal analysis. [europepmc]
• Systematic Review on Fractal Dimension of the Retinal Vasculature in Neurodegeneration and Stroke: Assessment of a Potential Biomarker. [europepmc]
• Brain Cortical Complexity Alteration in Amyotrophic Lateral Sclerosis: A Preliminary Fractal Dimensionality Study. [europepmc]
• Atypical Value-Driven Selective Attention in Young Children With Autism Spectrum Disorder. [europepmc]
• Poly(vinyl alcohol boric acid)-Diclofenac Sodium Salt Drug Delivery Systems: Experimental and Theoretical Studies. [europepmc]
• Dawn of a new era in industrial photochemistry: the scale-up of micro- and mesostructured photoreactors. [europepmc]
• Tunable Broadband Terahertz Waveband Absorbers Based on Fractal Technology of Graphene Metamaterial. [europepmc]
• Fgf10/Fgfr2b Signaling Orchestrates the Symphony of Molecular, Cellular, and Physical Processes Required for Harmonious Airway Branching Morphogenesis. [europepmc]
• Modality-specific tracking of attention and sensory statistics in the human electrophysiological spectral exponent. [europepmc]
• Alternative Controlling Agent of Theobroma grandiflorum Pests: Nanoscale Surface and Fractal Analysis of Gelatin/PCL Loaded Particles Containing Lippia origanoides Essential Oil. [europepmc]
• Application of Fractal Image Analysis by Scale-Space Filtering in Experimental Mechanics. [europepmc]
• Effects of Supercritical CO 2 on the Pore Structure Complexity of High-Rank Coal with Water Participation and the Implications for CO 2 ECBM. [europepmc]
• Fractal dimension: analyzing its potential as a neuroimaging biomarker for brain tumor diagnosis using machine learning. [europepmc]
• Fractal dimension analysis of resting state functional networks in schizophrenia from EEG signals. [europepmc]
• Alzheimer's disease: new insight in assessing of amyloid plaques morphologies using multifractal geometry based on Naive Bayes optimized by random forest algorithm. [europepmc]
• Identification of Geochemical Anomalies Based on RPCA and Multifractal Theory: A Case Study of the Sidaowanzi Area, Chifeng, Inner Mongolia. [europepmc]
• Pore Size Distribution and Fractal Characteristics of Deep Coal in the Daning-Jixian Block on the Eastern Margin of the Ordos Basin. [europepmc]
• Neuro-evolutionary evidence for a universal fractal primate brain shape. [europepmc]
• Optimizing the neural network and iterated function system parameters for fractal approximation using a modified evolutionary algorithm. [europepmc]
• Formulation and Characterization of Carbopol-Based Porphyrin Gels for Targeted Dermato-Oncological Therapy: Physicochemical and Pharmacotechnical Insights. [europepmc]
• Forward Computational Modeling of Respiratory Airflow. [europepmc]

Related

• Fractal Geometry: Mathematical Foundations and Applications.
• An Analysis of Fractal Features
• Some questions related to fractals
• Fractal universe
• Nombre d’or, fractals et symétries