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Fractal dimension

About: Fractal dimension is a research topic. Over the lifetime, 14764 publications have been published within this topic receiving 329050 citations.


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Book
01 Jan 1992
TL;DR: This book discusses Fractal Image Compression, the Causality Principle, Deterministic Laws and Chaos, and the Backbone of Fractals.
Abstract: Causality Principle, Deterministic Laws and Chaos.- The Backbone of Fractals: Feedback and the Iterator.- Classical Fractals and Self-Similarity.- Lim and Self-Similarity.- Length, Area and Dimension: Measuring Complexity and Scaling Properties.- Encoding Images by Simple Transformations.- The Chaos Game: How Randomness Creates Deterministic Shapes.- Recursive Structures: Growing Fractals and Plants.- Pascal's Triangle: Cellular Automata and Attractors.- Irregular Shapes: Randomness in Fractal Constructions.- Deterministic Chaos: Sensitivity, Mixing, and Periodic Points.- Order and Chaos: Period-Doubling and Its Chaotic Mirror.- Strange Attractors: The Locus of Chaos.- Julia Sets: Fractal Basin Boundaries.- The Mandelbrot Set: Ordering the Julia Sets.

1,920 citations

Journal ArticleDOI
TL;DR: In this article, the fractal dimension of the set of points (t, f(t)) forming the graph of a function f defined on the unit interval was measured using a self-similarity property.

1,825 citations

Book
01 Aug 1988
TL;DR: Fractal Modelling of Real World Images and a Unified Approach to Fractal Curves and Plants are studied.
Abstract: Contents: Foreword: People and Events Behind the "Science of Fractal Images".- Fractals in Nature: From Characterization to Simulation.- Algorithms for Random Fractals.- Color Plates and Captions.- Fractal Patterns Arising in Chaotic Dynamical Systems.- Fantastic Deterministic Fractals.- Fractal Modelling of Real World Images.- Fractal Landscapes Without Creases and with Rivers.- An Eye for Fractals.- A Unified Approach to Fractal Curves and Plants.- Exploring the Mandelbrot Set.- Bibliography.- Index.

1,752 citations

Book
27 Sep 2001
TL;DR: In this article, the authors present a mathematical model for chaotic multidimensional flows and fractal dimension calculation based on the Lyapunov exponents and the Hamiltonian chaos.
Abstract: Preface 1. Introduction 2. One-dimensional maps 3. Nonchaotic multidimensional flows 4. Dynamical systems theory 5. Lyapunov exponents 6. Strange attractors 7. Bifurcations 8. Hamiltonian chaos 9. Time-series properties 10. Nonlinear prediction and noise reduction 11. Fractals 12. Calculation of fractal dimension 13. Fractal measure and multifractals 14. Nonchaotic fractal sets 15. Spatiotemporal chaos and complexity A. Common chaotic systems B. Useful mathematical formulas C. Journals with chaos and related papers Bibliography Index

1,687 citations

Journal ArticleDOI
01 Apr 1984-Nature
TL;DR: In this article, a new method, slit island analysis, is introduced to estimate the fractal dimension, D. The estimate is shown to agree with the value obtained by fracture profile analysis, a spectral method.
Abstract: When a piece of metal is fractured either by tensile or impact loading (pulling or hitting), the fracture surface that is formed is rough and irregular. Its shape is affected by the metal's microstructure (such as grains, inclusions and precipitates, whose characteristic length is large relative to the atomic scale), as well as by ‘macrostructural’ influences (such as the size, the shape of the specimen, and the notch from which the fracture begins). However, repeated observation at various magnifications also reveals a variety of additional structures that fall between the ‘micro’ and the ‘macro’ and have not yet been described satisfactorily in a systematic manner. The experiments reported here reveal the existence of broad and clearly distinct zone of intermediate scales in which the structure is modelled very well by a fractal surface. A new method, slit island analysis, is introduced to estimate the basic quantity called the fractal dimension, D. The estimate is shown to agree with the value obtained by fracture profile analysis, a spectral method. Finally, D is shown to be a measure of toughness in metals.

1,651 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023548
20221,078
2021504
2020509
2019515
2018454