Chaos and Fractals Math

A fractal is a chaotic mathematic object which can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and are generally self-similar and independent of scale. In many cases a fractal can be generated by a repeating pattern, typically a recursive or iterative process. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken"/"fraction". Chaos theory, in mathematics and physics, deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions. Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder.

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• Images From Chaos - Gallery of chaotic and complex systems and attractors from the University of Zaragoza, Spain.
  
• 3D Mountains - Shows how to create fractal mountains, three-dimensional Mandelbrot and Julia sets, convex, stellated and polyhedra. Includes pictures, plots and mathematical formulations.
  
• Efg's Computer Lab: Fractals and Chaos - Software and information resource on the Mandelbrot set, geometrical explosion sets, and attractors. Includes diagrams and mathematical backgrounds.
  
• Introduction to Lacunarity - Analysis of the degree of gappiness of different sets. Includes mathematical aspects, results and publications.
  
• Dan Freeland - Explains the basics. Includes gallery and free software for exploring different sets and singularities.
  
• Kleinian Groups Pictures - Discusses the mathematical theory of Kleinian groups. Includes illustrations, examples, formalism and program source code.
  
• The 3x+1 Fractal - Paper that generalizes the Collatz problem to complex numbers. Includes insights, results and references.
  
• Fractal Explorer - Tutorial for beginners covering the Mandelbrot and Julia sets, as well as four-dimensional sets. Includes interactive generators and gallery.
  
• Mathematical Figures Using Mathematica - Collection of sets and attractors. Includes Mathematica source code, mathematical formulations and illustrations.
  
• Fractal Dimensions - Easy to comprehend mathematical approach to understanding the significance of the applied study of fractals and attractors. Includes didactic examples and illustrations.
  
• Wikipedia: Fractals - Free encyclopedia article covering historical aspects and mathematical formulations. Includes two and three dimensional illustration sets.
  
• Fractal Map - Interactive online Mandelbrot and Julia generator.
  
• Fractals Unleashed - Tutorial covering the different types of sets and attractors. Includes mathematical formulations, applets, programs, gallery and an art contest. [English, Russian, Ukrainian]
  
• Julia and Mandelbrot Set Explorer - Online navigator for various sets and attractors from the Clark University. Includes background and a short course on complex numbers.
  
• Introduction: Chaos, Fractals And Attractors - Introduction to chaos, attractors and dynamic systems theory. Includes mathematical formulation, images and references.
  
• Chaos, Fractals, and Arcadia - Article on the mathematical ideas lurking in the background of Tom Stoppard's play Arcadia. Includes examples, illustrations and references.
  
• The Dynamical Systems and Technology Project - Educational resource from the Boston University. Includes mathematical framework and formulation, animated illustrations and calculation spreadsheets.
  
• Bitshifters - Images generated by different commercial applications. Includes FAQs and tutorial.
  
• Homepage of Kristian Gustavsson - Weblog about the mathematical background of different sets and attractors in the complex plane. Includes downloadable generator and gallery.
  
• Project Fractales - Research group at the INRIA national research center, France. Includes research details, publications and software.
  
• Fractal Brownian Archipelago - Shows how brownian motion can model the shape of coastlines. Includes interactive demonstration and a collection of island set.
  
• Technocosm - Focuses on the visualization of three dimensional attractors. Includes formula derivations and image galleries.
  
• Newton Basins - Article about the basins of attraction for the Newton's method for finding roots of equations and their resulting representation in the complex plane. Includes mathematical framework and examples.
  
• Iterations and the Mandelbrot Set - Discusses how differently the iterations behave depending on which portions the coefficients are plucked from. Includes basics, concept, formulations and references.
  
• The Mandelbrot and Julia Sets Anatomy - Scientific publication about the anatomy of different sets and attractors and chaotic dynamics. Includes animated samples, articles and mathematical formulations.
  
• Fractal Foundation - Foundation with purpose of educating people about the mathematical theory and the interconnectedness of complex systems. Includes mission statement, mathematical framework, gallery and contact.
  
• Quantum Jumps, EEQT and the Five Platonic Fractals - Explains how quantum jumps generate new family of fractals on spherical canvas. Includes graphics in several formats, mathematical framework and bibliography.
  
• The Modular Group and Fractals - Explains the basics of fractals, Riemann Zeta, modular group gamma, Farey fractions and Minkowski question mark. Includes publications.
  
• A Fractal Christmas - Short Article from Inside Science News Service describing the basics. Includes illustrations and links.
  
• Fractals and Multi Layer Coloring Algorithms - Scientific paper of the University of the Basque Country, Spain, addressing the mathematical aspects of multi layer colorization. Includes examples and references.
  
• The Spanky Fractal Database - Collection of sets, attractors and related material for free distribution. Includes large categorized index of software, information and links.
  
• 3D Fractals Bicomplex Dynamics - Resource on the bicomplex generalization of the Mandelbrot set. Includes scientific publications, illustrations, news and downloads.
  
• Mathematics: Stilldreamer - Explains the basics of Sierpinski systems and other sets. Includes interactive example programs with source code.
  
• IFS Attractors - Index and definition of different attractors. Includes images, plots and glossary.
  
• Tetrabrot Fractal Videos - Collection of videos made by rotation, zooming, and cycling through the four-dimensional Tetrabrot sets. Includes basics, mathematical formulations and descriptions.
  
• The Dynamics of Deterministic Chaos in Numerical Weather Prediction Models - Explains a general systems theory for chaos, quantum mechanics and gravity as applied to weather patterns. Includes illustrations, scientific publications and references.
  
  

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