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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Fractal Geometry

An educational resource on the mathematical framework and formalism from the Yale University, covering the concept of self similarity. Includes topical examples, images, algorithms and software.

Mandelbrot, Benoit B.

Founder of fractal geometry. Includes biography, vita, publications, interviews, and reviews.

3D Fractals Bicomplex Dynamics

Resource on the bicomplex generalization of the Mandelbrot set. Includes scientific publications, illustrations, news and downloads.

Bitshifters

Images generated by different commercial applications. Includes FAQs and tutorial.

Chaos and Economics

Introduction to chaos, attractors and dynamic systems theory. Includes mathematical formulation, images and references, especially from an economical point of view.

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.

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.

Fractal Brownian Archipelago

Shows how brownian motion can model the shape of coastlines. Includes interactive demonstration and a collection of island set.

Fractal Dimensions

Easy to comprehend mathematical approach to understanding the significance of the applied study of fractals and attractors. Includes didactic examples and illustrations.

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.

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.

Fractals: The Beautiful Complex Patterns Generated by Simple Iterative Computations

Outlines a general systems theory for chaos, quantum mechanics and gravity as applied to weather patterns. Includes illustrations, scientific publications and references.

Homepage of Kristian Gustavsson

Weblog about the mathematical background of different sets and attractors in the complex plane. Includes downloadable generator and gallery.

Images From Chaos

Gallery of chaotic and complex systems and attractors from the University of Zaragoza, Spain.

Introduction to Lacunarity

Analysis of the degree of gappiness of different sets. Includes mathematical aspects, results and publications.

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.

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.

Kleinian Groups Pictures

Discusses the mathematical theory of Kleinian groups. Includes illustrations, examples, formalism and program source code.

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.

Mathematical Figures Using Mathematica

Collection of sets and attractors. Includes Mathematica source code, mathematical formulations and illustrations.

Mathematics: Stilldreamer

Explains the basics of Sierpinski systems and other sets. Includes interactive example programs with source code.

The Modular Group and Fractals

Explains the basics of fractals, Riemann Zeta, modular group gamma, Farey fractions and Minkowski question mark. Includes publications.

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.

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.

Technocosm

Focuses on the visualization of three dimensional attractors. Includes formula derivations and image galleries.

Tetrabrot Fractal Videos

Collection of videos made by rotation, zooming, and cycling through the four-dimensional Tetrabrot sets. Includes basics, mathematical formulations and descriptions.

Wikipedia Category: Fractals

Collection of encyclopedia articles about self-similar geometric objects with both aesthetical and scientific uses.

Wikipedia: Fractal

Free encyclopedia article covering historical aspects and mathematical formulations. Includes two and three dimensional illustration sets.

The 3x+1 Fractal

Abstract about paper that generalizes the Collatz problem to complex numbers.
(June 01, 2004)

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