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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.
http://www.3dfractals.com/
Resource on the bicomplex generalization of the Mandelbrot set. Includes scientific publications, illustrations, news and downloads.
http://bitshifters.com/fractal.html
Images generated by different commercial applications. Includes FAQs and tutorial.
http://www.bentamari.com/
Introduction to chaos, attractors and dynamic systems theory. Includes mathematical formulation, images and references, especially from an economical point of view.
http://math.bu.edu/DYSYS/arcadia/
Article on the mathematical ideas lurking in the background of Tom Stoppard's play Arcadia. Includes examples, illustrations and references.
http://www.efg2.com/Lab/FractalsAndChaos
Software and information resource on the Mandelbrot set, geometrical explosion sets, and attractors. Includes diagrams and mathematical backgrounds.
http://swiss.csail.mit.edu/~rauch/islands/
Shows how brownian motion can model the shape of coastlines. Includes interactive demonstration and a collection of island set.
http://math.rice.edu/~lanius/fractals/dim.html
Easy to comprehend mathematical approach to understanding the significance of the applied study of fractals and attractors. Includes didactic examples and illustrations.
http://www.fractalfoundation.org/
Foundation with purpose of educating people about the mathematical theory and the interconnectedness of complex systems. Includes mission statement, mathematical framework, gallery and contact.
http://classes.yale.edu/fractals/
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.
http://www.mi.sanu.ac.rs/vismath/javier1/
Scientific paper of the University of the Basque Country, Spain, addressing the mathematical aspects of multi layer colorization. Includes examples and references.
http://www.dd.chalmers.se/~f99krgu/main/index.html
Weblog about the mathematical background of different sets and attractors in the complex plane. Includes downloadable generator and gallery.
http://www.cps.unizar.es/~jlsubias/
Gallery of chaotic and complex systems and attractors from the University of Zaragoza, Spain.
http://groups.csail.mit.edu/mac/users/rauch/lacunarity/lacunarity.html
Analysis of the degree of gappiness of different sets. Includes mathematical aspects, results and publications.
http://www.cut-the-knot.org/blue/Mandel.shtml
Discusses how differently the iterations behave depending on which portions the coefficients are plucked from. Includes basics, concept, formulations and references.
http://aleph0.clarku.edu/~djoyce/julia/explorer.html
Online navigator for various sets and attractors from the Clark University. Includes background and a short course on complex numbers.
http://klein.math.okstate.edu/kleinian/
Discusses the mathematical theory of Kleinian groups. Includes illustrations, examples, formalism and program source code.
http://www.math.yale.edu/mandelbrot/
Founder of fractal geometry. Includes biography, vita, publications, interviews, and reviews.
http://mathforum.org/advanced/robertd/
Collection of sets and attractors. Includes Mathematica source code, mathematical formulations and illustrations.
http://aleph0.clarku.edu/~djoyce/newton/newton.html
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.
http://quantumfuture.net/quantum_future/papers/qfract/
Explains how quantum jumps generate new family of fractals on spherical canvas. Includes graphics in several formats, mathematical framework and bibliography.
http://technocosm.org/chaos/
Focuses on the visualization of three dimensional attractors. Includes formula derivations and image galleries.
http://www.javaspider.com/jfract
Collection of videos made by rotation, zooming, and cycling through the four-dimensional Tetrabrot sets. Includes basics, mathematical formulations and descriptions.
http://www.sciencedirect.com/science/article/pii/S0097849304000366
Abstract about paper that generalizes the Collatz problem to complex numbers.
http://math.bu.edu/DYSYS/dysys.html
Educational resource from the Boston University. Includes mathematical framework and formulation, animated illustrations and calculation spreadsheets.
http://www.ibiblio.org/e-notes/MSet/Contents.htm
Scientific publication about the anatomy of different sets and attractors and chaotic dynamics. Includes animated samples, articles and mathematical formulations.
http://www.linas.org/math/sl2z.html
Explains the basics of fractals, Riemann Zeta, modular group gamma, Farey fractions and Minkowski question mark. Includes publications.
http://en.wikipedia.org/wiki/Category:Fractals
Collection of encyclopedia articles about self-similar geometric objects with both aesthetical and scientific uses.
http://en.wikipedia.org/wiki/Fractal
Free encyclopedia article covering historical aspects and mathematical formulations. Includes two and three dimensional illustration sets.
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