Chaos and Fractals: A Search for Order (A brief introduction to the history and basic concepts of the topic. Excellent for beginners!)
Fantastic Fractals (an outstanding comprehensive site featuring tutorials on fractal images and music, an illustrated "Just for Kids" section, interactive online fractal generators, free fractal software, newsletter, and other resources.)
Our Fractal Universe: Mandelbrot and More (in English, French, or Spanish)
Fractals Unleashed (in English, Russian, or Ukrainian). Also has a "kid's section" and a wide variety of interactive features and many other resources.
The Fractory (An outstanding and mathematically detailed comprehensive site with excellent explanatory graphics!)
Robert L. Devaney's The Dynamical Systems Technology Project At Boston University (Funded by the National Science Foundation, this site was specifically developed to introduce these topics into secondary and college level courses. Also on this site is a section on Chaos, Fractals, and Arcadia which examines the use of concepts in chaos theory in Tom Stoppard's play "Arcadia" to facilitate interdisciplinary studies. This section also provides links to similar "Arcadia" related sites.)
Michael Frame's, Benoit Mandelbrot's, and Nial Neger's site for their course on Fractal Geometry At Yale University (Also funded by the National Science Foundation, this comprehensive site covers topics which include basic concepts, fractal types, and an extensive treatment of the applications of fractal geometry and concepts in many areas such as art, architecture, economics, literature, etc. The site also includes many illustrative and fascinating animations as well! Now listed on major search engines, it serves as an excellent companion site for the one listed above.)
The Mandelbrot - Prechter Debate (The key sites related to the controversy of who discovered the application of fractals to the analysis and prediction of fluctuations in financial markets and how fractals are used for this purpose.)
The Fractal Microscope: A Distributed Computing Approach to Mathematics in Education (An outstanding comprehensive and interactive site that should not be missed!)
Sprott's Fractal Gallery (One of the most outstanding fractal sites on the web with an enormous range and diversity of fractal imagery and information by one of the foremost contributors to fractal mathematics, applications, and visualization. Includes a constantly changing computer-controlled fractal display, 3-D anaglyph fractal images, Strange Attractor animations, and many other features of interest!)
The OpenCourseWare (OCW) project of The Massachusetts Institute of Technology (An educational initiative of enormous significance which is gradually making available all of its course materials on the web. This includes the course syllabus, lecture notes and graphics (in PDF format), and references. Be sure and read the terms of use and reproduction of materials. A truly bold and visionary undertaking and an educational resource freely available to all worldwide!)
Of recent major interest to engineers and those in the new interdisciplinary field of Informatics is the application of chaos theory to optical data transmission systems. In part, this is driven by research on encryption techniques for more secure communication. For more information on this topic, the recent book Chaos-Based Digital Communication Systems by F.C.M. Lau and C.K. Tse (Springer Verlag, 2003) is a highly regarded source. An excellent review in PDF format is available online (from Optics & Photonics News, October, 2004).
An area of current biomedical research is the application of chaos-related mathematics (i.e.,non-linear dynamics) to the understanding of life-threatening abnormal heart rhythms (arrhythmias). Additional information on "Cardiac Chaos" may be found by searching for the term "spiral fibrillation pattern" on Google.
John Mathews' Fractal Resources (A treasure-trove of mathematical resources - mainly on Mandelbrot and Julia Sets - developed for undergraduate research!)
Fractals by Jean-Pierre Louvet
. This outstanding site features topics by the author/artist often not found elsewhere (e.g., how color is applied to fractals, etc.) as well as his own extraordinary gallery of unique fractal images (in French and English). Its server also hosts the legendary Fractal FAQ - a highly informative and very recently updated resource which now includes a section on fractal music! Jacco's Fractal Links (A select and wide-ranging collection of categorized links covering everything from tutorials, galleries, unique sites, practical technical information, and much more! In Dutch but most links are in English)
The Virtual Museum Of Computing: Mandelbrot Exhibition
Fractal Website: Chaffey High School
Dave Strohbeen's Fractal Music Lab
Patrick Swickard's Fractals, Chaos, and Music
John A Maurer's A Brief History of Algorithmic Composition
Kristine H. Burns' Algorithmic Composition, a Definition
Harlan Brothers' Fractal Music Workshops (See description below under "Websites of Special Interest")
Fantastic Fractals (Features an interactive "Fractal Explorer" for several different fractal types.)
David E. Joyce's Julia and Mandelbrot Set Explorer (Also contains an excellent illustrated short course on complex numbers!)
David E. Joyce's Newton Basins (An in-depth site on this very popular fractal type with an online variable-parameter Newton fractal generator)
The Fractal Microscope: A Distributed Computing Approach to Mathematics in Education
Our Fractal Universe: Mandelbrot and More (in English, French, or Spanish)
The Fractory (An outstanding and mathematically detailed comprehensive site with excellent explanatory graphics!)
The Virtual Museum of Computing: Mandelbrot Exhibition
Evgeny Demidov's Interactive 3D Fractals (Using Java applets, this site allows a variety of forms such as fractal mountains, mandelbrot and julia sets, complex fractal polyhedra, etc. to be "dragged" by cursor and viewed from any point in three-dimensional space producing wonderful visual imagery. The mathematics behind it all are also included as is a link to a VRML fractal laboratory which provides yet another special 3D experience. Other interactive 3D applications in science and mathematics are also available at his site. For additional sites providing interactive Java and VRML displays of complex polyhedra (including kaleidoscopic polyhedra), see George W. Hart's Virtual Polyhedra, Vladimir Bulatov's Polyhedra Collection, and Zvi Har’El's Kaleido VRML collection)
F.Edward Boas' Fractal Letters
Fractals Unleashed (temporarily unavailable - - in English, Russian, or Ukranian)
Lars Kindermann's MusiNum -The Music in the Numbers
Lars Kindermann's Discovery Online - Fractal Music: Create Your Own Chaos
The Martin Junglas Homepage: Munisum and other mathematical Java Applets
The Fractint Homepage (freeware)Fractint Discussion List Resources
The Spanky"What's New" Page (A listing of new fractal-related sites and resources on the Fractint website.)
Iterations et Flarium: Tiera-Zon, Flarium 24 & Other Programs (freeware)
Martin Pfingstl's ChaosPro (freeware)
Dirk Meyer's 3-D fractal generator Quat (freeware)
Lars Kindermann's MusiNum (freeware)
Gustavo Diaz-Jerez's FractMus 2000 (freeware)
Arnold Reinder's a Music Generator (shareware)
Dave Strohbeen's ArtSong (shareware)
Dave Strohbeen's Fractal Music Lab
Fractalus (Home to many of the most outstanding fractal artists on the web. Also includes tutorials, contests, a Fractint mirror site, The Infinite Fractal Loop, and many other features.)
Art And Science Collaborations, Inc. (ASCI)The Museum Of Computer Art (MOCA)
Fractal Art Museum Enterprise (FAME)
Fractals by Jean-Pierre Louvet (in French and English)
Samuel Monnier's The Para-Mathematical Gallery (Ultra Fractal images)
Lee H. Skinner's Composite True-Color Fractals (Fractint)
Les St Clair's Fractal Homepage
Sylvie Gallet's Fractal Gallery (in French and English)
These "Online Artist Community Sites" provide personal gallery and biographical pages for many different categories of media and many other resources such as tutorials, contests, chat rooms, etc.
Fractal Artists' Ring: An online fractal community site directed by Mindy Sommers and Tina Coggins. The site features free personalized support, members' personal galleries, discussion boards, newsletter, special projects and much more. The site welcomes submission of all types of fractal art and is especially supportive of post-processing.
Renderosity: An online community site primarily for computer-related art including photography (free resources as described above).
Art Space 2000: An online community for exhibiting fine and applied art of all kinds (e.g., photography, painting, sculpture, algorithmic, etc.). Annual fee required.
The Society for Chaos Theory in Psychology & Life Sciences: A multidisciplinary society including researchers, teachers, artists, philosophers and all who share an interest in chaos theory and applications of nonlinear dynamics. Membership fee required (reduced for students). Includes newsletters, journal, annual conference and more. Free listserv subscription available.
Patrick Grim's course "Philosophy and Computers" at Stonybrook University is a unique fractal-related interdisciplinary site. (Co-author - with Gary Mar and Paul St. Denis - of "The Philosophical Computer," his chapter (co-authored with Paul St. Denis) "Fractal images of Formal Systems" attempts to use fractals as a form of visual and spatial representation of a number of formal logical operations similar to those used in computational logic and computers. The images associated with the above not only contribute to the understanding of formal logical systems but have aesthetic dimensions of their own (e.g., lattices, Sierpinski triangles, cellular color grids, etc. They may well have considerable significance for formal artistic and aesthetic systems of visual and spatial representation as well).
Redmond Shouldice: The Spiral Dance: Nature at Work and Play (A brief history of the mathematical origins and many forms of the spiral pattern in nature. Overall Winner of The RDS/Technology Ireland Young Science Writers Competition, April 1994).
R. Knott's Fibonacci Numbers and the Golden Section (A numerical series that produces patterns similar to those found in nature, i.e., shell spirals, flower petals, leaf arrangements, branching structures, etc., often seen in fractal art).
Robert S. Ferrero's A sketchbook on L systems (An excellent companion site to the above, this site offers an extensive overview of fractal L systems and the morphogenesis and modeling of plants and branching structures including important links and resources. It might be noted that Fractint has an extensive collection of L system geometric patterns including branching structures, grids, polyhededra, etc. whose order of complexity may be varied by the user).
George W. Hart's Virtual Polyhedra, Vladimir Bulatov's Polyhedra Collection, and Zvi Har’El's Kaleido VRML collection (These sites provide outstanding interactive Java and VRML displays of complex polyhedra (including kaleidoscopic polyhedra).
János Rohán's Astrojan Astronomical Picture Collection (in English and Hungarian). For astronomical images and references to images appearing on this site see his page on Text Files of Astronomical Images and click on NGC2 and Spiral Galaxies .
Eugene Savov's " Theory Of Interaction: The Simplest Explanation Of Everything" (Prior observation and his own astrophysical research has led the author to propose a new qualitative fractal-based theory of the origin of the universe. He presents his theory as consistent with known data with fewer assumptions than that of the widely-held "big-bang" theory and is therefore - he argues - a more preferable scientific alternative. Sample chapters in PDF format may be downloaded from his site).
Salvador Vera's Matematicas recreativas (Mathematical Recreations - in Spanish or English): An extraordinary directory of annotated links to a wide range of unique sites relating to mathematical topics including fractals. Fascinating and great fun!
Harlan Brothers' Fractal Music Workshops (For schools and colleges in the U.S. that wish to enhance their mathematics or music curricula, Harlan Brothers conducts a six-hour lecture/lab fractal music workshop for students and teachers at their home institution. Developed by him in conjunction with the NSF funded Yale University Fractal Geometry Workshops offered by Michael Frame and Benoit Mandelbrot to train educators, his workshop also uses special fractal music software developed for teaching and composition. In addition, his site also contains examples of fractal music illustrating the application of structural scaling and power laws in music composition about which he has published.
Both Chaospro and Chaoscope have the ability to use 256 color-maps (i.e.".map") such as those used by Fractint and Winfract. The following two programs allow for creating new or modifying existing color-maps for use with these programs:
(1) Originally for Windows 3.1, Ron Barnett's Makemap will work with all later Win32 platforms. Just unzip the files and keep them all together either in a separate folder or better still, in the folder where you have your ".map" files (freeware).
(2) Another more recently designed Win32 program is Paulo Guagliumi's Automatic Map Generator available in English or Italian (freeware).
(3) Paul N. Lee's downloadable collections of .map files, parameter files, instruction manual, and other useful files (e.g., sstools.ini) to use with Fractint or compatible programs (freeware). Great site to use when getting started using Fractint (FractInt)!
"Bring It In" is a simple and useful utility which converts an image file so it can be imported into Ultra Fractal 3.x for processing and transformation. The program site contains an interface screen-shot and illustrations (freeware).
Now a "web classic," Irfan Skiljan's IrfanView is a very fast Windows graphic viewer and editor which supports an especially wide range of file formats. It has thumbnail previewing, basic image editing functions, and is available in numerous languages with both slideshow and multimedia capabilities and many other features. For special image effects, it supports Adobe 8BF type Photoshop filters. A separate add-on software management package with filters is also available for download at his site (both freeware for non-commercial use).
Additional plugin filter management software and filter packages (many freeware) are available at Harold Heim's The Plugin Site. His free "Plugin Newsletter" provides a wealth of information about available plugin filters for all image editing programs as well as technical information about their use. Highly recommended!
DHTMLCentral.com (freeware): "PopupDescriptions" Script
Javascript Corral (freeware): "History Back Button"
JavaScript Kit - formerly Website Abstractions (freeware):"Full Screen Window Opener - I.E. 4x only"
The Javascript Source (freeware):"Link button" - - secondary storage site.