Now That You're Here

The primary purpose of this site is to exhibit my images and music compositions. In addition, I wish to provide some personal suggestions for creating fractal images and music depending on different levels of technical knowledge.

The "thumbnail" Galleries contain a number of fractal images or fractal-derived images created by programs requiring different levels of technical experience. The program used is indicated below each image and "clicking" on the image will link you to an enlarged version. Some enlarged images in Gallery-1 are also accompanied by fractal music compositions. In some instances, fractal images may have undergone additional processing with an image editor (i.e., post-processing) after being generated by the indicated program.

The MIDI music for this page was composed with the full MusiNum program discussed below. Media Player (available on Windows Update) is the MIDI player used for Internet Explorer. Crescendo (operating in detached mode) is the MIDI player plugin I am using for Netscape.  A basic version is available from the Crescendo homepage (freeware).

Consider the following as helpful hints drawn from my own "Fractal Adventures" rather than an extensive tutorial.

Please note that that the Resources page also contains links to all programs referred to below.

 

Getting Started

You don't need to download and install any programs to start making wonderful fractal images or music. You can immediately start to generate your own fractal images and set up a gallery using a website having an interactive online program. I suggest starting with the very easy to use Fractal Map and then proceeding to the Mandelbrot Explorer or the very informative Java application The Mandelbrot Set Explorer . Another choice -  just right for beginners and featuring several different fractal types - is the "Fractal Explorer" included in the comprehensive site Fantastic Fractals.  

Two of the most comprehensive and intermediate level instructional sites on the web are Our Fractal Universe: Mandelbrot and More (in English, French, or Spanish) and Fractals Unleashed (in English, Russian, or Ukrainian) which also has a "kids' section." Together, they also contain interactive applications for many different fractal types and are great for fractal enthusiasts of all ages!

For the more experienced, I suggest David E. Joyce's Julia and Mandelbrot Set Explorer which also contains an excellent illustrated short course on complex numbers as well as his in-depth companion site Newton Basins both of which contain online variable-parameter fractal generators. 

The Fractal Microscope is also a truly outstanding interactive site as is The Fractory among comprehensive instructional sites.

A very unique site - instructive and fun to visit - is Evgeny Demidov's Interactive 3D Fractals.  Using Java applets, this site allows a variety of forms (e.g, fractal mountains, terrains, 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.

It's noteworthy that a number of the fractal sites either referred to on this or the Resources Page  were developed under the sponsorship of ThinkQuest, an organization that promotes mentored student-developed instructional websites in all disciplines. 

Another interactive online application that is fun to use is F. Edward Boas' Fractal Letters which converts letters and text to a fractal format. "Fractals Unleashed" cited above also includes a similar application. Great for reports!

For those wanting to compose fractal music online, I suggest starting with Lars Kindermann's two companion sites. The first site, MusiNum - The Numbers in the Music , is an interactive tutorial where you can compose simple fractal music and learn how the MusiNum program works. The second site, Discovery Online - Fractal Music: Create Your Own Chaos , is the main online interactive site and allows you to compose fractal music of somewhat greater complexity.

An online Java version of MusiNum with reduced capabilities called Munisum developed by Martin Junglas is also fun and instructive to use. A full-featured version of the MusiNum program for Windows is available for download on the first site referred to above (freeware).

If The Fractal Bug Bites!

Now that you've been entering numerical values in the online image program and mastered the art of "zooming" - or composed some fractal music - it's time to learn a bit about the mathematics which produced your unique fractals. Explanations ranging from simple to advanced can be found under Introduction to Fractals on the Resources page. 

Perfect for beginners are Chaos and Fractals: A Search for Order offering a brief introduction to the basic concepts and history of the topic, and a former ThinkQuest entry Fantastic Fractals - now hosted elsewhere -  which is 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.

More advanced comprehensive ThinkQuest sites include Our Fractal Universe: Mandelbrot and More (in English, French, or Spanish), The Fractory, and Fractals Unleashed (in English, Russian, or Ukrainian) which also has a "kid's section" and a wide variety of interactive features and resources.

For the more experienced, Robert L. Devaney's The Dynamical Systems and Technology Project At Boston University  funded by the National Science Foundation is an excellent source of information 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.

An excellent companion website to the above - now listed on major search engines - is 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. 

A unique fractal-related interdisciplinary site is that of Patrick Grim's course "Philosophy and Computers" at Stonybrook University. 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.

For sites related to econometrics and the controversy of who discovered the application of fractal analysis to fluctuations in financial markets, see the Mandelbrot - Prechter Debate. Of significant current interest are "The (Mis)Behavior of Markets" by Benoit Mandelbrot and Richard L. Hudson, and John Matson's Scientific American blog "Benoit Mandelbrot and the wildness of financial markets."

Another excellent site for instructional purposes is The Fractal Microscope, one of the most outstanding comprehensive and interactive instructional sites available on the web! Visit it! 

Julien C. Sprott's Fractal Gallery is one of the most outstanding award-winning 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. It includes a constantly changing computer-controlled fractal display, 3-D anaglyph fractal images, Strange Attractor animations, among many other features of interest!

An educational initiative of enormous significance is the OpenCourseWare (OCW) project of The Massachusetts Institute of Technology 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). During an abnormally rapid heart rhythm termed "ventricular tachycardia," damage to ventricular muscle tissue intereferes with the normal pathways of electrical activity. Electrical activty flows in an abnormal recurrent circular spiral pattern producing contraction at each cycle. If this condition does not stop by itself, this spiral wave pattern may break down into smaller traveling spiral waves producing ventricular fibrillation. In this state, rapid desychronized contraction of muscle tissue occurs stopping the pumping action of the ventricles and resulting in cardiac arrest. Normal synchronous contraction may be restored with the rapid application of an electrical pulse(s) delivered from an external or implanted "defibrillator" device. Additional information on "Cardiac Chaos" may be found by searching for the term "spiral fibrillation pattern" on Google.

Often, similarities between fractal images generated as a means of artistic expression and images of natural phenomena can be quite striking!  One such area are images obtained in atmospheric and astronomical research.  An excellent site depicting such imagery is János Rohán's Astrojan Astronomical Picture Collection. 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 (in English or Hungarian).

Similarities as those indicated above and his own astrophysical research have led Eugene Savov to propose a new qualitative fractal-based theory of the origin of the  universe. In his book "Theory Of Interaction: The Simplest Explanation Of Everything," the author 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).

Another site that should not be missed is Jean-Pierre Louvet's Fractals which features many 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 (available in French and English). Its server is also host to the legendary Fractal FAQ - a highly informative and very recently updated resource!

Two sites providing exceptional but quite different kinds of listings of fractal resources and links are John Mathews' Fractal Resources which deals primarily with mathematical articles and sites related to Mandelbrot and Julia Sets and developed for undergraduate research, and Jacco's Fractal Links which is a select and wide-ranging collection of categorized sites covering everything from tutorials, unique sites, practical technical information, galleries, and much more (In Dutch but most links are in English)!

As part of his comprehensive site, Dave Strohbeen's Fractal Music Lab has  outstanding tutorials on fractal music generation.

For an historical overview of computational approaches to music composition, Patrick Swickard's Fractals, Chaos and Music , John A. Maurer's A Brief History of Algorithmic Composition, and Kristine H. Burns' Algorithmic Composition, a Definition are very helpful articles.

Downloading Your Own Fractal Program

If you're ready to download and install your own image program and are just starting out, I suggest InkBlot Kaos - one of Stephen C. Ferguson's many programs cited below - and the often recommended Aros Fractals (both freeware).

Another program for beginners is WinFract ("Fractint for Windows", version 18.21) discussed below and is  available from a link on the Fractint home page (freeware).

The fractal program Fractint deserves special mention at this point. This multi-featured freeware program is a collaborative effort of many programmers over the years (many of them are the most outstanding creators of fractal images on the web). They are collectively known as the Stone Soup Group and everyone should read the story of the origin of this name (freeware).

Fractint is the most extensively documented program on the web and for many fractal artists it is still the program of choice although many more advanced programs have since been developed. It is a Dos-based program which works extremely well in a Windows environment; however, it is quite technically advanced. One of its many interesting features is variable parameters for 3-D rendering as well as conversion of images to random-dot stereograms again with a number of options. It is still unsurpassed for the extraordinary number of fractal types available and the detailed information about each of them (just press F2 when viewing a particular type).

Fortunately, its Windows version and offspring Winfract ("Fractint for Windows," version 18.21) is based on the same code, has a graphic interface and - while not having as many features - is just right for both beginners and more advanced users. I suggest you might start with this program before giving Fractint a whirl. 

The gratitude of all Fractint and WinFract users is due Noel Giffin who has coordinated, maintained and enhanced this extraordinary site!

See Paul N. Lee's site listed below under "Useful Things to Have" for additional materials to use with the above programs and also help to get you started.

Although many consider Fractint to be the "Fractal Program of all Fractal Programs" - and deservedly so - it is limited in the range of colors (color palette) it can produce. It is not a so-called "true color" program. For the very advanced user, however, there are, auxiliary "psuedo" true color programs for Fractint that can synthesize a greater range of colors and are available on the Fractint homepage.

In the last few years, a number of genuine true color programs have been been developed with an expanded color palette that also can also perform a wide variety of mathematical transformations on the fractal image. These transformations can produce extraordinary images of forms that vary in color, shading, texture, transparency and depth.

Examples of such popular programs are Stephen C. Ferguson's many programs which include Inkblot Kaos, Tiera-Zon 1 & 2 and Flarium24 among  many other of his more recent programs (freeware).

A program which is also popular with beginners as well as those more technically advanced is the program Fractal eXtreme (shareware).

Another program that has an outstanding range of features, compatibility with Fractint, an excellent user interface and has received much praise is Frederik Slijkerman's Ultra Fractal (shareware). So advanced and innovative is this program that many outstanding fractal artists believe that it is fast becoming the successor to Fractint. An introduction to the program by Damien Jones and a comprehensive tutorial by Janet Parke - as well as other resources - are available from the "related sites" link on the Ultra Fractal website. An excellent Ultra Fractal guide by Dr. Joseph Trotsky can be found on the Fractal Artists Museum Enterprise Website (click logo on bottom of this page). Additional tutorials and extensive resources can also be found on Jacco's Fractal Links.

Martin Pfingstl's powerful and aptly named fractal program ChaosPro fills a special niche with regard to inter-program compatibility and functionality. It can import and read almost all of Fractint's major fractal types and color-map files. In addition, a rapid built-in formula compiler allows it to import and read all of Ultra Fractal's formula, coloring, and transformation files. It also features real time fractal exploration during which all modifications are immediately visible, true multi-tasking and multi-windowing, all-version Win32 platform support, and rapid creation of zoom animations. Also included is a unique application and elegantly designed interface for transforming 2-D to 3-D fractal images in various rendering modes.  An in-depth description of program features and tutorial are also available online. Users should comply with copyright restrictions on the use of imported files set forth when installing the program. Outstanding at any price (freeware)!

Also not to be missed are Terry W. Gintz's  classic programs Dofo-Zon, Fractal ViZion and Fractal Zplot as well as his many newer programs such as QuaSZ which renders 3-D fractal images and image slices for Mandelbrot and Julia Sets in a variety of modes (e.g., quaternarion, cubic Mandelbrot, etc.) producing very unique images. Other features include random formula generator, batch mode, and integrated video routines. Packages of programs are available at reduced cost (shareware/purchaseware).

An interesting 3-D program with a number of capabilities similar to the program described above is Dirk Meyer's Quat. The adjustable shading, highlighting, and other features give the 3-D fractal a special sculpted appearance making for many artistic possibilities. The fractal may be previewed and rendered from any point of view using a simple drop-and-drag interface. Optional dual images for stereoscopic viewing may also be rendered. See Gallery-7 (freeware).

Just recently released with new features, Nicolas Desprez's Chaoscope (in English or French) is a 3-D Strange Attractor program which renders images in five modes (i.e., Gas, Liquid, Light, Plasma, and Solid). In the unique Solid mode, shading and highlighting especially enhance the depth effect.  When rendering in color, mapping files such as those used by Fractint and Winfract provide coloration and other parameters. Attractor formulas include classic Lorenz, IFS, Icon, various polynomials, etc. Images may be dragged and previewed from any point in 3-D space prior to rendering. Batch processing is also included. A freeware animation program and manual (in English or French) are also available onsite. Formulas by Clifford Pickover and Julien C. Sprott - two of the foremost contributors to fractal mathematics and visualization - are included. Sprott's Fractal Gallery is an outstanding resource for use with the program!  An online manual makes it an easy to use by all. This is a unique and truly extraordinary "breakthough" program (freeware)!

These are only a very few of the available fractal programs on the web. For many more and for programs compatible with different operating platforms, see Paul N. Lee's Fractal Links and his fractal image program descriptions or the "What's New" page on the Fractint website.

For music composition, while the "MusiNum" program recommended above is one of several based on number theory and is great fun to use, equally so is the excellent and feature-laden music generator FractMus 2000 developed by the composer Gustavo Diaz-Jerez. It has twelve different types of algorithms (including fractal) to choose from and a very user friendly interface (freeware). Grab it!

Most fascinating are programs which scan different image features (e.g., color values, etc.) of your fractal image and - using your choice of musical parameters - generate a fractal-related composition unique to the specific image. For both the beginning, advanced and professional user, Dave Strohbeen's Windows program ArtSong is truly outstanding. The new version of this program now has an extraordinary range of functionality allowing for a variety of fractal and non-fractal compositional modes. In addition, a fully-functional program is available for download with a generous number of free usage trials prior to purchase (shareware).

Another very interesting program that scans almost any image feature (even tables of data) and composes fractal-related music is Arnold Reinder's a Music Generator (shareware).

For the above and many other types of fractal music composition software and how they work, I again suggest Dave Strohbeen's Fractal Music Lab and his program descriptions.

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. Originally 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.

Some Useful Things To Have

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!

Exhibiting  Your Work On The Web

Whether you have your own website or not, you may wish to attract a wider audience to view your work. Fortunately, there are now comprehensive online artist community websites available with categories for almost every kind of art. They not only provide personal gallery pages, biographical pages, technical information, but  often have contests, topic-related chat rooms, special events and a variety of other features as well.  A number of outstanding sites which provide some or all of these features - many with associated webrings - may be reached by clicking on the graphics below. Each has its own unique style and media categories which may be just right for you (see additional information on Resources Page).