If you're like me, trigonometry, analytical geometry, and calculus were about as interesting as watching grass grow.  Formulaic expressions were generally 2-dimensional representations, hand-drawn with chalk on a blackboard.  Introduced with a whopping $35 pricetag, electronic calculators were bulky, very basic 4-function (add, subtract, multiply, and divide) devices with LED displays.  Today's math classes had better buckle their seatbelts...

What's a "fractal," you ask...  Well, it's a complex graphical representation of a relatively simple mathematical formula.  They are generated pixel-by-pixel, according to colors corresponding to parameters defined for a given formula.  Fractals were first postulated about a hundred years ago, but until recently, technology constrained their treatment to only a theoretical level.  The combined advent of ever-faster processor chips, higher resolution computer screens, and of course the Internet, has brought fractals to the masses.  While reminiscent of Spirographs or kaleidoscopes, it's safe to consider them a new art form.  Actually, they're a perfect communion of art with science. 

Fractals have a quality of imperfect self-replication, or self-similarity, stemming from their use of one calculation's output as an input for the next.  When these iterative calculations are graphed, the edges of any fractal design can be "zoomed in" or magnified to reveal new and even more interesting designs.  These patterns are interpreted as the building blocks of nature, partly explaining for example why pine trees, fern leaves, and snowflakes look like they do.  At the core of certain types of these designs lies the "Mandelbrot Set,"  the so-called "Thumbprint of God," which at first glance appears to be a squashed bug, or the silhouette of somebody's big fat butt.  Fractal designs are so richly appealing, it's easy to get lost in them.  "Enchanting" is how one enthusiast describes them.

Several freeware application programs are available online, which fractal devotees use to work their magic.  Although the many formulae, parameters, and color maps used within these programs to generate fractals are either guarded strictly, or shared with varying degrees of hesitation, the final designs are typically made freely available for all to appreciate.  Fractals are inevitably assigned etherial-sounding, though entirely apt, names containing words such as "Maelstrom" or "Infinity." 

As a collector and armchair critic yet to formally throw my own hat into the fractal ring, it is with great pleasure that I share selected favorites from certain Cyber-Picassos, each with their own distinctive style.  I've included links from each artist's name, where his / her homepage can be accessed.  There, you can search out full-sized copies of these and other gorgeous fractals.  Bookmark me before you go, so you can come back real soon!  Enjoy!

Linda Allison presents Gumbycat's Cyberhome, my favorite website of all time. There, Linda imparts secrets of fractal painting!

"To me, fractals are an art form. The mathematics provides the form, but it is up to the individual creator (artist) to choose the boundaries, colors, and manner of presentation of each fractal." Melissa Binde is a versatile fractal artist, implementing diverse techniques, and combining with non-fractal designs via layering.

"My work with fractals is more inventive than it is artistic. I enjoy inventing new rendering methods that show off the mathematical beauty of fractals." Paul Carlson also reflects a great variety of methods. Paul's very generous with his formulae, too!

"One of my hobbies is creating computer art. Computer art encompasses a variety of methods. Some artists use a computer paint program which simulates conventional artist tools on the computer screen. Others use the computer to modify and enhance images from other sources. I do not use those techniques. These images were created, pixel by pixel, by computer programs I wrote. The images contain abstract forms and colors. They mean whatever you choose for them to mean. There is no wildlife art here. I do not do ducks. "

"Synth Art? No, it has nothing to do with the synthesizers or music (yet). That's how I call my graphical works, based on a free use of the fractal images and other algorithmically-generated patterns. It's an experience in synthesis of the 'cold' and 'perfect' computer images with the real-world pictures and pieces of 'traditional' (but also computer-assisted) art."

J.P. has a style which alternates between kaleidoscopic views and undersea textures. Magnifique!

"I like to think of the fractals I render as an alternate reality, a Fractal Reality. The images offer a look into an infinite variety of alternative places, worlds, time, and dimension... an ever changing realm without any bounds to its exploration."

Dr. J.C. (Clint) Sprott is University of Wisconsin's Physics Department's answer to Timothy Leary. With textbooks (including "Strange Attractors: Creating Patterns in Chaos") to his credit, Clint appears to be the only fractal artist actually making money from his work! Julia sets and Mandlebrot sets are kids' stuff to a guy dealing in chaos demonstrations, cellular automata, coupled logistic map lattices, bifurcation diagrams of logistic equations, 3D quadratic maps and anaglyphs, stochastic web maps, quadratic map basins, iterated function systems, and strange attractor symmetric icons. That's not to be confused with cyclic symmetric attractor anaglyphs... of course.