When wandering at the vegetable department of a supermarket, did you ever pay attention to a fresh and clean cauliflower and get intrigued by it? If not, take a look at Fig. 5 now, or simply buy a fresh and clean cauliflower, then zoom in and out using your eyes at its surface structure. Despite the elegant spiral arrangement of the small buds, what more can you see? The whole cauliflower consists of smaller cauliflowers, and the smaller cauliflowers in turn consist of even smaller cauliflowers, so on and on ...! If, as you look closer, your size shrinks according to the size of the cauliflower buds you focus on, can you tell whether you are looking at the whole cauliflower? A small bud of it? A smaller bud on a small bud? ... Most certainly you cannot, because you are looking at a self-similar structure, a scaling-invariant object, ... a fractal.

For definition, a fractal object is self-similar in that subsections of the object are similar in some sense to the whole object. No matter how small a subdivision is taken, the subsequent subsection contains no less detail than the whole.

Image (right): Fig. 5. Cauliflower -- a living fractal.

Fractals are ubiquitous in nature. Look up, you can see the forever changing shapes of fractal clouds. If you have rich imagination beyond your eye sight, your mind will see the fractal distribution of matter in the universe. Look around, you see fractal trees, fractal mountains, fractal coast lines. Look inside your body, you see the fractal architecture of arteries and veins, fractal bronchi, fractal nerve system ... all perfect examples of Nature's efficient and optimal use of space and materials. Look at the hierarchical structures of a bureaucratic system and a society; the word "fractal" never fails to pop up.

"Fractal" is a powerful word, a powerful concept, an unifying theme of a vast and diversified set of shapes, structures, and organizations adopted by Nature.

Image (left): Fig. 4. Mandelbrot set -- The Icon of Fractals.

It is Benoit B. Mandelbrot, a mathematician at IBM, who first put all the dangling and scattering pieces of fractals together in the late '70s and '80s, coined the very word "fractal," and founded a new geometry -- Fractal Geometry, the Geometry of Nature. Rightfully, he has earned his position in the Hall of Scientific Giants. The first fractal image from one of his toy mathematical models, the Mandelbrot Set, has become the icon of fractals (see Fig. 4).

As a typical Western scientist, Mandelbrot didn't stop at the metaphysical level of simply formulating concepts, as many ancient Oriental sages had often done. He had developed and revitalized many rigorous mathematical methods and had heavily relied on computers and computer graphical techniques. Because of the latter, he broke ranks with conventional mathematicians, and was quite often at odds with REAL mathematicians.

One mathematical method he employed is iteration, which is one of the connections between Chaos theories and Fractal theories. Mandelbrot views a fractal object as a set of points in a space, and these points are connected and bonded by certain relations or rules. If the rules are known and one wants to unveil the fractal object, one can simply pick up a random point in the space, apply those rules to it, and observe its trajectory. This is analogous to discovering the phase portrait, or global picture, of a chaotic system by repeating the same set of equations again and again. If the rules indeed correspond to a fractal object, after many iterations, the trajectory of the test point converges to a finite object and the search is ended. If one needs a real life analog, think about the digging of a dinosaur skeleton fossil.

Image (right): Fig. 5. Barnsley's IFS fern -- a successful example of modeling Nature with mathematics.

Mathematician Michael F. Barnsley is just such a paleontologist of fractal "fossils." He developed the so-called Iterated Function Systems (IFS) to model and generate both real-life and abstract fractal images. Shown in Fig. 5 is his benchmark fractal fern. Only 28 numbers are needed to contain the rules of generating this fern; this means succinct storage of information. Barnsley realized this potential and further developed a system to compress computer images -- a good example of the transition from pure ideas to practical applications.

Image (above): Fig. 6. Fractaiji -- a hybrid of Fractals and Taiji.

I have just said a few bad words about ancient Oriental sages. Now let me make some compensation by hybriding Lao-tzu's Taiji with Mandelbrot's fractals. The result is Fig. 6, a fractaiji.

Enough has been said, and it is time for some pictures. In the following, I will just make three quotes as regards whether fractal images, and computer graphic arts in general, are really art. The first quote is a con and the remaining two are pros. See the pictures and form your own opinions. Enjoy.

"Fractal images are incomplete art, of course, since they are abstract and not culturally rooted." -- P.W. Atkins, 1990

"The distinction between art and science is contrived. Both are processes of discovery and both use a variety of tools and techniques." -- Computer Graphics World, 1989

"Nature is relationships in space. Geometry defines relationships in space. Art creates relationships in space." -- M. Boles and R. Newman, Universal Patterns, 1990

Reed

Window

Complex

Dragon

Spike

		Great fleas have little fleas
		        upon their backs to bite 'em,
		And little fleas have lesser fleas,
		        and so on ad infinitum.
		And the great fleas themselves, in turn,
		        have greater fleas to go on;
		While these again have greater still,
		        and greater still, and so on.

					-- Jonathan Swift

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