The universe around us is not linear but fractal in nature. That is, we see the same pattern appearing time and again, no matter what the scale it is examined on. Look at a river with its tributories. Each tributory is itself a river, with smaller tributories, which in turn are themselves rivers with smaller tributories, and so on down to creeks. Or a fern frond; the main frond of which consists of a two rows of sub-fronds, each of which consists of two rows of smaller fronds or leaves. Or the pattern of a coastline: bays and peninsulas contain smaller bays and peninsulars, right down to grains of sand.
Fractals were discovered and described by a mathematician by the name of Benoit Mandelbrot.
The Koch Snowflake
A good example of a simple fractal. Take an equilateral triangle. Add a smaller but identical triangle in the middle of each of the three sides. Repeat ad infinitum.
The Sierpinski Triangle
The Mandelbrot Set
The classic fractal; a mathematical equation named after its discover, Benoit Mandelbrot. What makes this equation interesting is that it is recursive; that is, it continually refers back to itself, so that each successive step employs as one of its parameters the outcome of the preceding step. When the Mandelbrot equation is fed into a computer, and the computer is instructed to paint the various points that arise on a screen or a print-out, the most astonishing patterns emerge.
Explore the Mandelbrot Set
Mandelbrot Explorer - an interactive site that lets you explore the Mandelbrot set. You can also download images from the gallery (no thumbnails here unfortunately).
It seems that Africans were the ones who first incorporated fractals into their art, culture, and philosophy. This has only become evident recently, through the work of Ron Eglash, a mathematician who visited Africa to study the fractal patterns in villages and African society ( Ron Eglash - African Fractals in Buildings and Braids - Gallery Ezakwantu)
Since the 1980s, Eglash has been studying the complex fractal systems found in Sub-Sahrana African braiding, architecture, textiles, sculpture, painting, carving, metalwork, religion, games, craft, technologies, and philosophy. ( Blog post - African Fractals (The Liberator Magazine)).
The following books explore the indigenous insights and contributions of Africans to the mathematics of fractals, and computer simulations, as well as fractals in African culture.
African Fractals: Modern Computing and Indigenous Design - Ron Eglash, Rutgers University Press (1999)
Africa Counts: Number and Pattern in African Cultures - Claudia Zaslavsky, Lawrence Hill Books; 3rd edition (1999)
Fractal patterns often appear in occult representations of reality. Tibetan Buddhist mandalas for example will sometimes show a central Buddha-figure, surrounded by smaller secondary Buddhas, each of which are in turn surrounded by even smaller Buddhas.
One occult system that is especially fractal in its approach is Kabbalah. According to the Jewish occult system of Kabbalah, everything in existence can be classified in terms of, and reduced to, ten fundamental archetypal principles or essences, called Sefirot. Originally, the Sefirot were conceived of as the ten fundamental qualities of the manifest and knowable Godhead, as opposed to the unmanifest, unknowable En Sof or "infinite". But later they were applied to classifying everything in existence, whether divine or mundane. This is especially so in the case of non-Jewish or Magical, Hermetic Kabbalah (usually spelt "Qabalah").
But what makes Kabbalah a fractal theory of reality is its oft-stated assertion that each of the ten Sefirot that together make up creation is composed of ten Sefirot, each of which is in turn composed of ten, and so on ad infinitum.
The Fractal Music Project - links, software, and some examples of fractal music. check it out!
The Fractint Homepage - The definitative fractal-generating program, a freeware for PC's - part of the Spanky Fractal Database (see below)
The Spanky Fractal Database is a collection of fractals and fractal related material for free distribution on the net. Maintained by Noel Giffin
Fractal Information - by Neal Kettler