"In very simple terms, fractals are geometrical figures that are generated by starting with a very simple pattern that grows through the application of rules. In many cases, the rules to make the figure grow from one stage to the next involve taking the original figure and modifying it or adding to it. This process can be repeated recursively (the same way over and over again) an infinite number of times.

The fractals' growth mechanism can be visualized very easily with a simple example. Start with a + sign and grow it by adding a half size + in each of the four line ends. Repeat the exact same process recursively as many times as desired. We'll call this the Plusses fractal:

Notice how the + sign grows into a rhombus (popularly known as diamond) in very few simple steps."

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty" -- Bertrand Russell, from The Study of Mathematics: Philosophical Essays

"It is interesting to note that Bertrand Russell wrote this in 1907, which applies so well to fractals, about 70 years before they were discovered by the Polish-born French mathematician Benoit B. Mandelbrot in the 1970's [1] Mandelbrot's fractal geometry provides a mathematical model for many complex forms found in nature such as shapes of coast lines, mountains, galaxy clusters, and clouds."

"One interesting property that fractals can have is that of self-similarity. The name sounds complex but the idea is very simple. What self-similarity means is that each small portion, when magnified, can reproduce exactly a larger portion.

[Note 1] -- The word fractal was coined by Mandelbrot in 1975. It is also important to note that several images, now considered fractals predate the work of Mandelbrot."

Fractals, an Introduction
http://ejad.best.vwh.net/java/fractals/
reprinted without permission

"Many formations in nature, although both constituted and caused by dissimilar phenomena, are not only similar to look at, but have identical mathematical descriptions. This would suggest that together they form a higher overall order outside that limited by our concept of linear cause and effect... This order, reverberating down into the microscopic and subatomic levels, both structures and reflects our consciousness."

Jill Purce, The Mystic Spiral, Journey of the Soul
reprinted without permission

"You do not see spherical clouds, or cubical clouds, or icosahedral clouds. Clouds are wispy, formless, fuzzy clumps. Yet there is a very distinctive pattern to clouds, a kind of symmetry, which is closely related to the physics of cloud formation. Basically, it is this: you can't tell what size a cloud is by looking at it."

"This 'scale independence' of the shapes of clouds has been verified experimentally for cloud patches whose sizes vary by a factor of a thousand. Cloud patches a kilometer across look just like cloud patches a thousand kilometers across. Indeed, this statistical self-similarity, as it is called, extends to many other natural forms... including moutains, river networds, trees, and very possibly the way that matter is distributed throughout the entire universe. In the term made famous by the mathematician Benoit Mandelbrot, they are all fractals."

"We've now established the uncontroversial idea that nature is full of patterns. But what do we want to do with them? One thing we can do is sit back and admire them. Communing with nature does all of us good: it reminds us of what we are. Painting pictures, sculpting sculptures, and writing poems are valid and important ways to express our feelings about the world and about ourselves. The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instict is to try to understand it -- to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions. There is a little of all these instincts in all of us, and there is good and bad in each instinct."

Ian Stewart, Nature's Numbers
reprinted without permission