The Mandelbrot set is probably the most famous and well known fractal. When people use the term 'fractal art' most of them are referring to imagery found around the Mandelbrot set. It was first discovered by Benoit Mandelbrot in 1979. Since then a whole new genre of art has sprung up, based on the many patterns that can be found around the Mandelbrot set. All the imagery on this site is from the Mandelbrot set.
Technically the Mandelbrot set is a superset. It's a mapping of all the connected Julia sets in the complex plane (see image T2). Each point in the complex plane can be considered the 'seed' for an entire Julia set. I'm not going to delve into the details of Julia sets here - if you're interested there's plenty of information about them on the 'net. Suffice it to say that one important attribute of a Julia set is whether it's connected or not. This attribute led Mandelbrot to write the first computer program to graph this. He wanted a visual map of all the connected Julia sets.
A black and white image of the Mandelbrot set can be seen below, picture T1. The area in white approximates the Mandelbrot set (I call these black and white pictures negatives, more in later chapters). Interestingly, all the connected Julia sets are themselves connected, and take the form of the Mandelbrot set. The small white specks around the main mandelbrot shape are small copies of the parent mandelbrot shape (more on these in chapter 3). Well, I probably shouldn't use the word copy, because each one is slightly different. There are an infinite number of these copies in the border region of the Mandelbrot set. All of these copies are connected to each other, but on a scale too small to be apparent in the image. This is one of the things that makes the Mandelbrot set so interesting. The entire set is a single connected organism - its many parts are similar, sometimes to the point of being indistinguishable, but nothing is ever exactly the same.
The Mandelbrot set is just one of many visualizations made possible by the computer. It's important to understand the images one sees are the result of a process, almost an inquiry. It's amazing, because all of the apparent complexity found in the Mandelbrot set is the result of a very simple feedback process. In fact about the only thing easy to describe about the Mandelbrot set is the equation used to create it.
Actually most of the imagery one sees about the Mandelbrot set isn't technically part of the Mandelbrot set, but is rather what I call froth. Froth occurs on the forever undefined boundary of the Mandelbrot set, and upon closer examination usually turns out to be something close to nothing. Below is another image of the Mandelbrot set, picture T2. This picture shows the Mandelbrot set in black, and the froth in shades of gray, around the Mandelbrot set.
The Mandelbrot set is a mapping in the complex plane. A complex number is a convention created by mathemticians. There are two components to a complex number, known as the 'real' and 'imaginary' parts. Thus a complex number can be viewed as a two dimensional point, and can be plotted on an (x,y) graph. Computer programs that plot the Mandelbrot set usually scan through a grid of points, and treat each (x,y) pair as the two components of a complex number (variable C). With another complex variable, Z, started at zero, the computer then iterates through a very simple feedback loop, the mathematical shorthand of which is Z^2 + C, (Z squared plus C, where Z is the iterant, and C is the point being sampled - both complex numbers and using the rules of complex arithmetic). The computer does this for a certain number of iterations, or until the value of Z exceeds two. In this case, the value of Z accumulates (feeds back on itself) every iteration - the value of C is constant (for a given point).
As you can see from picture T2, the entire Mandelbrot lies within a radius of two (Z <= 2) from the origin of the complex plane. A computer program typically 'samples' each point within a rectangular region of the Mandelbrot set. The sampling process involves the computer evaluating the magnitude of a complex number (Z) after each pass through a feedback loop (as defined above), waiting to see if it ever exceeds a value of 2. Any point that exceeds a value of two is said to have 'escaped', because its value will rapidly explode thereafter due to the squaring term in the feedback loop. There is no way to determine ahead of time how many iterations will be required before a point escapes - it can take anywhere from 2 to inifinity-1 iterations. Points that are part of the Mandelbrot set never escape; the computer (at least in theory) could iterate Z forever and it would always remain bounded in a fractional space. These points are known as 'prisoners', and comprise the prisoner set (Mandelbrot set). Since the computer obviously can't iterate forever, we tell it to give up after a certain number of iterations if a given point hasn't yet escaped. Such a point would be considered part of the Mandelbrot set. I usually color the Mandelbrot set in black, with all the color shown in the froth.
One thing that's fascinating about the Mandelbrot set is there's no end to how many times one can magnify it. Since a feedback loop invloving x and y themselves is involved, the only limits are the precision of the arithmetic involved. On each magnification new detail emerges. I wrote the program which created all the images on this site, and one of its unique features is the ability to zoom very far into the Mandelbrot set - much further than programs that use the built in (floating point) arithmetic on the PC.
Now I can explain how image T2 is colored. The region around the Mandelbrot set is colored based on how many iterations a point had to go through before it escaped. All the underlying dynamics Z took as it went throught the feedback loop are thrown away, and only the number of iterations (times the computer ran Z through the feedback loop) it went through before it escaped are counted. The points in the outer ring only went through 2 iterations. The next band in (towards the mandelbrot set, black) went through three iterations, and so on. There are an infinite number of these bands, or encirclements, piled up towards the mandelbrot set, each one on a slightly smaller scale.
This is the essence of froth - these forever shrinking encirclements about the Mandelbrot set. I take a very literal, old-school interpretation of the froth. Since the computer can only display a limited number of colors, I simply map the froth (which is an integer value) to a smaller number of colors. This leads to the cyclical bands of color around the Mandelbrot set.
These days there are many fractal art programs that allow one to take a much more liberal interpretation of the Froth. It is possible to transform it many different ways. It's also possible to interpret the dynamics Z goes through during the iteration process, leading to many interesting visualizations. All the images on this site are 'old-school' as far as fractal art goes, and directly map the froth to a cyclical color map.
You might ask how the patterns that emerge can be considered art, seeing that they unfolded on their own, with no human guidance or control. I'd answer that despite this fact, there are several reasons to consider it art. First off, it takes a person to choose which part of the fractal to explore, where to look, the composition, etc. Also, how one interprets the froth is completely arbitrary - how it's colored and patterned can be transformed many different ways.
Next chapter, basic anatomy of the Mandelbrot set.