Design and Mathematics -- fractals
"As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to reality."
-- Albert Einstein
fractals.
it's very hard to describe fractals -- but this website i found is a lot better at describing this phenomenon:
http://library.thinkquest.org/3493/frames/fractal.html
what is a fractal?
"Basically, it is a rough geometric figure that has two properties: First, most magnified images of fractals are essentially indistinguishable from the unmagnified version. This property of invariance under a change of scale if called self-similiarity. Second, fractals have fractal dimensions, non-integer dimensions."
graphical representation of fractals
"Graphically, fractals are images created out of the process of a mathematical exploration of the space in which they are plotted. For this page, a computer screen will represent the space which is being explored. Each point in the area is tested in some way, usually an equation iterating for a given period of time. The equations used to test each point in the testing region are often extremely simple. Each particular point in the testing region is used as a starting point to test a given equation in a finite period of time. If the equation escapes, or becomes very large, within the period of time, it is colored white. If if doesn't escape, or stays within a given range through out the time period, it is colored black. Hence, a fractal image is a graphical representation of the points which diverge, or go out of control, and the points which converge, or stay inside the set. To make fractal images more elablorate and interesting, color is added to them. Rather than simply plotting a white point if it escapes, the point is assigned a color relative to how quickly it escaped. The images produced are very elaborate and possess non-Euclidean geometry. Fractals can also be produced by following a set of instructions such as remove the center third of a line segment."
there are different "sets" of fractal images -- these are some of the most well known fractals:

The Mandelbrot set

The Julia set
fractals and design
"One of the most useful applications of fractals and fractal geometry in in image compression. It is also one of the more controversial ideas. The basic concept behind fractal image compression is to take an image and express it as an iterated system of funtions. The image can be quickly displayed, and at any magnification with infinite levels of fractal detail. The largest problem behind this idea is deriving the system of functions which describe an image.
One of the more trivial applications of fractals is their visual effect. Not only do fractals have a stunning aesthic value, that is, they are remarkably pleasing to the eye, but they also have a way to trick the mind. Fractals have been used commercially in the film industry, in films such as Star Wars and Star Trek. Fractal images are used as an alternative to costly elaborate sets to produce fantasy landscapes."






