In the August 1985 issue of Scientific American, A.K. Dewdney described a "computer microscope" which looks in "at the most complex object in mathematics." He was writing about the Mandelbrot Set, and his beautifully illustrated article (most of the images come from Heinz-Otto Peitgen) remains an excellent introduction to fractals.
I like the dramatic opening sentence of Dewdney's article. It lends metaphoric life to a mathematical construct that is at once abstract and eminently graphic: "The Mandelbrot set broods in silent complexity at the center of a vast two-dimensional sheet of numbers called the complex plane."
Dewdney goes on to describe a computer program for zooming in on the Mandelbrot Set. Today, dozens of fractal-generating computer programs are available, but they all operate in same basic way that Dewdney outlined for his readers in 1985.
If you'd like to read Dewdney's article, it's available in PDF format here.