I was very disappointed by Clarke's very long and poetic, but ultimately wrong and insulting description of the Mandelbrot set. This description occurs twice (once in the story as told by little Ada and again in an appendix in Clarke's own voice). His description is wrong because it involves only the distance of a point in the plane to the origin (the absolute value, if you are thinking of the plane as the complex numbers). If that was true, the Mandelbrot set would be symmetrical around the origin, which it obviously is not. I could be generous and say that he is trying to avoid introducing the complex numbers because he doesn't want to scare the readers, but it actually seems to me as if he himself doesn't understand what he is saying. (It would have been possible to avoid introducing the idea of complex numbers without being wrong by defining the map (x,y) -> (x^2-y^2,2xy) which is the same as squaring the complex number x+yi.) And, mostly, I am offended on behalf of my friends in complex dynamics who investigate the Mandelbrot set and other geometric consequences of chaotic dynamics by Clarke's implication that all there is to do with the M-set is look at it on a computer. Like many other areas of math research, computers are useful for providing clues as to what might be true, but in the end it is a human mind which must analyze and explicitly prove the claims in mathematical terms. A good deal was known about complex dynamics before Mandelbrot noticed this set on a computer, and much more is known since from the really deep work of these mathematicians (not at all something that would have been easily done "as soon as man learned to count" as Clarke claims in a completely ridiculous example of hyperbole.)

For a better view of the Mandelbrot set, I suggest you look at Boston University's interactive description.