Fractal Theory: Fractal Dimensions
The Mandelbrot set and Julia sets are fractals. What this means is that the boundary between the black area that is the Mandelbrot set and the surrounding area that isn't the Mandelbrot set is not a simple line or a curve (one dimensional), but it also isn't a filled-in circle or square (two dimensional). It is so convoluted, folded, and detailed, that it is considered to have fractional dimension.
When you double the magnification of a fractal, the length of the curve, and hence the area covered, does not merely double. All previously visible portions of the curve double in length, but new bumps, curves, and fjords in the boundary become visible and add to the length.
The Mandelbrot set has been proven to have a fractal dimension of two. That means that each time you double the magnification, the length of the boundary increases four times. It also means that the Mandelbrot set is as complicated as a fractal can get. The length of the boundary of the Mandelbrot set is infinite -- it can be any length you want, if you measure it with a small enough measuring stick.