be (0.75)^2 + 0.5 = 1.0625. All these calculations can be done on a handheld
calculator. Carrying on we find that the answer gets bigger and bigger.
Let’s try another value of c, this time c = – 0.5. As before we start at x = 0
and substitute it into x^2 – 0.5 to give – 0.5. Carrying on we get – 0.25, but this
time the values do not become bigger and bigger but, after some oscillations,
settle down to a figure near – 0.3660...
So by choosing c = 0.5 the sequence starting at x = 0 zooms off to infinity,
but by choosing c = – 0.5 we find that the sequence starting at x = 0 actually
converges to a value near – 0.3660. The Mandelbrot set consists of all those
values of c for which the sequence starting at x = 0 does not escape to infinity.
The Mandelbrot set
This is not the whole story because so far we have only considered the one-
dimensional real numbers – giving a one-dimensional Mandelbrot set so we
wouldn’t see much. What needs to be considered is the same formula z^2 + c but
with z and c as two-dimensional complex numbers (see page 32). This will give
us a two-dimensional Mandelbrot set.
For some values of c in the Mandelbrot set, the sequence of zs may do all
sorts of strange things like dance between a number of points but they will not
escape to infinity. In the Mandelbrot set we see another key property of fractals,
that of self-similarity. If you zoom into the set you will not be sure of the level of