The Mandelbrot Set
If we iterate a function – findetc with the nth iteration written– it may have one of these behaviours:

The iterated function may tend to infinity: Ifthen

The iterated function may tend to a limit:then

The iterated function may be constant:so

The iterated function may cycle:
The Mandlebrot set concerns the family of quadratic functionsIf we define the iteration relationthen the Mandlebrot set illustrates those values offor which does not iterate to infinity.
Features of the Mandlebrot Set

The Mandlebrot Set is a connected subset of

Is symmetric under reflection in theaxis.

Meets the real axis in the interval

Has no holes in it.

Little copies of the whole Mandlebrot set appear all over the diagram in a manner characteristic of fractals.