Newton's method for approximating the roots of a function |

Newton's method becomes interesting for functions with
multiple roots. Then a natural question is which initial guesses x_{0}
will iterate to which root? For a given root, the collection of all such guesses
is called the basin of attraction of that root.
Here is a simple example. |

Cayley extended Newton's method to complex numbers and began the study of the basins of attraction of the roots of complex functions. The graphcal approach to this extension of Newton's method is difficult: the graph of a complex function is a four (real) dimensional object. Here the formula is useful, because it can be applied to complex numbers as well as to real numbers. |

Next we shall see how applying this method to a family of
cubic polynomials led to the discovery that the Mandelbrot is *universal*, in a
sense ubiquitous.

Return to the Mandelbrot set and Julia sets.