# 6c: Why do you start with z=0?

## Description

This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by
others.

# 6c: Why do you start with z=0?

Zero is the critical point of z^2+c, that is, a point where

d/dz (z^2+c) = 0. If you replace z^2+c with a different function, the

starting value will have to be modified. E.g. for z->z^2+z+c, the

critical point is given by 2z+1=0, so start with z=-1/2. In some cases,

there may be multiple critical values, so they all should be tested.

Critical points are important because by a result of Fatou: every attracting

cycle for a polynomial or rational function attracts at least one critical

point. Thus, testing the critical point shows if there is any stable

attractive cycle. See also:

1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the

Role of Critical Points, _Computers and Graphics_ 16, 1 (1992), pp. 35-40.

Note that you can precompute the first Mandelbrot iteration by starting with

z=c instead of z=0, since 0^2+c=c.

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