Brian Prentice wrote:
....critical points. ....explain what they are and why they are important to the generation of your fractals.
An initial value z0 is usually chosen from the so-called "critical points" of the function fp at which the derivative vanishes. There are several explanations for "critical points" which have been around for many years: SPANKY FRACTAL DATABASE http://spanky.triumf.ca/www/fractint/mand_lambda_type.html sci.fractals Usenet Group's FAQ http://www.faqs.org/faqs/fractal-faq/section-6.html (see Q6c) And others http://library.thinkquest.org/3493/frames/chaos.html#critical http://math.bu.edu/DYSYS/FRACGEOM2/node2.html The discussion of "Critical Points" requires a basic understanding of calculus. Specifically, you need to understand the concept of the _derivative of a function_. You may have wondered how one decides what value should be used for the initial value of z. It turns out that the best choice for the initial value of z is a _critical point_ of the fractal equation. A _critical point_ is defined as a value that satisfies the equation: f'(z)=0, where f'(z) is the 1st derivative of the fractal equation. That is, we take the derivative of the fractal equation, set it to 0, and solve for z. Whether this is easy, difficult, or even possible, depends on the fractal equation. (See http://www.fractalsciencekit.com/program/maneqn.htm for the rest of the above with examples.)