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This paper is WAY beyond my ability to truly understand, but I sorta get the general ideas.
I'm posting this because with all the new theoretical research happening here with the Mandelbulb, the details of King's investigations might be relevant in some way. I've read some of his other philosophical writings and he's a very creative thinker.
BTW, if there are any cool discoveries in this paper, can someone put them in layman's terms? :dink:

http://www.dhushara.com/DarkHeart/DarkHeart.htm (http://www.dhushara.com/DarkHeart/DarkHeart.htm)
"Exploding the Dark Heart of Chaos"
Chris King March-Dec 2009
Mathematics Department, University of Auckland

Abstract: This paper, with its associated graphical software and movies, is an investigation of the universality of the cardioid at the centre of the cyclone of chaotic discrete dynamics, the quadratic ‘heart’ forming the main body of the classic Mandelbrot set.
Using techniques investigating and exploring the continuity, bifurcations and explosions in its related Julia sets, we demonstrate its universality across a wide spread of analytic functions of a complex variable, extending from the classical quadratic, through higher polynomials and rational functions, to transcendental functions and their compositions.
The approach leads to some interesting and provocative results, including decoding dendritic island periodicities, and multiple critical point analysis, leading to layered Mandelbrot set 'parameter planes', and intriguing issues of critical point sensitivity in the irregular structures in the Mariana trenches of the more complex functions."

http://www.dhushara.com/DarkHeart/DarkHeart.htm (http://www.dhushara.com/DarkHeart/DarkHeart.htm)
(http://www.dhushara.com/DarkHeart/Images2/35.png)
(http://www.dhushara.com/DarkHeart/Images2/z%5e3-z+c.png)

Well, it's certainly interesting. I haven't had time to really read through it yet, but there's definitely some good stuff there.

In the interests of interpreting it for you: the parts I've seen seem to be largely interested in defining the structure of escape-time fractals based on the critical points (locations at which the derivative is zero) in the iterative function. The earlier stuff--finding the locations and shapes of the central cardioid and period-2 bulb in the Mandelbrot set--looks pretty routine to me, I've seen that before somewhere. But it looks like he's taken the idea and applied it to more exotic functions.

I like the thoroughness of the exploration--he's got good examples of a whole range of different types of function, and examines them with an eye to how the iterative function affects the pattern. What's interesting is that he contrasts the simple, well-behaved z^2 + c with things like sin(z), and notes that even though the widespread behaviour is much different we have some significant local similarities.

Though I have to say, it's definitely not rigorous, more experimental than theoretical--I'd like to see some proof of some of these claims, or at least a rigorous formulation of them. I'd like some general claims about how the locations of critical points affect the shape of the fractal, but this is a very good start.

Ok thanks miquel, I will take a look.

One thing I have to say though in the interest of understanding is go look up albrecht Duerer . Especially in relation to the cardiod.

I have to say that we mathematicians have forgotten the use of rhetoric and the need to communicate. When I was taught pie charts I thought they were a waste of time initially! I had no idea how alienating maths is! Pie charts and bar charts etc are an attempt to communicate data sets to anyone, but very little attempt is made to communicate what a particular mathematician may be playing with.

You know that autistic kid playing in the corner with his bricks? That essentially is what a mathematician is!