Interesting thread. I've read the stuff regarding the use of the Newton method for finding the position of the "cusps" but my point of view I don't really care that much.
Now I'm sure that the purpose of my topic has been misunderstood, so I will try again. Please ignore the distraction which I created by mentioning my seahorse cusp generation algorithm (which currently relies upon conjecture, not Newton's method). I got ahead of myself in excitement in an attempt to explain a "more simple", alternate, and completely valid view of what the Mandelbrot Set represents.
In the strictest definition, Benoit Mandelbrot's mapping method is a surjective mapping from the entire complex domain onto the Boolean set {false, true} which is then graphed into two dimensions. The recurrence
z_new = z^2 + C
where C is varied on the complex domain
is Newton's method being applied to a family of functions. Recall that Newton's method for numerically finding root(s) of f(z) is to iterate
z_new = z - f(z) / f'(z)
I setup the "obvious to me" differential equation
z^2 + C = z - f(z) / f'(z)
which in standard form becomes
y'(z)(-z^2 +z - C) - y(z) = 0
where C is an independent variable.
The solution to that equation is
Lacking a better name, I shall refer to that function as MP(z, C). ("MP" = Mandelbrot Parent; I welcome suggestions for a better name.) Note that MP(z,C) has no roots. It is well known that Newton's method of root finding "behaves poorly"
in some situations.
In the classic Mandelbrot graphing method, C (element of the complex domain) selects a unique function; Newton's method is then applied to that function with an initial guess of zero as the root. If the iteration of Newton's method for C escapes to infinity, then C is not in the Mandelbrot Set; otherwise, C is in the Mandelbrot Set. If C is not in the Mandelbrot Set, then color[false] is indicated for that C. If C is in the Mandelbrot Set, then color[true] is indicated for that C. Assuming that color[false] = white and that color[true] = black, this would result in a black Mandelbrot fractal on a white background. Refer to the first generated image of the Mandelbrot Set
Colors weren't used. Instead a '*' was displayed if C was in the Mandelbrot Set, and presumably the background was implicitly white.
What this means is that each C element of the Mandelbrot Set represents a specific function(z); if Newton's method is applied to that function, the iteration will not escape to infinity. Thus, in my valid view, the Mandelbrot Set represents a set of
functions.
As I found specific examples helpful in math classes, I shall choose C = -0.75 (primarily because David Joyce discusses that iteration here
http://aleph0.clarku.edu/~djoyce/julia/julia.html)
To "plug" C = -3/4 into MP(z, C), I shall feed
http://www.wolframalpha.com/ this:
exp(-(2 tan^(-1)((-1+2 z)/sqrt(-1+4 C)))/sqrt(-1+4 C)) for C = -.75
Before simplification, MP(z, -3/4) looks like this:
I hope that you see how each different C selects a specific f(z), as that is key to understanding what I am trying to communicate.
Back to the example, Mathematica simplifies MP(z, -3/4) into
f(z) = e^(-tanh^(-1)(0.5-z))
If one performs Newton's method on that function, the results will simplify to the iteration z_new = z^2 - 3/4
Feel free to confirm that theory/fact with Mathematica or whatever means. Or just enter
z - e^(-tanh^(-1)(0.5-z)) / (d/dz(e^(-tanh^(-1)(0.5-z))))
into
http://www.wolframalpha.com/ z^2 - .75
just as I expected. (There would be a bug in Mathematica otherwise.)
While I do seem distracting with my interesting side notes, another way of seeing which function Newton's method is being applied to with Mandelbrot's method is to just enter the differential equation into wolframalpha:
z^2 - 3/4 = z - f(z) / f'(z)
Now this is interesting since Mathematica expresses the solution in algebraic form rather than exponential form:
So if Newton's method is applied to
f(z) = sqrt(2 z+1)/sqrt(3-2 z)
the resulting iteration will be
z^2 - 3/4
which is listed as an alternate form for
z - (sqrt(2 z+1)/sqrt(3-2 z)) / (d/dz((sqrt(2 z+1)/sqrt(3-2 z))))
Note that f(z) = sqrt(2 z+1)/sqrt(3-2 z) has a root at -1/2, and then read the already posted URL
http://aleph0.clarku.edu/~djoyce/julia/julia.html and ponder.
There is much more to share and discuss, but we cannot proceed until the basic concept of what I am trying to convey is understood. If you can understand undergraduate calculus, then you can understand my different view of Mandelbrot's method.
(This is an "old new theory" which I never published.)
Any questions? Does anyone understand?