A Different View of the Mandelbrot Set

[kram1032]

Very interesting stuff happening in this thread Does anything interesting happen if you try the same thing with higher-order methods or, as an opposite, the lower order secans-method?

[Roquen]

We're talking about multiple things at once which is leading to miscommunication:  First is finding a closed form equation, from which you've drawn the conclusion that adding 1/4 is of some particular interest and from which you can inspect where the cusp will fall.  And secondly designing domains.  For folks with Mathematica, you'd perform these solutions like this:

Code:

r = z^2 + C == z - f[z]/f'[z]
DSolve[r, f[z], z]
DSolve[r, f[z], z, GeneratedParameters -> (Subscript[c, #] &)][[1, 1, 2]]

Where the two DSolve statements are really the same, the second simply prettifies the displayed result.

Regarding, the iteration which you mentioned of

z_n+1 = z_n² + C - 2

my conjecture predicts that will result in a fractal with the cusp centered at -2.25

Given iteration form: z_n+1 = z_n² + T(s), for input sample point of 's' we have in this case T(s) = s-2,  solve for s: T^-1(c) = (s+2).  So the needle is at T^-1(-2) = 0 and the cusp at T^-1(1/4) = 2.25 (not minus).  Unless I'm screwing up somewhere.

Your initial visual example was: T(s) = sin(s) + .25.  If you replace that with T(s) = sin(s) - 2 you get different fractals, but where the first had a cusp, the second will a tip of the needle.  The same will hold for any other desired point of interest.  So T(s) is the domain seen from the iterative functions perspective and T^-1(c) is what we'll see in the visualization domain.  So if in the visual domain we want to translate what we see we'd do T(s) = sin(s-t) + f, where 't' is the visual translation and 'f' is whatever point we're mapping to where sin(x) is zero.

The complex plane can be extended to the projective plane. Loosely speaking, by adding a single infinitely distant point to the set of all complex numbers (google for "Riemann sphere" for a nice geometric interpretation). From that perspective, Mandelbrot iterations don't ever diverge, but converge on the infinitely distant point instead.

Actually I'm in the process of throwing together a Fragementarium example that does exactly this via the domain transform of T(s) = 1/s + k.  So this maps some point 'k' to infinity.  The classic examples are k=0,-2,1/4: so origin, tip of needle and cusp.