Strange Mandelbrot Set

[element90]

I assume that the pictures are for the critical point zero ?

Finding the critical points using f'(z) = 0 for the first formula results in a critical point at zero and the other 10 critical points are dependent on the position in the complex plane. However, the critical point at zero is useless because after the first iteration all subsequent iterations result in 1.

I've checked, those pictures use c as the initial value of z so result in 'perturbed' fractals as c (the location in the complex plane) is not a critical point.

I've added a variation of these formulae to Saturn.

z = (z^alpha + beta)/(z^gamma + delta) + epsilon + zeta

with the following default values:

alpha = 10
beta = position in the complex plane
gamma = 8
delta = position in the complex plane
epsilon = position in the complex plane
zeta = -2
z0 = 0

Pictures soon ...

At some point formulae of this type will be added to Neptune (my multiple critical point fractal program), I have to implement critical points dependent on the location in the complex plane. The critical points will have to be evaluated for all locations instead of once.

[element90]

As promised pictures using this formula:

z = (z^alpha + beta)/(z^gamma + delta) + epsilon + zeta

called "Mandelbrot Division" in the development version of Saturn and Titan.


https://copy.com/i2fXfJpsi8Sx

https://copy.com/KbokiLLRNZV9


https://copy.com/2i1WpFjehFrx

https://copy.com/GCLyGBzkaxzn


https://copy.com/d8ArqmPLoVT0

https://copy.com/NC1aLSDZmCq9


https://copy.com/ATF8y3kyG4TL

https://copy.com/AszlNwGvJuDq


https://copy.com/jlVGSIIdBxiK

https://copy.com/qu11GLfWVXph

[jdebord]

Thank you element90 for the explanations and of course the wonderful pictures !

BTW, my previous formula for the iterated derivative f'(c) was erroneous. It has been corrected.

[David Makin]

With respect to critical points - an idea I had for a different Newton formula is probably transferable - in that I set the user params so they specified the roots and the formula constructed the Newton to be iterated accordingly.
Obviously one could do the same by allowing the user to specify critical points as user parameters - this would also give an interesting way for user-controlled morphing (as with my Newton formula).