• February 27, 2021, 02:41:49 PM

### Author Topic:  General cubic complex  (Read 492 times)

0 Members and 1 Guest are viewing this topic.

#### gerrit

• 3f
• Posts: 2333
##### General cubic complex
« on: February 16, 2021, 07:49:42 AM »
$$z \leftarrow z^3 + az^2 + b$$ is the most general form (up to a conjugation). For parameter space images I use critical orbit one (z=0) or two (z = -2a/3). Images are made on a plane in 4D (a,b) space. Let c be the complex "pixel" then
$$a = Ac + a_0\\ b = B c + b_0$$
with $$(a_0,b_0)$$ center of image in 4D space, and
$$A = e^{i\theta}\sin(\phi)\\ B = \cos(\phi)$$ determines the "angle" of the 2D subspace through $$(a_0,b_0)$$.
Altogether 6 variables to select plane of view. UF offers a 2D control surface to control pairs.

Here's a funny look at the full set through some particular plane, and a zoom into a seemingly normal cusp. They use critical orbit #2.

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #1 on: February 16, 2021, 09:33:08 PM »
Here's another view, with a zoom to where the big components almost (?) touch. First critical orbit.

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #2 on: February 16, 2021, 10:28:33 PM »
You can use the parameter space images to pick 2D Julia sets, there is one for every (a,b) point in 4D space.
If you pick a pointwith both critical orbits escaping, Julia set is totally disconnected, of neither escape it's connected, of one escapes and the other not it's disconnected but with connected parts (blobs).
Here's one and the UF code I use. I save DE in final z which I use for coloring.
Code: [Select]
CubicM {init:   c = #pixel   float theta = real(@angles)*2*#pi   float phi = imag(@angles)*2*#pi   float psi = 0   wcmult = exp(1i*(theta + psi)) * sin(phi)   zcmult = exp(1i*(psi)) * cos(phi)   cz = @centerz   cw = @centerw   if(@corbit == 0)      z = 0   else      z = -2*(wcmult*c+cw)/3   endif   dz = 0   acoeff = (wcmult*c+cw)   bcoeff = (zcmult*c+cz)   bool hasEscaped = falseloop:   dz = dz*(3*z^2 + acoeff*z*2)+ wcmult*z^2 + zcmult   z = z^3 + acoeff*z^2 + bcoeff   if |z| > @bailout      hasEscaped = true      z = #magn * z*log(cabs(z))/dz      z = z^@dePower   endifbailout:   !hasEscapeddefault:  title = "CubicM"  center = (0, 0)  float param bailout    caption = "bailout"    default = 1e10  endparam  complex param centerw    caption = "center w"    default = 0  endparam  complex param centerz    caption = "center z"    default = 0  endparam  complex param angles    caption = "thetaPhi"    default = 0  endparam  float param dePower    caption = ""    default = 1  endparam  int param corbit     caption = "0 or 1"    default = 0  endparam  switch:    type = "CubicJ"    centerw = centerw    centerz = centerz    angles = angles    dePower = dePower    seed = #pixel    bailout = bailout}CubicJ {init:   c = @seed   float theta = real(@angles)*2*#pi   float phi = imag(@angles)*2*#pi   ;float psi = @rot*2*#pi   float psi = 0   wcmult = exp(1i*(theta + psi)) * sin(phi)   zcmult = exp(1i*(psi)) * cos(phi)   cz = @centerz   cw = @centerw   dz = 1   acoeff = (wcmult*c+cw)   bcoeff = (zcmult*c+cz)   z = #pixel   bool hasEscaped = falseloop:   dz = dz*(3*z^2 + acoeff*z*2)+ wcmult*z^2; + zcmult   z = z^3 + acoeff*z^2 + bcoeff   if |z| > @bailout      hasEscaped = true      z = #magn * z*log(cabs(z))/dz      z = z^@dePower   endifbailout:   !hasEscapeddefault:  title = "CubicJ"  center = (0, 0)  float param bailout    caption = "bailout"    default = 1e10  endparam  complex param centerw    caption = "center w"    default = 0  endparam  complex param centerz    caption = "center z"    default = 0  endparam  complex param angles    caption = "thetaPhi"    default = 0  endparam  float param dePower    caption = ""    default = 1  endparam  switch:    type = "CubicM"    centerw = centerw    centerz = centerz    angles = angles    dePower = dePower    bailout = bailout}

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #3 on: February 17, 2021, 10:17:54 PM »
Parameter space image using second critical orbit (first is all black, i.e., non-escaping) and Julia set of point near cusp of the minibrot.

#### FractalAlex

• Fractal Frogurt
• Posts: 477
• Experienced root-finding method expert
##### Re: General cubic complex
« Reply #4 on: February 17, 2021, 10:20:10 PM »
Another nice addition to the collection. Nice work! Besides, some your fractals would be nice to explore in Kalles Fraktaler, if we manage to solve some issues that might occur.
"I am lightning, the rain transformed."
- Raiden, Metal Gear Solid 4: Guns of the Patriots

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #5 on: February 19, 2021, 04:29:35 AM »
Another nice addition to the collection. Nice work! Besides, some your fractals would be nice to explore in Kalles Fraktaler, if we manage to solve some issues that might occur.
Cubic is already in KF, not sure how useful it would be to have this plane DOF added. Also deep zooming is not really where the interesting stuff is at here.

Another parameter-space critical orbit #2 image and a zoom to the bottom "hole".

#### claude

• 3f
• Posts: 1784
##### Re: General cubic complex
« Reply #6 on: February 19, 2021, 04:55:16 AM »
$$z \leftarrow z^3 + az^2 + b$$ is the most general form (up to a conjugation). For parameter space images I use critical orbit one (z=0) or two (z = -2a/3). Images are made on a plane in 4D (a,b) space.
would be nice to explore in Kalles Fraktaler, if we manage to solve some issues that might occur.
Built in formula is RedshiftRider 1: a*z^2 + z^3 + c; set Factor A in formula dialog; set Seed to second critical point if desired (you need to do the calculations by hand, sorry)
Can only plot c-plane (aka b-plane); can't plot a-plane (could be an interesting maths exercise to see how to do it / if it's possible with perturbation, but I'm not doing it tonight); other planes would probably be possible somehow if the a-plane is possible

#### FractalAlex

• Fractal Frogurt
• Posts: 477
• Experienced root-finding method expert
##### Re: General cubic complex
« Reply #7 on: February 19, 2021, 02:09:20 PM »
And how do I calculate the second critical point?

#### claude

• 3f
• Posts: 1784
##### Re: General cubic complex
« Reply #8 on: February 19, 2021, 04:08:46 PM »
.
And how do I calculate the second critical point?
See first post in this thread...

#### FractalAlex

• Fractal Frogurt
• Posts: 477
• Experienced root-finding method expert
##### Re: General cubic complex
« Reply #9 on: February 19, 2021, 04:14:46 PM »
Oh oops... sorry.

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #10 on: February 20, 2021, 05:42:08 AM »
$$z \leftarrow z^3 + az^2 + b$$ is the most general form (up to a conjugation). For parameter space images I use critical orbit one (z=0) or two (z = -2a/3). Images are made on a plane in 4D (a,b) space. Let c be the complex "pixel" then
$$a = Ac + a_0\\ b = B c + b_0$$
with $$(a_0,b_0)$$ center of image in 4D space, and
$$A = e^{i\theta}\sin(\phi)\\ B = \cos(\phi)$$ determines the "angle" of the 2D subspace through $$(a_0,b_0)$$.
Altogether 6 variables to select plane of view. UF offers a 2D control surface to control pairs.
Actually, without loss of generality you can set a0 = 0, as a translation in c-space can always get you there. (In generic case at least.) So only 4 parameters, or 2 complex ones. If UF would support two mice you could really interactively explore.

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #11 on: February 21, 2021, 09:08:18 AM »
This Julia set was found by zooming in on an embedded Julia set in a critical orbit #2 image, which looks like the regular embedded Julia from the Mandelbrot set but polluted with black (non-escaping) shapes in between the spirals. At this location critical orbit #1 was all non-escaping, so if I'd pick a black pixel as Julia set seed it would be totally connected (as all critical orbits are bound, applies to all polynomials), but if I choose an escaping point as seed the Julia set will be disconnected. Experimentally the Julia set visually looks indistinguishable if I vary the seed c by a tiny bit going from escaping to non-escaping. That means it will not be possible to decide if the Julia set is connected or not by looking at the image, which is attached.

The second image is a zoom onto the connection between the two main islands. Could be connected or not.

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #12 on: February 22, 2021, 12:14:13 AM »
If you zoom in on the center of the previous image, eventually you see (first image) it is in fact disconnected. But with a tiny tweak of the seed c (small enough so the previous image looks the same) you get the second image, which is connected. If you'd select a seed c such that by zooming into this Julia set you just get the same spiral appearing over and over forever you would have hit an uncomputable point. This visualization of uncomputable points is of course also possible in the normal quadratic case but there you find either a filled Julia or a hole in the center.

#### claude

• 3f
• Posts: 1784
##### Re: General cubic complex
« Reply #13 on: February 22, 2021, 10:54:15 AM »
either a filled Julia or a hole in the center.
in quadratic Mandelbrot, dendrites occur too - but they are at Misiurewicz points which are algebraic (not uncomputable).  probably other cases can be (sub)classified like Siegel disks and Cremer points...

#### gerrit

• 3f
• Posts: 2333
##### Re: General cubic complex
« Reply #14 on: February 22, 2021, 06:15:23 PM »
in quadratic Mandelbrot, dendrites occur too - but they are at Misiurewicz points which are algebraic (not uncomputable).  probably other cases can be (sub)classified like Siegel disks and Cremer points...
I don't think you could decide a Misuirewicz point that way in the quadratic case. If you zoom into the center of the Julia you'll see dendrites as deep as you go but they could evaporate or have a filled Julia set in the center or not, so if the point you're given was actually Misiurewicz you'll never find out this way.
« Last Edit: February 22, 2021, 08:13:39 PM by gerrit »

### Similar Topics

###### New Theory of Super-Real and Complex-Complex Numbers and Aleph-Null

Started by M8W on Fractal Mathematics And New Theories

2 Replies
660 Views
January 06, 2018, 08:40:15 AM
by M8W
###### General

Started by mclarekin on Color Snippets

0 Replies
244 Views
March 21, 2019, 05:36:39 AM
by mclarekin
###### general formula for kompassbrots

Started by hgjf2 on Fractal Mathematics And New Theories

0 Replies
106 Views
September 26, 2020, 08:30:25 AM
by hgjf2
###### Question about usage in general

Started by Xerilon on Mandelbulb3d

11 Replies
1075 Views
August 01, 2018, 04:44:11 AM
by Kalter Rauch
###### General information about Mandel Machine

Started by Dinkydau on Mandel Machine

4 Replies
886 Views
May 05, 2018, 01:15:57 PM
by julofi