Author Topic: General cubic complex (Read 491 times)

gerrit · « **on:** February 16, 2021, 07:49:42 AM »

Didn't see any previous thread about this, so made a new one.
\( z \leftarrow z^3 + az^2 + b \) 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.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/general-cubic-complex/4040/

gerrit · « **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 · « **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 = false
loop:
   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
   endif
bailout:
   !hasEscaped
default:
  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 = false
loop:
   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
   endif
bailout:
   !hasEscaped
default:
  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 · « **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 · « **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.

gerrit · « **Reply #5 on:** February 19, 2021, 04:29:35 AM »

Quote from: FractalAlex 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.

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 · « **Reply #6 on:** February 19, 2021, 04:55:16 AM »

Quote from: gerrit 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.

Quote from: FractalAlex on February 17, 2021, 10:20:10 PM

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 · « **Reply #7 on:** February 19, 2021, 02:09:20 PM »

And how do I calculate the second critical point?

claude · « **Reply #8 on:** February 19, 2021, 04:08:46 PM »

.

Quote from: FractalAlex on February 19, 2021, 02:09:20 PM

And how do I calculate the second critical point?

See first post in this thread...

FractalAlex · « **Reply #9 on:** February 19, 2021, 04:14:46 PM »

Oh oops... sorry.

gerrit · « **Reply #10 on:** February 20, 2021, 05:42:08 AM »

Quote from: gerrit 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.

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 · « **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 · « **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 · « **Reply #13 on:** February 22, 2021, 10:54:15 AM »

Quote from: gerrit on February 22, 2021, 12:14:13 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 · « **Reply #14 on:** February 22, 2021, 06:15:23 PM »

Quote from: claude on February 22, 2021, 10:54:15 AM

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.

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Author Topic: General cubic complex (Read 491 times)

gerrit

General cubic complex

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