### Julia and Mandelbrot sets w or w/o Lyapunov sequences

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#### marcm200

• Fractal Freak
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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #30 on: April 27, 2019, 10:20:03 PM »
@claude: Thanks. "Binary colouring" and kind of a binary answer too - "No. No. Yes."

Just experimenting a bit with formulas for tricomplex numbers and quadratic Julia/Lyapunov sets, shapes and the repeated log colouring. Seems to reveal some structure on the surface of the interior of the Julia set.

EDIT: I was imprecise here, the upper row and the rightmost object on the last row show the exterior of the corresponding Julia sets.
« Last Edit: April 28, 2019, 10:25:45 AM by marcm200, Reason: Clarifying what the image shows »

• 3f
• Posts: 2083

#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #31 on: April 28, 2019, 12:36:31 AM »
I like the red swirly one.

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #32 on: April 28, 2019, 10:23:21 AM »
Thanks.

Here's a bigger version (the largest I have right now, takes quite some time to compute the interior using maxit of 2000) and some other points of view. The upper left shows it without colouring, just dimming. Still a nice shape but the repeated log adds quite some dynamics to the surface.

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #33 on: April 29, 2019, 04:10:25 PM »
Some more interior sets with sequence AB. This time I also used two functions, I randomly defined:

(x,y,z)²=(x*(x-2y),y²-x*(x.z),-2xz)
and
(x,y,z)²=(x*-z²,2xy,y²-sqrt(x²+y²+z²))

I'm trying to get a "feel" what type of definition changes the behaviour, but am just at the beginning there.

The middle row is very interesting as it shows a large object and then a smaller copy in one of the corners (which are the corners of the 3D cube the observer is looking onto in the line of the cube edge to its center). Although quite a nice fractal feature there (only seen in very few cases), I'm checking my software for a weird mistake I might have made (although - a mistake leading to shadow of a smaller copy of an object - from the top of my head, I'm not sure how to code that if I wanted to).

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #34 on: May 02, 2019, 02:07:55 PM »
Finally the thread's title is justified: A 6-dimensional M-set of quadratic tricomplex Julia sets with sequence AB.

The two c-values were seen as a 6-dimensional vector (cA: x,y,z, cB: x,y,z). If the corresponding Julia set in the 3D dynamical space contains the origin, the cA/cB 6D parameter space point is colored white (this was probably clear to everyone before that paragraph, but I have to write it down to not get confused with 6D/3D and 2x 3D and parameter/dynamical space).

However, to visualize it, I had to randomly fix 3 of the 6 coordinates to random values between -2 and 2. The other 3 were mapped to the axis of the 3D cube I am actually computing and visualizing.

I omitted the color for now since I wanted to see the shapes of the (sub)-M-sets, only dimming to get a 3D appearance. I'd like to have a coloring like: "how multidimensionl is that point", so basically how many other 6D points are there in the complete M-set that have these 3D-sub-coordinates.

But a rough estimate revealed that I'd need about 200 TB space to just compute a very tiny 2006 pixel space. So that's out of the question for now. But if I think about it: it's not that much, the physicists measuring the black hole image probably needed more - and reflecting on how often my first computer crashed when its 40 kb free space were full... but that's beside the point. So I'm stuck with no color for now.

Here are some shapes I found, not necessarily representative, but interesting to me.

What I realised in general, is, many cubes had some sparcely spread out flat objects, rather 2D looking, maybe parts of a 6D tentacle that barely touches that 3D sub-space, just like a tennis ball on a sheet of paper, and the 2D ant only sees a point.

I would like to know whether the M-set itself is connected, but I don't see how to answer that without computing the whole 6D set. Has anyone ideas in that regard?

The formula used for power-2 is a modifed version of David Makin's formula:
(x,y,z)² := (x²-y²-z², 2xy, -2z(x-y) (Makin had 2z in the last term).

EDIT: I forgot to explain the coordinates in the image: The 6 coordinates were given in the order cA: x,y,z, cB: x,y,z. Stated values were fixed constants, "_" stands for a value that is mapped to one of the viewed axis (range -2 .. 2).

#### marcm200

• Fractal Freak
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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #35 on: May 05, 2019, 11:49:28 AM »
The last of the random-walk 3D/6D M-sets. There is a tendency (however only a very loose rule of thumb) that the objects are more interesting to look at and show some regular features if the fixed coordinates of the 6D vector actually comprise cA or cB as a whole.  Mixing seems to distort the objects very strongly.

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #36 on: May 23, 2019, 08:58:07 AM »
I used trafassel's coloring method for his gallery image Mandelbulb Julia (or a modification thereof for the 2D case).

Quadratic Julia set with Lyapunov sequence AB or ABAABB.

During iteration for a point, three sums were computed (d is usually 1):

red = red + d*re(z) / |z|²
green = green + d*im(z) / |z|²
blue = blue + d*(re(z)*im(z) ) /|z|²

Since I did not want to store the values for a histogram or do a two-pass calculation to normalize, I use a circle to shift the values. As long as one is negative I add 255, as long as one is above 255 I substract 255. So finally the values were in the RGB range and could be used directly for coloring.

As by construction the method is very sensitive to the iteration number. I used it here only for interior points with a max it of 250 (bailout 10^6). But that dependency gives one a degree of freedom to fine tune. And it is a nice way to add fine structure to the interior.

Here are some results. First row: sequence AB, 2nd row sequence ABAABB, third row AB but the sums were calculated with an alternating sign (d changes from -1 to 1 and back with each iteration).

It's interesting to see that some images (upper left, lower left) contain two different shaped objects that are attached to each other and alternate - and have a different fine coloring, as if they stem from different origins.

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #37 on: July 10, 2019, 10:48:43 AM »
This is probably nothing new but when reading 3DickUlus' post about alternating Julia seeds (https://fractalforums.org/code-snippets-fragments/74/julman-coupled-Mandelbrot-functions/2897/msg15285) and gerrit's hint, that alternating seeds (like in the Lyapunov sequence AB here) in the quadratic case is basically a quartic polynomial, I started to think in more detail than before about the orbits and what this de-factorization means.

Since in the de-factorized quartic case one does basically two iterations of the quadratic case, the orbits have fewer points, so they might look different depending on the "step" size between two consecutive orbit points. If one uses orbit characteristics as a coloring method (like min or max distance to the origin) this might result in differently colored images.

I tried it in a very basic way for the quadratic Julia set using sequence AB and the values cA=0.626-0.4432i and cB=-0.0648-0.5576i.

The first image shows the orbit of the (randomly chosen) interior point 0.187+0.78i in the quadratic case doing the iteration z=z²+c(sequence counter in sequence AB) (left side) and the de-factorized quartic case with iteration z=z4+2*cA*z²+cA² + cB. The small image in the middle shows the whole Julia set and the even smaller turquois rectangle in the upper right part indicates the point whose orbit was to be followed.

The orbits are quite different in points visited as expected.

So I used min distance (actually the repeated log of that value until it droips below 1) as an index to a color palette.

The images look different in overall coloring. But what's quite interesting to see, was, that in the quartic case (right side) there's a difference between the smaller and larger objects on how close they come to the orbit, that difference was not seen in the quadratic case.

Has anyone done this in more detail and more parameter choices?

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #38 on: July 22, 2019, 04:57:28 PM »
I recently read about average speed in quadratic iterations (I'm trying to find the reference) being the distance of successive orbit points. I wanted to use that in kind of an (anit)-Buddhabrot fashion: Following the orbits of interior and/or exterior points but not incrementing a counter every time a pixel is hit, but rather adding the distance value |zn+1-zn| itself.

The images below show some examples of Julia and Mandelbrot sets (without Lyapunov variations for now).

One can nicely see some hotspots in the first two rows (Julia sets), I suspect those to be periodic points of an attracting cycle (would make sense since many orbits pass near such points so the sum should be high - and I'm not performing periodicity checks, so if a cycle is entered way before max iteration, it will repeatedly add terms).

Currently a pixel has two sums: when an orbit of an exterior starting point passes through or one of an interior starting point, so I can experiment with combining them in different ways. Then a repeated log is computed until the value dropps below 1 and is used in a color map.

I like how that coloring method reveals a lob-sidedness of the Julia sets despite their membership symmetry.

#### marcm200

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#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #39 on: October 29, 2019, 08:34:34 AM »
"2D or 3D?"

A hybrid Julia set, sectionally defined, one-letter sequence A.

If z_old < 0 use David Makin's formula as iteration, else use Tyler Smith's
cA={ 0 , -0.9 , 0 }

I can't quite figure out whether there are true 3D parts. The views in the image show two perpendicular principal planes and a lot of spread out points.

Is there an analytic way to determine if there is a 3D connected component with volume? Or if everything is equivalent to a 2D surface or even flat?

#### marcm200

• Fractal Freak
• Posts: 774

#### Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences

« Reply #40 on: December 06, 2019, 10:24:33 PM »
If one sees the periodic points of an attracting cycle as gravity centers that attract every point in the complex plane according to a law depending on distance, cycle multiplier and a constant, one can alter the shape of the set.

The algorithm:

1. Find the critical points.
2. Construct the critical orbits to find all attracting cycles.
3. Construct the standard Julia set.
4b. For every filled-in Julia set interior point z, calculate the translation vector v(z):

z_translated := z + v(z)

v(z) = sum over all periodic points pp of all cycles:

direction vector from z to pp
* GAMMA (the "gravity" constant)
/ [ distance between z and pp ] ^ LAWEXPONENT (the decay of the gravity influence)
* (1-cycle's multiplier) (the "attraction" strength)

4c. Compute the new pixel coordinates of z_translated and color as interior

With large GAMMA or LAW values the images transform somewhat to electrostatic field lines (like in the kid's experiment with iron filings on a sheet of paper).

I like the insect best. It looks like a fisher's net trying to catch some deep-sea Mandelbrot creatures.

(I changed the topic's title as lately there aren't any Lyapunov variations).

#### marcm200

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#### Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences

« Reply #41 on: December 16, 2019, 10:09:28 AM »
"Fairy tale tower"

Some explorations with random formulas: c-parameter-space images following the origin, escape radius is the 2-cube. Colors only indicate geometrically disjoint regions.

{x,y,z}new := { sin( R*x²-S*y²-T*z² ) , 2xy , 2z(y-x) } + { Cx, Cy, Cz }

The first image looks a bit like the towers of a castle from a fairy tale.

A quick sweep with a sparse grid through some Julia sets showed more or less point clouds.

• Fractal Fluff
• Posts: 352

#### Re: Julia and Mandelbrot sets using Lyapunov sequences

« Reply #42 on: December 16, 2019, 10:09:33 PM »
I recently read about average speed in quadratic iterations (I'm trying to find the reference) being the distance of successive orbit points. I wanted to use that in kind of an (anit)-Buddhabrot fashion: Following the orbits of interior and/or exterior points but not incrementing a counter every time a pixel is hit, but rather adding the distance value |zn+1-zn| itself.

maybe :
https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/av_velocity

• 3f
• Posts: 1558

#### Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences

« Reply #43 on: December 20, 2019, 06:31:08 PM »
Triskelion has fixed points at the cube roots of unity and parameters that exactly determine the derivatives at those fixed points. Setting the derivative effectively picks a point in the Mandelbrot set: in the cardioid for |derivative| < 1, on its boundary for = 1, and outside it for > 1, with the argument determining the angle from the center. So a positive real derivative just above 1 will give a disconnected elephant valley Julia, a negative real one just below -1 will give a "dragon" from the front bulb, -1 + epsilon i will give a disconnected seahorse valley one, and so forth.

You can pick a point in the Mandelbrot set directly:

1. pick the point C
2. find the fixed point Z = Z^2 + C, ie Z = (1 - sqrt(1 - 4 C)) / 2
3. find its derivative: D = 2 Z
4. set c = 2 s + D

Then proceed with the f() g() h() formulation at the top of the UFR code.  I implemented it in Fragmentarium, seems to work ok, though my colouring skills are not hot enough for the plane-filling ones...

doesn't work in all cases, I think Advanced Maths iterative algorithms like polynomial mating are needed.
« Last Edit: January 13, 2020, 03:15:36 PM by claude, Reason: wrong »

#### marcm200

• Fractal Freak
• Posts: 774

#### Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences

« Reply #44 on: January 13, 2020, 12:08:36 PM »
This very unusual Julia set shape shows coordinate planes - I wonder whether those have volume (and a cycle might in principle be detectable by cell-mapping)- or if any one of the spread out voxels has interior.

It reminds me of a 3D Lyapunov image I computed a while ago forum link, also depicting planes and some point-cloud object.

{x,y,z}_new := { 4xyz , 4x²y²-z² , x5-y5-z4 } + { 0, 0 , -0.5 }

Colors are based on trafassel's direct RGB summation method.

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