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Hi All,

In the last two weeks, I've been spending way too much of my free and (not so free) time trying to find the holy grail of 3-d mandelbrot sets. Up to today, I really hadn't found anything original to share. An idea I came up with is to choose the squaring formula based on location of the (x,y,z) coordinate. This includes changing that choice for each iteration as the triplet is being squared (cubed, e.g.) and translated.
Here's two results I've gotten so far.

One I call Mandelettuce - this uses the cubic White-Nylander form, but only phi (asin(z/r)) is being tripled, not theta. The distance criteria is distance to the axis. The closest axis defines which one to define phi.

The second one uses the standard quaternion form which normally just produces axial symmetric 2-D mandelbrot. Which axis X, Y, Z will be the dominant (i.e., X-axis: x-x-y*y-z*z) is determined by which one is the shortest distance to the axes again.

It seems like this new method is "cheating" as it may be similar to a symmetry operator that might be used in an IFS. But, I guess you could say this a hybrid method.

One of the issues to resolve is how best to choose the proper axis and how to position the axis into geometric shapes (e.g. tetrahedral symmetry)

-mike

Wow, Mike, we're thinking along similar lines.  Beautiful images.

Here is one more thought I had today, while considering those possibilities.

Try viewing it like this.  In 2d, Z is being rotated away from [1,0,0].  I think of it as kicking a ball.

To generalize that concept to 3d, what about this?

Take your x and z components.  "kick the ball" separately in relation to the x and z axis, then add the two resulting vectors together.  The magnitude of the two vectors would be in the same proportion as that between x and z.  So when z=0 in C, you get the standard mandelbrot in the xy plane.  Although to be honest, I suspect you'd also get the standard mandelbrot in the zy plane.  Would look like a twin hydra or something.

Sorry I'm so busy at work, I didn't have time to turn that into a formula.

Ted

   I found another species of Mandelettuce.

  I'm using "triplex algebra" with two possible attractors:
  The standard "z^2" if y*y>z*z ,
  otherwise, perform "z^2" operation with flipped y and z for theta and phi formulas.

wow :D
Great idea!
How does that whole thing look zoomed? :)

Very pretty pic.  That just begs for some of twinbees renders with an iteration of 100.

When I zoom in, there's some nice features but it seems like it's far from an ideal fractal as there is some chaos in 1-D but not 2-D.
I dialed up the iterations to 50.

I just wanted to show what happens if you simply apply a flip of y and z coordinates if the absolute value of one is bigger than the other using the standard z^2 Mandelbulb. It's nothing special, but maybe it'll help us visualize better the solution.

Here's another variant.
Six possible attractors chosen on distance to axis and whether the coordinate of that axis is positive or negative. Attractor is quaternion Mandelbrot. It seems to have infinite detail, although here I cap at 50 iterations. The closeups are detailed, almost noisy, if anyone's interested.

There's actually a term for what the basic shapes are in my multi-attractors: spherical polyhedra.

-mike

Similar variant as above but with ambient occlusion and max iterations = 10.
The problem with this species is that the features seem to get very thin.

Wow -- can you arbitrarily select the coordinates of the attractors? If so, please try out the following twelve coordinates. (  phi = 1.6180339887, or (1+sqrt(5))/2  )

(phi,0,1)
(phi,0,-1)
(-phi,0,1)
(-phi,0,-1)

(1,phi,0)
(-1,phi,0)
(1,-phi,0)
(-1,-phi,0)

(0,1,phi)
(0,-1,phi)
(0,1,-phi)
(0,-1,-phi)



This generates a shape with icosahedral/dodecahedral symmetry. It has an automorphism group of order 120 (60 rotations * 2 reflections). By comparison, those cubic fractals have an automorphism group of order 48 (24 rotations * 2 rotations). A dodecahedral fractal would be one of the most elegant and organic geometrical objects in three dimensions, resembling pollen grains, radiolaria or Herpes viruses. The golden ratio is also designed to look aesthetically pleasing, so you might win an art competition by rendering it! (The human body, Parthenon and sunflower phyllotaxis all incorporate the golden ratio, to name a few.)

Calcyman - the reason I've only done cubic so far, is because it's simplest to do. To do tetrahedral or the dodecahedral as you suggested I will need to rotate (x^2-y^2-z^2,2*x*y,2*y*z) to the various points, and I will need to calculate distances to each one on the points to determine which formula to use.
The thing that puzzles me is why I don't get bulbs at each attractor. It could have to do with the position and size of each attractor.

By request, the icosahedral attractor. I found a simple shortcut instead of computing the rotated quaternion attractors. The idea is that z^2 can also be thought of as a reflection of the pole [e.g.,(1,0,0)] to (x,y,z) with lots of sphere normalizations.

nice :D
Like a lot of flowers!

Impressive! Much less 'artificial' than the cubic counterpart -- it's a shame that polyhedral symmetry and Mandelbulb-esque detail are almost mutually exclusive.

A quaternionic fractal could have the even more impressive icosian symmetry, which has 14400 automorphisms. The icosians are a group of 120 vectors, which are closed under multiplication. In the context of fractals, this means that squaring any icosian will produce another icosian. In a geometric context, the icosians are the vertices of the 600-cell, or the centres of the faces of the 120-cell (dodecaplex).

Using multiple "attractors," I've found several variations.
This one is interesting:

this one actually looks like it might be worth for a zoom... :)

I truncated the norm by mistake on the last picture. If you let the fractal reach its true extent it has some disconnected flowers. I also added some color... finally. No more "all blues." Not sure if zooming would bring out much as there may be some non-isotropic stretching.

nice, however, the colours are so saturate that they actually kill some detail...^^

kram1032: It's not the colors' fault, I think. The un-truncated fractal doesn't have the "furriness" of the truncated one. Instead, there's a lot of thin flakes like the first icosahedral one I showed. It seems that radial scaling is too fast compared to attraction to the vertices.

With my updated rendering code, here's the fractal from the first post revisited. Some flat parts and some chaos.
Algorithm:

   if ((x*x>y*y)&&(x*x>z*z)) {
    x1 = x*x-y*y-z*z;
    y1 = 2*x*y;
    z1 = 2*x*z;
   }
   if ((y*y>z*z)&&(y*y>x*x)) {
    x1 = 2*x*y;
    y1 = y*y-x*x-z*z;
    z1 = 2*y*z;
   }
   if ((z*z>y*y)&&(z*z>x*x)) {
    x1 = 2*x*z;
    y1 = 2*y*z;
    z1 = z*z-x*x-y*y;
   }
  x = x1+a;y=y1+b;z=z1+c;

(http://www.fractalforums.com/gallery/1/803_08_02_10_4_16_20.jpeg)
(http://www.fractalforums.com/gallery/1/803_08_02_10_4_16_57.jpeg)
(http://www.fractalforums.com/gallery/1/803_08_02_10_4_17_41.jpeg)

Nice :)

You know, even if that what you said before was the case, hyper-saturate colours tend to kill some amounts of details...

THe new renders with less saturation are way better with that :)

kram1032: There's a balance of saturation vs. detail. I guess I find the bright colors appealing, reminds me of candy.
Here's the 6-way attractor rendered with better methods:
(http://www.fractalforums.com/gallery/1/803_09_02_10_2_45_14.jpeg)
Ignore the diagonal line going through the center. That must be an artifact of my "if" statements.  ::)

With some nice organometallic textures, those could be the merge of nature and technology, when on one had the robots get intelligent enough and on the other hand nature fights back and they suddenly notice that both have the same goals and start a symbiosis O.o

Ok, I guess I have too much fantasy today :)

Anyways, very nice and somewhat cuborganic :)

Actually reminds me of virus capsids which somebody attempted to render for me in a koch snowflake form.

Using a Mandelbub order 2 transformation with interchange of y and z depending on z*z < or > y*y, the Julia with seed (-0.75,0,0)
has some nice bulb properties. Essentially, the interchange removes the poles, but introduces some discontinuities at |y| = |z|.
Careful examination suggests the true 3-D Mandelbrot is stil elusive. For example, the bulbs get squashed in the y-z plane.

(http://www.fractalforums.com/gallery/1/803_15_03_10_2_58_28.jpeg)

that's VERY close :D
How does the whole Mset of this look like?

The M-set of this one is actually the Mandellettuce I posted several months earlier on this thread.
Julias look better in some cases even though (I think) they tend to be more self-similar and less diverse as you zoom in.
I'll try to get a new pic of the M-set of this one tomorrow, but as you'll see it's very much lacking. 

Cool, mandel-pumpkin?  O0

Very interesting!

You certainly got very close.

I have wondered myself if Tglad's folding trick could somehow be used on the sphere to let any poles vanish during a Mandelbulb computation. But even then the coordinate system would still introduce some distortion.

Very cool msltoe  O0

msltoe, wow that looks amazingly close to the holy grail we're looking for - and probably the best so far. Congrats!

Can you zoom in to the 'equivalent' of seahorse valley on one of those sphere-lets at say 10-50 iterations?

All,

 Thanks for the compliments. This is an interesting and encouraging discovery, but the crown jewel has yet to be discovered.
 
 Twinbee: I don't think zooming in this Julia set will bring anything exciting. If you consider the 2-D Julia variant, the zooming in just brings in more and more smaller bulbs, with no mixing of antennae-like structures.

 One interesting application of this "mirroring" method is that it produces more aesthetically pleasing 3-D Julia set renderings by removing the z=+/-1 poles. The mirroring is continuous, but not analytic.

 I experimented with some variants to fix the squashing of the bulbs that start in the x=0 plane. Here's the latest formula:

 if (z*z<y*y) {                                                                                                                     
  r = x*x+y*y+z*z+4*y*y*z*z;
  r1 = sqrt(r);
  theta = 2*atan2(y,x);
  phi = 2*asin(z/r1);
  r = x*x+y*y+z*z;
  x1 = r*cos(theta)*cos(phi);
  y1 = r*sin(theta)*cos(phi);
  z1 = -r*sin(phi);                                                                                                   
 } else {                                                                     
  r = x*x+y*y+z*z+4*y*y*z*z;
  r1 = sqrt(r);
  theta = 2*atan2(z,x);
  phi = 2*asin(y/r1);
  r = x*x+y*y+z*z;
  x1 = r*cos(theta)*cos(phi);
  y1 = -r*sin(phi);
  z1 = r*sin(theta)*cos(phi);                                                                                                                       
 }
x=x1+a;y=y1+b;z=z1+c;

The Julia variant (-0.8,0,0) looks a little less distorted than the previous one:
(http://www.fractalforums.com/gallery/1/803_16_03_10_2_39_29.jpeg)

But the M-set lacks the beauty and symmetry of the J-set:
(http://www.fractalforums.com/gallery/1/803_16_03_10_2_49_38.jpeg)

It's not all that bad, actually :)

I'd be really interested in seeing a "julibrot" projection of that last one....

Ok. so maybe I'm getting a little carried away with symmetry operations, but this 8-fold variant fills the spherical surface better than the 4-fold. I've done higher n-folds, but they're less exotic. There's something really strange about how the translational (a,b,c) field (aka Mandelbrot formula) works in 3-d space. Once we figure this out, the "true" 3-D Mandelbrot should be within reach.
(http://www.fractalforums.com/gallery/1/803_17_03_10_4_13_12.jpeg)

It's surreal!, but almost everything here is ;)

I like it a lot anyway...

For the M-set, if only the y-z behavior could be fixed, we'd be set. Although, this has been the problem since day 1.

Spectacular! Wow, how close is that? Distortion on the spheres is either fixed, or very minimal for the smaller spheres maybe.

EDIT: Wouldn't mind seeing it with a lower iteration...

Lower iterations are very different than Mbulbs  in that they are not bulbous, rather look more like a low iteration 2D Mset. I'll try to get a pic posted tonight.

Various maximum iterations. My numbering scheme may be off by 1:

(http://www.fractalforums.com/gallery/1/803_19_03_10_12_38_27.jpeg)

The code:

  psi = atan2(z,y)+pi*2;
  psi2 = 0;
  while (psi > pi/8) {
   psi -= pi/4;
   psi2 -= pi/4;
  }
  cs = cos(psi2);
  sn = sin(psi2);
  y1 = y*cs-z*sn;
  z1 = y*sn+z*cs;
  y = y1; z = z1;

  x1 = (x*x-y*y)*(1-z*z/(x*x+y*y+z*z));
  y1 =   (2*x*y)*(1-z*z/(x*x+y*y+z*z));
  z1 = 2*z*sqrt(x*x+y*y);

  y = y1*cs+z1*sn;
  z = -y1*sn+z1*cs;

   x=x1-0.8;
   y=y+0;
   z=z+0;

Note the slight change in the Mandelbulb equation which seems to cause less distortion, also z1 can be positive or negative yet produces the same Julia set.

N=5 is nice, it's like it is struggling to get out of its shell...

More iterations please, I want to get OUT!!!  O0

Because of my success with the Julia variant, I've been obsessed on cracking the code using symmetry operations to cull out the right stuff.  I've found three M-set  variants so far.

The following is the closest hit, but still a far way off. The general idea is to vary between + and - rsin(phi) over even and odd iterations.

The 16-fold symmetry was done for a specific reason. The un-symmetrized version has a little bulb in a small angular span. The symmetry operation was used to extract that out but discard the boring stuff.

I conjecture that the real M3D, if it exists, requires minimal or no symmetrizing.

(http://www.fractalforums.com/gallery/1/803_20_03_10_3_36_55.png)

I'm using a very high number of iterations (N=200). If you dial that down, you lose the smooth bulbs. But, maybe, there would be less fuzz and more of something to discern when zooming in. The bulbs look promising from a distance, but upon zooming in lack the symmetry seen in the Julia set.

That looks nice :)
What about distance estimation? :)

It keeps getting closer! Good for you on not giving up - the Mandelbulb would look like a barren desert compared to the delights we can expect if you found it.

I agree that one wouldn't think any symmetry operations to be needed. Actually, I wonder if you could zoom into these areas highlighted by red boxes - we could already start to be seeing some strange and wonderful things.

EDIT: Just read your end quote. To be honest, I'd still like to see some zoom-ins to the earlier Julia version. I know the Julia has less interesting detail than the Mandelbrot equivalent generally, but I'm still curious :)

Twinbee, KRAFTWERK, kram1032, ...:

 I appreciate your encouragement as I stumble away. I also believe that a true 3-d mandelbrot will have incredible complexity to explore for years to come. Right now, you could say I'm simulating what we think it should look like, which ultimately will help us visualize and find the real result.

 Twinbee: I'm constantly modifying the formulas , so I haven't had a chance to do too many zooms, but I zoomed into the minibrot that you suggested and sure enough it looks a bit different than its parent:
(http://www.fractalforums.com/gallery/1/803_21_03_10_2_28_16.png)

I've also done some zooms of the Julia x8 set which do actually show a bit of complexity mainly, I think, because I'm not using the exact formula. This is one I posted in the gallery a few days ago:
(http://www.fractalforums.com/gallery/1/803_17_03_10_10_34_52.jpeg)
I shared that formula so anyone can pursue it. It would make a nice movie.

-mike

Mike, what would a four-fold symmetry look like instead of the current 16-symmetry?

It looks to me like you've found it.  You've really found it.

Ted

Ted,
 
  The "golden mean" seeded Julia, for sure, is an important breakthrough. The mandelbrot-like objects you see though are just good illusions generated by lots of symmetry. The four-fold M-sets began with one the first posts on this thread. The original four-fold J-sets are about ten posts up.
 I can't believe I've literally spent half my little daughter's life (whose now 8 months old) on this problem.   :evil1: At least, we've found some new shapes/ideas along the way...
 
-mike

Hi Msltoe,

I think you have a cool formula which makes a giant step towards the goal of a 3D Mandelbrot and/or Julia. I have put some experimental code into my UF Raytracer and can duplicate your Julia that has a seed of (-0.8,0,0). I read earlier in the thread that there were problems with the y z plane, or at least something involving the y and z variables. If I take you Julia and use a seed of (-0.8,0.1,0) I start to see fissures and cracks in the object. Increasing the y value even more will create a real mess. I also found something very interesting. If after decrementing psi and psi2, psi is subtracted from psi2, the resulting image is a quaternion! This will happen for virtually any symmetry value. A quaternion can also be obtained by increasing the symmetry to a large value, say 512. Here are some illustrative images:

Fractalrebel: Thanks for implementing and evaluating the formula. I've seen the fissures, too. I guess that means that the formula is not continuous for b or c != 0. Perhaps, the true formula is? The psi/psi2 thing you see sounds like the infinite symmetry version, which is *too* roundy.

 What puzzles me is why the Julia b & c =0 sets look so good yet the M-set version doesn't. It could be that the formula is only asymptotically correct for b & c = 0. The fissures might be removed by making the symmetric elements continuous at their intersection points. Think back to the simplest "z*z<y*y" version: The two parts visually merge at the two planes z*z=y*y, but the formulas do not. The quaternion is one such solution but it's too smooth. Some of the quaternion variants might also satisfy this condition but probably would look identical w/ and w/o the symmetry operator.

I agree that making the elements continuous at their intersection points will probably solve the problem. Then we will have the true 3D Mandelbrot/Julia. Do you have any thoughts on how to do that?

I'm not sure if continuity alone will do it, actually... this feels like a "cusp" problem. I think it needs to be not only continuous, but with a continuous derivative, too. That way we won't have any corners in the iterating function, which will probably smooth out the result, too.

Here is another Julia, this one with a seed of (0.375,0,0).

Reesej2,

You are probably right that continuity for both the function and the derivative are necessary.

fractalrebel: Nice zeolite! As long as b=c=0, things look good.

reesej2: I, too, agree that continuous derivatives are important. In fact, I think the joining at y*y=z*z is continuous. I tried using an interpolation function like 1/(1+exp(-beta*(alpha-0.5))) where alpha = z*z/(y*y+z*z) and it always seems to have some artifacts. I've also looked at symmetrizing the y and z formulas, but so far nothing works. This is a tough problem, but well worth it.

-mike

msltoe, that big zoomed in pic looks ace. Until I get the formula up and running myself, is there any chance of that (or even more zoomed in), but with less iterations? I'm curious as to what the surface would look like. Same or bigger resolution would be great too.

Similar closeup of the Julia x8 with 15 iteration limit
(http://www.fractalforums.com/gallery/1/803_26_03_10_1_01_04.jpeg)
My renderer currently gives a "muddy" look.

Interesting, and thanks.

Here's an idea which might be worth exploring; trying to create a similar Julia, but in 2D, and then try to see the differences between that and the 2D Mandelbrot, and then try to use that information to find the difference between this 3D Julia and a potential 3D brot.

The 2-D Julia looks a lot like the 3-D one. The best 3-D Mandelbrot version I can find is by alternating signs of z for even/odd iterations using the same formula.

The thing that's so special about this 3-D Julia is that the surface is always positively indented. This occurs because I've symmetrized out the poles and because only one coordinate, x, is being translated at each iteration. The M-set at low iterations gets "teared" which tells me that it is isn't right. Something about the way the y and z coordinates interact allows this to happen. It's not obvious how or whether this problem can be resolved. But, considering how far we've come, anything's possible.

I think, somehow this still wasn't really tried but it was proposed a few times...
how would a default Mbulb look like, which has *8 for the second angle, rather than *2?

r² phi*2 theta*8...
or alternatively
r^8 phi*2 theta*8

ehrm, stumbeld on this thread because of a recent gallery posting, doesnt this qualify as the holy grail ?
why havent we accepted it ?!
http://www.fractalforums.com/theory/choosing-the-squaring-formula-by-location/msg14371/#msg14371


(http://www.fractalforums.com/gallery/1/803_20_03_10_3_36_55.png)

It's right above the picture in that same post, and it's kind of valid, but only if you're picky.

Hello msltoe.

I just wanted to let you know that I managed to get a nice 3D print out of your beautiful formula.
Thank you for the formulas my friends!!!  :beer:

I remember so well when I followed this thread and was blown away by that julia, this makes me extra happy now when I can wear it on my finger.

(http://nocache-nocookies.digitalgott.com/gallery/16/thumb_1002_29_05_14_3_05_41.jpeg) (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16167)

Klick the image for full view! ^^^

Johan: Thanks for bringing this formula to life into a beautiful wearable ring.   

Oh, You have no idea how greatful I am for this approval Mike (?).
As I just said to jehovajah: this is the essance of this forum. I just love it, I love you all!  :beer: O0

  That does look cool.  I bet a lot of us are thinking about building a CNC machine at some point- want to work with metal, wood, and stone instead of plastics. 

Msltoe one day please share with us the tetrahedral variants :)

I am having fun revisiting these as DarkBeam digs them up. I made the formula continuous with some changes but the nice Julias are similar.
WebGL thingy: https://www.shadertoy.com/view/MtfGWM (https://www.shadertoy.com/view/MtfGWM)

If the glsl is confusing I could write it out plainly.

Great idea I was looking for the continuous version!

Great stuff! Nice variety of shapes, which are even more beautiful when rotating and morphing. Those WebGL things are great.

I just dove into adding formulas to Mandelbulber v2, so yes please share the formula in the easy form:

newx = ...
newy = ...
newz = ...

This one is an addition to the ones already in MB3D?

MsltoeFoldQuat.m3f
MsltoeSym2.m3f
MsltoeSym3.m3f
MsltoeSym4.m3f
Msltoe_Sym.m3f

Can't check them all, since only Sym2, Sym3 and Sym4 have a description of the formula.

Sym2:

Sym3:

Sym4:

This appears to be different since it is based on rotating yz into -pi/8.0, pi/8.0. What I changed was making this the absolute value so the edges of the "pie slice" align with the next slice. Then I don't rotate it back which may totally ruin the original idea but the julias around y=z=0 look similar.

Mmm not sure. Cannot help.

Those were early formulas by Jesse. :(

awesome dudes, this one is really one that looks like one would expect

but errh, what do you mean by continuous version !? ???

Chris when a formula is discontinue the distance estimation fails ... you can't derive.
So continuous functions are always preferred :angel1:

i see, thank you ;) undifferentiable functions are problemetatic, right sir :)