How to recognize the holy grail fractal?

[mfg]

Dear Friends,
Thank you for your posts in this thread and messages from frakmonk and trafassel.

Let us separate two issues in our discussion:

a) On the one hand are the properties that a 3D 'holy grail' fractal set should fulfill.
b) A very close, but different problem, is what are the necessary properties required for the algebra that generates such fractal.

Let me for the moment concentrate on a), that is, the way the set looks or should look:

a1) I think we have consensus that the 3D set must exhibit small copies of itself, this property (with or without some additional characteristics) has been called auto-similarity, self-similarity, universal, tunability, etc.
a2) This is certainly not enough, because some 3D fractals like Sierpinsky 3D triangles and others exhibit smaller copies of itself but do not have the richness expected for a 3D 'holy grail' fractal.

Let me turn to a different question that may shed some light into the problem:
The mandelbulb produced with Triplex algebra comes close to a 3D 'holy grail' fractal but is not it. Why? maybe ...

+ because the quadratic iteration is not as intricate as expected and eighth power iteration is a peculiar sweet spot as mentioned by Daniel White?

+ because it has 'whipped cream' regions?

+ because the two hypercomplex axes in addition to the real axis are not quite symmetrical?

What do you think?

[cKleinhuis]

ok, fernandez, this is my five cents:

+ because the quadratic iteration is not as intricate as expected and eighth power iteration is a peculiar sweet spot as mentioned by Daniel White?

the eight power has just been producing the most intricating results, especially the formations of the bulbs are visible self-similar, even in 3d, because
those bulbs are clearly conststing of smaller copies of itself, the quadratic iteration is intricate as well, but it does not show the self similarity as in the
power8 version

the power2 variant clearly provides a method for creating a 3d object that is not just an extrusion or rotating of the plane, and shows visual advances
directly compared to the quaternion mandelbrot2 variants


+ because it has 'whipped cream' regions?

the whipped cream is something we have to concentrate on, although it already has been shown that the surface of the bulbs exhibit a fractal
dimension of 3, which basically means that the surface (area) already IS infinite, the border of the mandelbrot in 2d has as well a fractal dimension
of exactly 2, which basically means that the surface (border) of the mandelbrot is infinite as well

ok, now let us look at a cut through a slice of the mandelbulb exhibiting a whipped cream area
first we need to clarify that those "whipped cream" areas are not because a too low iteration used for displaying, as a mandelbrot at low iterations
e.g. just 10 or so looks really like a whipped cream area, but this is not meant, the whipped cream areas of the mandelbulb do not exhibit more
complexity when increasing iteration count

so, as far as i know the definition of a space filling curve (->fractal dimension =2) is that every cut of such a curve must exhibit equally infinite
lengthes, so, each 2 points of the mandelbrot border are connected with an infinite long line, this property would not apply to a sliced mandelbulb
exhibiting the whipped cream, some parts of the slice may have complex parts that when connecting them together is as well an line of infinite length
but there are parts that would just be connected through a curved line, such a curved line with no extra detail exhibiting on each sub-slice is
not what we call a fractal

so, in my words i am formulating a mandelbrot holy grail as this:

every slice of any cut of a holy grail candidate's border has to be a complete space filling curve, with that i mean that every 2 points connected on such a
slices border MUST have infinite distances between them ( in theory when iterating an infinite amount of times )


+ because the two hypercomplex axes in addition to the real axis are not quite symmetrical?
what do you mean by this ?! all of the axes are symetrical orthogonal i was thinking of a triplex method defined
by non-orthogonal axis just incorporating another parameter to play with, but this would in my eyes just produce weird results

[mfg]

Dear Crhistian, It is very interesting that you formulate the holy grail fractal in terms of space filling curves.

However, we should recall that not all Julia sets are fractal even if they lie within the M-set. Take for example the origin that produces a circle whose circumference is 2 .

This brings us to the following reflections ...

a3) So far we have dealt with the parameter representation, that is the M-set; But have said very little of the corresponding
  filled in Julia set. 2D quadratic iteration Julia sets, although conformal, etc. do not exhibit small copies of the larger Julia set. They do however, exhibit self-similarity in the Douady sense, that the shape of the boundary does not depend on the place where you look nor the magnification.

a4) The Julia set for the origin should be equal to the constant unit magnitude surface. For example, the
  set for the origin in the complex plane is a circle with unit radius. Indeed, the magnitude of complex numbers is given by the sum of the squares of the real and imaginary parts. This equation equal to constant one also reproduces a circle. I suggest that this should also be true in the 3D case:

“The Julia set for the origin should be equal to the constant unit magnitude surface in 3D”

What other properties do you all think we should request for the 3D holy grail Julia sets?

[cKleinhuis]

Hi mfg the julia set for the mandelbulb already is a unit sphere

So julia sets belong to the corresponding fractals soon i will publish a visualisation of mandelbrit and julia iterations

some things will become visible

the first to say about julia sets is that they inherit much less complexity because they share the same seed values
for every pixel

the thing with the unit circle for seed 0 should be better viewn as visualisation of a strange attractor strange attractors are what somehow represent magnetic areas around that the points somehow are dragged to strange attractors are the points in the mandelbrot set that posess perfect periods the bulbs of the mandelbrot contain longer periods e.g. The -1 location, the i location or -i seeds

i have prepared a nice visualidation if it where some thing become pretty clear and how a threed transform should look like, but the bupb formulas we have resemble already very nicely the transforms from 2d

[cKleinhuis]

Arrh sorry the i and-i seeds obvously dont provide (short) periods

[Alef]

In my opinion holy grail could be a wrong target. Here alredy are bunch of 3D candidates, but they didn't gained much atention, and I think it's becouse it's hard to zoom 3D raytracer to the right point, raytracers fails on too small elements and xxx-brots aren't so spectacular, so pattern box sets are more popular. And unless you upload algebraic proof to arxiv.org it is judged by it's visual value. Maybe 3D prints would bring up different trend of fractals.

Recently I looked throught my old pictures and found one, what actualy looks what you could expect from true 3D version of stalk on mandelbrot set and re-rendered it. Actualy it was simplest formula so far, but using built in quaternion math.

 //manipulate numbers so that x^2 >< y^2 >< z^2
z= quaternion ( real(z) , imag(z), part_j(z) + part_k(z), 0.5*part_k(z) - part_j(z) );   
z= z*z+C; //using quaternion numbers


Oh, there were a thread with more ugly rendered pictures:
http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/

Maybe from this you can try to get formula
x= ... +Cx
y= ... +Cy
z= ... +Cz

And expand it to more powers.

IMHO reverding target could be bringing natural forms into world of 3D fractals. Romanesco broccoli is prime example of fractals and fibonacci numbers in nature. But so far I haven't seen any raytraced 3D romanesco broccoli, with pseudo-fuchsian (what an ugly name) being the closest.



Alsou an Aloe polyphylla looks pretty much like possible fractal with limited iterations (depth).


Maybe they both could be represented by self simmilar spirals in XY plane and height function in z axis with greatest height in center of spiral. If they would be repeated, you could have sacred groove of romanesco broccolis, and then they could be hybridised with abox/ kali's amazing surface / intpowerfold / menger.

Chaos Pro parameter of fractal above:

Mbrot_3Dspiral  {
  credits="Asdam1;12/9/2011/22/16"  commentTemplate="Saved on $month$\
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  CommentText="Saved on Feb, 28 2013 at 00:04:21\nDate: Feb 28, 2013\\
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  roughness=+0 shadowMapMultiplier=+4 AORadius=+5 AOStrength=+3
  AOSamples=+150 FogEnabled=+1 FogAutoAdjust=+0 FogRed=255
  FogGreen=255 FogBlue=255 FogFront=+0.01 FogBack=+0.2 FogDensity=+2
formula:
  filename="Malinovsky3D.cfm" entry="qMandelbrot3D" p_bailout=1200
  p_settype="Mandelbrot Set" p_testInverted=0 p_julia=-0.45/0.5/0/0
  p_perturb=0/0/0/0 p_addinverse=0 p_coefInv=0.125 p_coefInv2=0
  p_coefInv3=0 maxiter=40 highresmult=8 backtrace=8
inside:
  filename="NumberSeekerColouring.ccl" entry="WaveTrichrome"
  p_palette="Fractal Explorer like" p_seed=1.5 p_orbits="None"
  p_switchrb=0 p_colmethod="With counter" p_naturalise=0
  p_postfn="1- None" p_lightR=0.7 p_scalarR=0.7 p_lightG=1.2
  p_scalarG=1.2 p_lightB=0.25 p_scalarB=0.25 density=1.5 solid=0
  background=16777215
dimensional:
  observer=-0.66830283965108/-0.45558266622361/-0.050793673043921
  topview=-0.051249313829431/0.99792207538851/-0.039051751360133
  viewpoint=-0.66826037553426/-0.44775798225608/0.14910129543772
  backClippingPlane=0.2 viewangle=36
lighting:
  lightModel=0 light0Shadow=yes light0Pos=1000/1000/-1000
  light0Shiny=0.5 light1Enabled=yes light1Shadow=yes light1Shiny=0.5
  light4Shadow=yes light4Shiny=0.5
gradient:
  smooth=yes colormodel=CM_RGB knotmode=all dragknotmode=global
  Offset=0 knotrgb=(0,100,100,100) knotrgb=(50,220,50,132)
  knotrgb=(101,120,245,194) knotrgb=(151,21,140,61)
  knotrgb=(202,220,50,132)
}

[mfg]

Fractal generation with scators will be started in a new post.

[macawscopes]

It depends on what algebra you're using!  They each have their own special fractal

<a href="https://vimeo.com/moogaloop.swf?clip_id=155587929&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA" target="_blank">https://vimeo.com/moogaloop.swf?clip_id=155587929&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA</a>

[macawscopes]

On the other hand I'm not sure why division should be necessary.

me neither!