Welcome to Fractal Forums

For technical reasons apparently, Trifox thought it would be best to continue the "search for the Holy Grail...", started by ILLI elsewhere, so, welcome to elsewhere.

It's slightly renamed, based on the whims of whoever winds up opening it, and this time it was yours truly, only because it was the only way I had left to answer a question left before it was closed.

For trafassel- in the old thread, page 13, post 190, can be found the scheme in its basic workings.  I really really really can't make heads or tails out of Mbulb-friendly code!  But if it could do what newx, newy, etc. do in post 190, it would have my blessing.  Again, if I ever get the chance to try it with ChaosPro, I'll try that first.  If you have a really deep understanding of the workings of Mbulb & think you can get it to do this, sure, give it a shot.  Regards.

sorry for inconvenience, i (we) are not sure about how to handle overlong threads yet,

here is the link for the specific thread entry you meant:
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-%28search-for-the-holy-grail-continues%29/msg16411/#msg16411

and the whole originating thread:
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-%28search-for-the-holy-grail-continues%29/

please continue as acustomed
 :angel1:

Trifox-  Thank you so much for the links.  Wish I knew how to do it myself.  I'm pathetic w. the aspects that make things handier, so it's v. appreciated.

One thing I noticed about locking a thread is that entries can no longer be modified, and I've seen more stupid mistakes I've made and never caught before in my haste...oh, well...

I can either reference the GOOD downloads from here, or repost them.  Which is preferred?

better reference them, it is also not quite sure how to handle overlong threads, in my eyes they tend to become unusable, on the other hand
they have all the information in one thread, no matter how long a thread becomes, in this particular thread it was just an intervention of me....

I implemented the formula:

newx=(x*x)-(y*y)-(2*z*w)+a
newy=(2*x*y)+(z*z)-(w*w)+b
newz=(2*x*z)-(2*y*w)+c
neww=(2*x*w)+(2*y*z)+d

Some pics where w (and d=0)   - XYZ Variant.

trafassel-  It's interesting, but it's gotta be wrong, since no view I've come up with has that boxy nature.  The formulaic was of course a functional guide, and may have to be adapted to a given generator's syntax requirements.  If I haven't mentioned it before, the character of most pix I've done w. the corrected version highlights what appears to be intersections of escape sets causing erosion of M-set shapes with other M-set features, i.e. crossing cusps, cut off bulbs, etc.  But ALL edges in 2-d, even for eroded minis, have edges that correspond to some edge parts of M. Easier to show than explain, but I cannot upload from where I am for some time.

Trifox-  I think it would be best to reference as needed then, only when needs arise.

To review, the old thread kindly linked to here by Trifox in post 1 contains many observational errors caused by my own code error.  To save you time, I've decided to highlight HERE those posts THERE that are correct and contain reliable info.

Undisputably GOOD info only begins there w. post 251.  The corrected (FractInt '.frm' 2d slice-view formulae) code is just above some sample pix there.  Those pix and the others in post 256 were done w. it.

In general, you must take into acct. that previously generated pix & their coords. are just plain wrong & merely artifactual.  Much conceptual data is right and pure math cited is correct, but the body of observational data is still remaining suspect due to the code error.  Controversy over division in the space, only further distracting, can be disregarded to the extent that it does not apply to M.  In all other respects, I'd hoped to be at least passably entertaining.

Also, for danger of it being lost to obscurity, please do not miss post 243, which may help you cope w. the loss of our Founder.  It never made it to the more specific thread opened for remembrances.

Trafassel-  Let me add that every dimensional subset (2d or 3d "slices") of this 4d object has rough edges or surfaces, nothing smooth in any view.  My recent complaint has been that any 3d generating programs are sparse on both how they work and how to use them effectively.  We are forced to guess whether an algebraic must be translated into a trigonometric (in which case I would probably NEVER be able to help...) or whether a simple 3-d diagrammetric map is given a perspective view (again, trigonometrically) by a programmer who just wants to show off...inadvertently making a formula the programmer hadn't counted on- go haywire...which is why, for now, with so little time at my disposal, I stick w. FractInt and generate incremental slices in 2d to get an overview of the object.  Hope that helps...later!

In the XYW-Space again the rough edges appears (as you wrote).

But with higher iterations it looks a little bit smoother.


 

Here's an alternative approach:

A complex number (x+i*y) can be written as:

m*(cos(a), sin(a)) = m*(x/sqrt(x^2+y^2), y/sqrt(x^2+y^2)) = m*(x/m, y/m)

where m is the magnitude (sqrt(x^2+y^2)) and a is the angle (atan2(y,x) in C).

Now consider instead the direction cosines - namely cos(ax)=x/sqrt(x^2+y^2) and cos(ay)=y/sqrt(x^2+y^2) (obviously ax is a and ay is 90-ax), so we have:

m*(cos(ax), cos(ay)) = m*(x/sqrt(x^2+y^2), y/sqrt(x^2+y^2))

This can easily be extended to more dimensions, so for 3 with m= sqrt(x^2+y^2+z^2) we have:

(x+i*y+j*z) = m*(cos(ax), cos(ay), cos(az)) = m*(x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x^2+y^2+z^2))

Also note (x/m)^2 + (y/m)^2 + (z/m)^2 = 1
Edit: Just a reminder that m = sqrt(x^2+y^2+z^2) here.

Now consider squaring this form, taking the most generic format then for complex numbers we have:


So extending to 3D we can use:

This gives up to 6^5 possibilities to try :)
I kinda like the look of these, but haven't tried any yet:


However I suspect the only 16 that really make sense are:

Trafassel-  I have to give D.M.'s post here a good study, but to be sure of what I told you, I looked for & found a symmetry that might resemble your pix just a little little bit.

2d plots of a by b where c=d have x-axis symmetry.  As |c=d| increases, the middle of the whole m-set you have when c=d=0 is cut away, in the way of a mirrored sunset over a still lake, so that the last you would see of it is when |c=d|>.793909 at approx. a=-.207107, b=0.  There, the filament end of the period 3 bulb finally perishes below the horizon, to continue the analogy.  If you stacked the incremental c=d views, it might look a little something like your pix, maybe moreso when the whole is turned 90 degrees on the real axis, but certainly without the flat areas of bulbs that emerge in your pix visible at the surface.  If you INTENDED such a view as I just described, your program rebelled on you, and maybe D.M.'s advice will fix it.  WAY beyond me, but I'll try to figure what's going on there...

Hi everyone!

I've been pondering a bit about the 3d spheres distribution on the main cardioid.
Since all attempts so far don't quite replicate the outer pattern seen in the 2d set (not without leaving many areas flat or distorted) I tried to find a simple way to transpose it in 3d: going from lower to higher periods and applying simple addiction/subtractions to simulate a 3d pattern.
The results so far are interesting- there seems to be a secondary "chain" going towards the period 3 sphere.
I'm not a mathematician, and I'm not good with numbers, so this is probably very wrong: but even if it's wrong I think there could be a way to predict the position and the period of the spheres looking at the 2d slice, even if we don't know the right formula for the higher dimension analogue.

So.. uh.. Keep up the good work!

(http://imgur.com/5s8zP.png)

(http://imgur.com/sztL6.png)

dyukv-  There IS in fact a complex analogue for every point within the 4d m-set object, so that any 4d coordinate has a 2d equivalent that it can be converted to.  That fact pretty much defines its limits in the space.  This was obfuscated by the bad code I spoke of.  The paper I wrote gets into this, but it might be better for you to wait for the updated version, that I will upload as soon as I can get to equipment that will allow it.

Also, I must reiterate that it is commutative, and therefore NOT in any way shape or form quaternion.  The difficulty comes with the notion that an extension of complex must be quaternion in the minds of those who programmed maybe ALL the 3d programs that exist to begin with.  All it has in common with those is the need for 4d for its full expression. 

David- W. my sadly underdeveloped trig. skills, I can only *assume* that your 3d treatment of post 9 would handle projection of j=sqrt(i) from Re & Im correctly.  But then how would the ij axis be worked in at other times?

I read author's notes on Chaospro, & I GET IT that trig. is resorted to in 3d progs. as a timesaving measure.  x*y*z= a lot of locations to test & store.  I would still rather do that by my method, where combinations of 3 of 4 axes can be mapped effortlessly and reliably, leaving the machines to crunch thru it by brute force if necessary, so as not to wait months for a reliable visual...

Otherwise we try to find trig. solutions to each that may wind up conflicting w. each other.  Also, I don't see the connection between your math and your code, but only because I don't know Mbulb code...so I'm in no position to tell whether your implementation reflects the formulaic in post 4 accurately.  My bad...

Mbulb is no doubt a true wonder, but I can only go on faith alone that only 4 of its many potential 3d expressions will accurately map j & ij with complex components.  Many many thanx for taking that on!

(by bunneemail) explanation to follow:

Due to fear of terrorists and hackers and bears (oh my!), I cannot upload from the public machine I'm writing on now.  But my bunny can from elsewhere!

Another pinch point:

The first 2 pix show a plot of c by d for a=-1 and are centered on c=0, d=.25.  The large circular feature in the first corresponds to the period 2 bulb in M, radius .25.  Picture 2 magnifies the same location by 100.

3rd is of a by d, with d=.25 @ ctr., for b=c=0.  Pretty cool, huh?

A conversion of the coords. algebraically corresponds to -1.176776695+.176776695i (approx.)

For more insight, consult latest version of my paper below.  Sorry, v. outta time!

(via bunneemail)

To continue last post, a couple more views of -1+0i+0j+.25ij, and the M equivalent and its julia, converted to complex.

Jesse recently informed me of his intension to make these formulae available in the Mbulb collection as "CommQuat", if I've got it right, and if it turns out right, I think.  Look for it, as I myself suffer from limited access, as you may know. I'd love to see results from it!  Thanx in advance... 

Has the search expired?

For those interested, I've done some curious work that can be found on this site under: Theories and Research->Is there anything novel left...which has 3d & 4d application potentials.

Found a new approach based on 4-param constant, 2 real, 2 imag, anyone interested?

In case there are any Python fans present, it's "...something completely different."   ...

Since I do everything in 2-d anyway, I've decided to post this new bit in the "New Theories and Research" section, under "Is there anything novel..."   -look at all the cobwebs!

So I was wondering- is there anything in the voluminous fractalforums "grail"-related archives about initial z value providing a 3rd and 4th dim. if you know offhand?

If you mean complex fractals with 2 dimensions as start z and 2 as c then that object (and related ones other than z^2+c) has been around since the early days of Fractint i.e. at least 15 years (probably a lot longer - I started late, only 1999) and is generally referred to as a Julibrot.

Here's an animation displaying 3D slices of the said 4 dimensions - in each case two are "fixed" as the correct axes and the third visual dimension taken as a line in the plane of the remaining two, the animation is produced by rotating the line from matching one of the spare dimensions around to the other e.g. x = real(starz), y = imag(startz), initial frame z = rea(c) intermediate frames z = line in plane of c at angle a (going from 0 to 90), final frame z = imag(c).

Unfortunately the results have as much or even more "whipped cream" than quaternions, in fact I personally suspect that the surface is only ever fractal in one given direction.

Here's the animation:

http://www.youtube.com/watch?v=gr-ul7sZDwc (http://www.youtube.com/watch?v=gr-ul7sZDwc)

Hello Fracmonc
Just want to inform you that your CommQuat is one important third of my mug-awarded Fractal Wasp Troll.

(http://nocache-nocookies.digitalgott.com/gallery/6/thumb_1002_19_04_11_2_19_06.jpeg) (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=6811)

Thank you for your great work!
Johan

David- I had wrote my own formula q+d & made a series of 2-slices (as I do) for a range of incremented real z0. Off z0=0, I found that a distance for real z0 yielded the same for pos or neg, so that those 3d had the corresponding symmetry.  I thought that Julibrots might be a bit different, but, I guess, no...

Kraftwerk- I didn't know it was a source of your prizewinner, and am that much happier 4 U.  That you could make it look that inviting is satisfying to me in the extreme!  And thanx!

I thought about mentioning it to you before, sorry about me being so late.
Glad you appreciate it, I for sure appreciate your work even if I have some problems following the maths! :)
This is why I did my series "The Mathematician", "The Programmer" and "The Artist", this is what fractalforums are all about, a beautiful symbiosis!
I have used the CommQuat in some more of my images, I will send you all links if you are interested...
You are welcome, and again, thank you!

Kraftwerk-  I'm sorry for not answering your last post sooner.  Yes, I'd love to see more, naturally.  I understand about not being prompt, but we have to meet other demands...the math is not so bad, for + &-, very simple, for x, a bit complicated, but div. is a bear, and needs review.

Later!

No problem fracmonk, if I do not follow the math all the time, but that is my problem, I am a novice learning more for every day here.   O0
Links to more images including the CommQuat:
http://MANDELWERK.deviantart.com/art/Royal-Monstrous-Coagulation-211226065
http://MANDELWERK.deviantart.com/art/The-Rose-with-the-Pearl-253973564
http://MANDELWERK.deviantart.com/art/The-burden-of-intellect-256020712
http://MANDELWERK.deviantart.com/art/Fighting-Fractal-Gods-201017720

Now I will not hijack your thread any more!  O0

K-  I'm only peeking in here on the run-  thanx 4 the linx, & perish the thought of "hijacking" the thread, (it's not "mine").  Anyone with something to say should ALWAYS feel free to make a contribution to the "cause"...

Take care, O.K.?

Here I tried to get to 3D mandelbrot:
http://www.fractalforums.com/3d-fractal-generation/re-true-3d-mandelbrot-type-fractal/msg38660/ (http://www.fractalforums.com/3d-fractal-generation/re-true-3d-mandelbrot-type-fractal/msg38660/)

Asdam- Thanx for your interest.  I am more and more interested in the function of numbers in our natural reality, and the properties they themselves have.  It gives me kind of a bias though.  I find that my formula here does create a redundancy in complex results, just projecting the same ones off the plane, so I lately tend to stick to 2d.  Never be discouraged, though.  Mandelbulb uses an actual 3d system that makes geometric sense, but my bias is against the inability to apply all properties when sticking to the 4 basic (binary)operations.  Not even sure if I'm making sense in that...

I think, if mandelbulbs would be mathematicaly correct, they too would be smooth rotation surface. Could be that having some manual rotations creates all the bubles and surface pattern, but realy they have just XY plane turned all around. Mandelbulbs alsou do not have elaborate zoom in pattern of mandelbrot.

What you think about this: http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/  (http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/)

Yes, saw it before.  I think what a lot of people are looking for are treatments that don't stretch and smear, to put it maybe too simply.  But depending on what your 3rd D is, that elongation of edge detail may be inevitable and true.

Simple, but almoust like using trigonometry. If there could be something simple making 2D formulas working in 3D, there would be much more different quaternions. I tried to iterate that manualy, but didn't got, what the formula realy do. Maybe something like tricorn aka mandelbar in zx axis.

I think, swirls and tentacles going to all sides (perfect 3D) would be pretty imposible unless someone will find some new numbers with square value different than i.

Asdam-  M & Tricorn can be intersected on the real axis.  I've done it in slices in 2d, as I always do, set up so the 2 planes cross each other @ 90 degrees.  A bit blobby...

That in 3D turned out different then were expected. Mandelbrot in XY and tricorn in ZX axis were fractal which moving or rotating completely changes geometry, and looks pretty strange. Could be the 3D engine;)

Maybe no (true) 3D mandelbrot. But in searching for this forum folks foung hudge number of other fractals. Hmm, maybe they should be convinced to search fork some Ark of Covenant of fractal world, some true 3D/4D Milk Way galactic fractal or true 3D/4D equation based dragon curve.

Since many 3d progs use unary trig. funcs AND may have been written w. other formulae in mind, I've never been inclined to trust results either.  That DOESN'T, however, mean that you can't...