Goals for a 'True' 3D Mandelbulb

[JColyer]

i've been digging into fractals off and on for years...always attempting to generate the same type of images found in the SoF and BoF books.  I figured out a decent way to capture the attractor basins without having to know a) what type of basin it is (attractive, parabolic, sigel) and b) without having to change/write custom trap code for each of these. 

now i've been exploring this 3d mandelbulb for a few days and just recent went from the 8th power version back to the classic 2nd power version.  i thought it was kinda funny that everyone's looking for this 'perfect' 3d version of the classic 2d mset, which although I can't back it up with detailed maths, in my mind could not exist using a '3d' version of the standard formula.

that being said I've been exploring the very 'deformed' looking 3d mandelbulb (2nd power) set and I think it's stunning, in many respects far more interesting then the 8th order ones.  i've applied my attractor mapping to the interior of julias for the 8th order set and the 2nd order set and it really seems that the 2nd order julias - internally - are far more interesting.

As of last night i've visually confirmed the existence of the standard attractive basin, possible a sigel-'sphere' and late last night may have stumbled upon a parabolic case attractor.

There are a couple of images over in the new 3d mandelbulb renders gallery of some of the stuff I've done in the past couple of days.  my images are not a shiny and polished as the other images being shown, but then again I'm not raytracing or using CUDA (yet).   I'm actually using a "off the shelf" mri/ct scan imaging application to take pre-rendered data clouds and doing stanard volumetric rendering...

JC

[LesPaul]

No, I believe I'm correct in saying that in the replicas the main mini-cardioid will have some period say p (in general the smaller the minibrot the higher the value of p) and then all the mini-bulbs (and mini-cardioids) on the minibrot have a period of n*p where n is the period of the corresponding full-size bulb - someone please correct me if I'm wromg.

That's the general trend that I've seen.  I don't know if it has been proven, or if it is even possible to prove.  There are places near the edge of even the main cardioid where orbits become wildly chaotic.

Thinking about orbits in 3D is an interesting way to judge a 3D Mandelbrot.  Finding nice, periodic loops in 3D would be encouraging.

[David Makin]

I figured out a decent way to capture the attractor basins without having to know a) what type of basin it is (attractive, parabolic, sigel) and b) without having to change/write custom trap code for each of these. 

Any chance of letting us know the algorithm ?
Any chance it opens the door to calculating a distance estimate for the inside more efficiently ?

[Dranorter]

No, I believe I'm correct in saying that in the replicas the main mini-cardioid will have some period say p (in general the smaller the minibrot the higher the value of p) and then all the mini-bulbs (and mini-cardioids) on the minibrot have a period of n*p where n is the period of the corresponding full-size bulb - someone please correct me if I'm wromg.

That's the general trend that I've seen.  I don't know if it has been proven, or if it is even possible to prove.  There are places near the edge of even the main cardioid where orbits become wildly chaotic.

Thinking about orbits in 3D is an interesting way to judge a 3D Mandelbrot.  Finding nice, periodic loops in 3D would be encouraging.

I seem to remember this (or something close) having been proven! I will look up some of the papers tonight. Hopefully I can actually look at (if not understand) the proof, but who knows.

[JColyer]

I figured out a decent way to capture the attractor basins without having to know a) what type of basin it is (attractive, parabolic, sigel) and b) without having to change/write custom trap code for each of these. 

Any chance of letting us know the algorithm ?
Any chance it opens the door to calculating a distance estimate for the inside more efficiently ?

Well yeah as long as nobody makes fun of me for it's simplicity. 

First a little background - like most everyone else here my interest was captured way back when all this was new and hard a shit to render even in 2d due to CPU slowness and a lack of sophisticated graphics.  When I saw the pics in SoF I became totally entranced with trying to reproduce the color plates that showed the basins of attraction of these.  My thinking was okay I understand that the mset and it's julias are basically just crumpled/deformed circles - but what the hell was going on inside them to distort them into these things???

Not being a mathelete or math geek by any means I could never understand how to engineer code that could classify what type(s) of attractors were at work and certainly could not write any of the more exotic orbit traps (parabolic, siegel, herman ring cases, etc.)  So I was stuck and kept hacking at it off and on over the years. 

At one point I tried to do something different to capture images of Siegel disks based on the description of the orbits and actually rendering the orbits of single points from a julia containing Siegel disks.   So i tried just calculating the distance traveled by a given point during it's orbit - and it worked I got an image that looked almost exactly like the ones in BoF/SoF - Success at last!

By mistake I had left this non-trapping orbit trap code in placed, changed the julia to one with a parabolic basin of attraction and well wadda ya know it too rendered up looking extremely similar to (what I've always assumed) were level-set decompositions of the julia set interiors.  And as I've said in other postings, this translates and works perfectly well with the 3D mandel bulb and 4D Quanternion/Hyper complex sets as well.

The basic premise is that one iterates all points in the formula to a fixed number of iterations (in the case of my recent 3d work, 50 iterations is fine) unless of course the point escapes - which in that case it's thrown out.  During the iteration I keep a running accumulation of the distance between Zn and Z (as a double or float) and that's the number that I use as my "level set" value in the renderings.

That's really about it, there's a couple of variations on a theme that work better for one formula/julia than another, but they all essentially revolve around the idea of capturing the distance traveled during the orbit.  I would like to take this further by learning more about the 'real' maths that are supposed to be used and now that I finally found this forum that might just happen!!

Now the caveats are that of course it's not really a level set decomposition, if you're dealing with a formula that can generate multiple types of basins in one image (Volterra-Lotka/Normalized-Q plane come to mind) then it can't tell the difference between them and of course it's not as "pure", but hey screw purity, I want pictures damnit!

And now the moment you've all been waiting for...

dist += Maths.Dist(Z,Zn)

Where Maths.Dist is a bog-standard distance between two points calculation.

So here's a 2nd order mandelbulb julia with just it's outermost (noisy) surface stripped off...using the data from the above highly complex piece of code   Sorry about the noise and artifacts in the render I'm limited to 512x512x512 data cubes and the volume renderer is made for rendering CT/MRI data not point clouds of mathematical goodness!

JC

[lkmitch]

Quick question about attractors and self-similarity : in the 2D M-Set, do the attractors in minibrots have the same properties as the main shape and surrounding circles ? (i.e fixed point in the cardioid, period 2 in the main circle, etc...)

No, I believe I'm correct in saying that in the replicas the main mini-cardioid will have some period say p (in general the smaller the minibrot the higher the value of p) and then all the mini-bulbs (and mini-cardioids) on the minibrot have a period of n*p where n is the period of the corresponding full-size bulb - someone please correct me if I'm wromg.

Yep, Dave's right.  For example, the "west midget," centered at about -1.75 on the spike, has a period of 3 for its cardioid.  The largest of its disks, on the spike just left of the cardioid, has a period of 2 x 3 = 6.

[David Makin]

There are places near the edge of even the main cardioid where orbits become wildly chaotic.

Given that *the* Set includes an infinite number of infinitessimally small points of infinite period that's only to be expected

[LesPaul]

I didn't realize it until just now, but I think the search may be over (for me, anyway)...  Allow me to piece together evidence from a few different threads...

Recently, David Makin produced this beauty:
http://www.fractalforums.com/gallery/0/141_02_08_09_4_15_15.jpg

It has the general look of the Mandelbrot, but perhaps without the rounded "bulbs" that you might expect.  But, it has a cross section which is the 2D Mandelbrot:
http://www.fractalforums.com/gallery/?sa=view;id=700

What you would also hope to see, upon zooming and increasing the iteration count, is the repeating/spiraling/etc, that is present in the 2D set.  Well, just a few days ago, bib produced this magnificent cutaway image:
http://www.fractalforums.com/gallery/1/492_20_11_09_8_03_11.jpg

I'm sold.  I don't really see any criteria that this 3D set fails to meet.  That image is the definition of what I'd expect a 3D Mandelbrot zoom to produce.

[TedWalther]

Which formula did he use for that image?

Ted

I didn't realize it until just now, but I think the search may be over (for me, anyway)...  Allow me to piece together evidence from a few different threads...

Recently, David Makin produced this beauty:
http://www.fractalforums.com/gallery/0/141_02_08_09_4_15_15.jpg

It has the general look of the Mandelbrot, but perhaps without the rounded "bulbs" that you might expect.  But, it has a cross section which is the 2D Mandelbrot:
http://www.fractalforums.com/gallery/?sa=view;id=700

What you would also hope to see, upon zooming and increasing the iteration count, is the repeating/spiraling/etc, that is present in the 2D set.  Well, just a few days ago, bib produced this magnificent cutaway image:
http://www.fractalforums.com/gallery/1/492_20_11_09_8_03_11.jpg

I'm sold.  I don't really see any criteria that this 3D set fails to meet.  That image is the definition of what I'd expect a 3D Mandelbrot zoom to produce.

[twinbee]

I'm sold.  I don't really see any criteria that this 3D set fails to meet.  That image is the definition of what I'd expect a 3D Mandelbrot zoom to produce.

You'd be surprised. As cool as those images are (and they are amazing), they still have 'whipped cream' in at least one axis. The real thing would look 1000x better still, and have detail for every pixel.

[LesPaul]

You'd be surprised. As cool as those images are (and they are amazing), they still have 'whipped cream' in at least one axis.

That's only true if you look at the parts outside the cutaway.  And whipped cream there is probably to be expected...  When you look at a 2D Mandelbrot, you're looking at it top-down, and you're seeing the entire set at once.  You can't do that with a 3D set unless you have 4D eyeballs.  And, if you did, you wouldn't need a cutaway to see all the complexity.

Similarly, imagine a 2D creature standing in the plane of the Mandelbrot set, on the outside looking in.  He'd be unable to see the spirals and deep complexity, as well as the self-similarity.  His view would be blocked by the "edge" made up of the last iteration, which would look like a fairly smooth, curvy surface, and he'd probably call it 2D whipped cream. 

You're right that the actual complexity in that cutaway exists primarily in only two dimensions, but it seems that the only reason for that is the choice that was made for where to place the clipping plane.  After seeing that, I feel pretty confident that cutaways here: http://makinmagic.deviantart.com/art/Attempt-at-real-3D-Mandy-126306420 would reveal the same kind of complexity.  It's already a fairly complex structure, even at a low iteration and it's complex in all three dimensions.

Time for me to splurge on some 3D fractal software, I guess!  (Or better yet, write some  )

[David Makin]

Which formula did he use for that image?

Ted

Hi Ted,

http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg7358/#msg7358

Of course strictly speaking using 4D math where the 4th dimension *has* to exist (as in the above) could probably be considered "cheating" with respect to creating a "true 3D" Mandelbrot. I mean that even if both the start value and constant term for the 4th dimension in the above math are zero then at some point in calculation if the second and third dimensions are non-zero then the next 4th dimension value will not be zero, so however you look at it this is truly 4D.

[mikeross]

This is my first post on fractal forums -- so a one sentence introduction: Im an artist and computer programmer, with some college-level math background, and only a brief foray into the world of fractals in highschool -- when I wrote some Mandelbrot exploration and other fractal programs (one on a TI-81!)

Ive been thinking about a sort of "cheating" way to eliminate the whipped cream effect in the current 3D mandelbulbs.  What about finding a way to measure the degree of whipped-ness of a region of the mandelbulb, then compressing/pinching space in this region to to directly counteract the stretched-out look?  I don't have any deeper thoughts than that, but curious if anyone has tried something like this. Some intermediate steps might be figuring out how to measure and describe whipped-ness precisely in 2D systems, then generalizing to 3D space.  Im guessing this is something like measuring the fractal dimension...

Mike

[LesPaul]

This is my first post on fractal forums -- so a one sentence introduction: Im an artist and computer programmer, with some college-level math background, and only a brief foray into the world of fractals in highschool -- when I wrote some Mandelbrot exploration and other fractal programs (one on a TI-81!)

That's funny, some of the first fractal code I wrote as on a TI-81, too. 

Ive been thinking about a sort of "cheating" way to eliminate the whipped cream effect in the current 3D mandelbulbs.  What about finding a way to measure the degree of whipped-ness of a region of the mandelbulb, then compressing/pinching space in this region to to directly counteract the stretched-out look?

Well, you could say that everyone's cheating in one way or another.  One way to reduce the smoothness is to increase max iteration.  The more, the better -- it just costs you dearly in terms of rendering time.

But I think many would feel that directly altering the set to make it look a certain way would be cheating too much.  The beauty of the set is that it just springs forth from a simple equation, without any manipulation.  It just is what it is.

Just opinion, though, and you might actually discover some interesting things about the nature of the surface by trying to measure the "whipped-ness."    I have no suggestion as to how you would go about it.