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Author Topic: Is there a limit in exploring 3D fractals?  (Read 7816 times)
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Madman
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« on: December 03, 2010, 10:52:50 PM »

When I started exploring the Mandelbulb and Mandelbox and some hybrids, the "possibilities where infinite"  wink

But now I find that discovering something new and original gets more and more difficult. More and more, I find myself exploring "broccoli" ( Let me describe what I call broccoli: It's the failed combination of formula's and parameters, which must be the fractalic equivalent to white noise; no matter how far you zoom in, it always looks like the same unorganised mess) or finding stuff that somebody else already discovered.

So i'm starting to think (and sometimes wake up sweating in the middle of the night...): There's an end to what we can find in 3D fractal explorations!!

Of course, I may be mistaken. But maybe I'm not. What do you think? Is there an end to it?
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David Makin
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« Reply #1 on: December 04, 2010, 12:34:31 AM »

It sounds like maybe:

1. You're forgetting to reduce the camera size when zooming deeper ?
2. You're forgetting the possibilities that using cutting planes can produce ?
3. You're only using higher powers in Mandelbulbs ? (>4) - remember that z^2+c is *always* more varied than the higher powers for plain z^p+c.
4. You need to try and weight bulb/box mixes more towards the box than the bulb.
5. You need to try more non-standard variations of the classic bulb formula - I think there are still several in my UF formula that are as yet unimplimented in Jesse and Buddh's programs (probably because the math in some is truly silly but they do produce interesting variations).

Another variation in my UF formula is that I added standard escape-time IFS style transforms which can be mixed with the other formula types to interesting effect (these options are a little more predictable than when mixing KIFS with boxes/bulbs) so it wold be nice to have these in the other software too.

Also we're still at the beginning of this genre/medium (I discount previous investigations into 3D/4D forms such as quaternon/hypercomplex/bicomplex) and some of the options give possibilities that allow reasonable computing times for more complicated 3D formulas actually involving 3D transcendental functions.

In addition nowhere near all the 3D counterparts of traditional 2D fractal formulas have been tried yet even discounting those involving transcendentals (I mean even just using the standard triplex math).

« Last Edit: December 04, 2010, 01:10:26 AM by David Makin » Logged

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ker2x
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« Reply #2 on: December 04, 2010, 01:38:13 AM »

We're still exploring 2D fractal and 3D fractal is very new. Yes, there is a lot of deja-vu in the now "standard" mandelbulb and mandelbox but i do not see a limit yet.
latest demonstration : inner view of mandelbox  tease

And think about it, the 3D fractal are more or less (more than less) based on the mandebrot formula. There is tons and tons of other fractal formula to play with smiley

I'm playing with "2D" Buddhabrot Fractal  (plain good old z = z▓ +c) and i'm not bored yet smiley
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/ker2x's-buddhabrot-gallery/
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kon16ov
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« Reply #3 on: December 04, 2010, 02:09:57 PM »

Howdy,

For what it's worth, I think that there's truth the infinite repetition you'll find if you do the same things every time (which I doubt you're doing, but bear with me...), kind of like going to the Seahorse Valley every time and trying to find new things -- they're there, but it's not always immediately evident the variations that start cropping up in the angles of the bulbs, etc.  That said, I've never been one to not doink with stuff, so what I've been doing, of late, is really messing with the numbers and using a lot of decimals. 

One of my favorite structures, of late, has been the result of a 14.275 power bulb.  I have also enjoyed looking at a hybrid of the standard Trig (real) ( Float power:: 7.4659, z-multiplier: -0.5) with a Bulbox that's pretty much default, each 1 iteration.  It's all wicker-y and has twisty stuff and a couple of other features that are not something I've seen before.  I might post examples to my gallery, at some point, but the images are still uploading to my server and they're ~25MB each...maybe later...

So, I guess, my not-in-any-way-shape-or-form-scientific-or-professional opinion, would be to take solace in the infinite monkeying that can occur with the numbers.  And, David, I'll have to look at the formulas because if there's something I truly enjoy, it's "silly math." smiley

Phil
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Madman
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« Reply #4 on: December 04, 2010, 04:20:15 PM »

Woha! This is going in a different direction then I intended. Thanks for all your advice guys! There's definitely a couple of hints and tricks I didn't think of, so I'll probably use them sooner or later. And after Dave's remarks, I realised that I haven't done much in UF for quite a while now. Time to correct that!

Rereading my post, I guess you could interpret it as a (rather pathetic ;-)) call for help, but that's not what I intended. So let me try to rephrase and see if I can synchronise my writing with my thinking grin. I guess that from a mathematical point of view, there's no end to the variety you can achieve by zooming more into either mandelbrots, -bulbs or -boxes, but if you look from a more artistical view I tend to find that at a certain point you get to a level where "the more things change, the more they stay the same". Am I making sense here?

Let me put it another way: If you google "fractal art", you 'll find that 95% of the hits show spirally thingies. Some of them are quite beautiful and have probably taken hours to render, but you know that with a little work, you can make something close to it yourself or at least something with the same "feel".

So let me rephrase my question: Do you think that there will be a time when every new picture will look like one that has been rendered previously or at least has the same "feel" as something rendered previously? Something that you will immediately recognise as, say, a scale -1.34 Mandelbox, rotated x,y and z degrees, scaled and then sphere folded? Something that can no longer surprise you?

If you think I'm rambling, let me know. I'll just shut up then cheesy
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bib
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« Reply #5 on: December 04, 2010, 09:03:04 PM »

After some exploring I share the feeling that a lot of 3D fractal are now similar, simply because each newcomer starts to explore the same parameters, more or less in the order they were discovered here. But I also think that there is still an enormous potential for the curious ones to explore new paths. After the first exploration phase of the boxes by hardcore enthusiasts we are soon to enter a new era of more diversification in terms of audience, that will result to a new wave of creativity, not only from a formula standpoint, but also from an artistic, material, mixed-media, printing, sculpture, why not architecture and design, street-art, event-driven standpoint.

Anybody hot for a 3D fractal animation projection bombing?
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msltoe
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« Reply #6 on: December 05, 2010, 02:55:36 AM »

I'm a little concerned that 3-D fractals will never quite have the same diversity that the Mandelbrot set has because of the number of types of conformal transformations available in 3-dimensions is limited (rotations, scaling, inversion, and translation).

Even with this limitation, one of the things we haven't figured out how to visualize well, yet, is the aesthetically appealing escape time "coloring". Our current 3-D fractal images are just outlines of a single constant escape time, or ET function of a single resolution parameter.

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David Makin
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« Reply #7 on: December 05, 2010, 03:35:41 AM »

<snip>
Let me put it another way: If you google "fractal art", you 'll find that 95% of the hits show spirally thingies. Some of them are quite beautiful and have probably taken hours to render, but you know that with a little work, you can make something close to it yourself or at least something with the same "feel".

So let me rephrase my question: Do you think that there will be a time when every new picture will look like one that has been rendered previously or at least has the same "feel" as something rendered previously? Something that you will immediately recognise as, say, a scale -1.34 Mandelbox, rotated x,y and z degrees, scaled and then sphere folded? Something that can no longer surprise you?

> Do you think that there will be a time when every new picture will look like one that has been rendered previously or at least has the same "feel" as something rendered previously?

Yes, absolutely, this is *bound* to happen *assuming* you remove the term "every new picture" and replace it with "the majority of new pictures" - already image searching for "Mandelbulb" or even "Mandelbox" is basically like searching for "dog" on say a stock photography site and of course that goes quadruple at least for "Mandelbrot" or "Julia" or "Fractal" wink
This is also true historically of traditional forms of art displayed in "real" galleries - it's just that the "internet effect" rapidly speeds up the process *and* increases the amount of dross required to be sifted through to find the good stuff especially when doing general searches either of the entire web or very large free gallery sites such as DA or R'osity.

> Something that you will immediately recognise as, say, a scale -1.34 Mandelbox, rotated x,y and z degrees, scaled and then sphere folded?

There are many, many images of 2D fractals that experienced fractallers will recognize as being say the standard Mandelbrot in elephant valley or some such *but* even then the *good ones* are likely to be such that reproducing a similar final result is very difficult simply because of the likelihood that the image is multi-layered with very particular combinations of colouring methods, palettes and merging and trying to reproduce something close would take as long or longer than the original did *even for the artist who created it* without actually starting from the parameters of the original !
With 3D the rendering and in particular the colouring methods are not yet as sophisticated as those for 2D and in fact I expect the colouring algorithms used are likely to be less significant in this genre than in 2D fractals, probably being replaced with ever more sophisticated base formulas, better lighting including virtually unlimited numbers of light sources etc, using algorithms that allow more and more different fractals in the same scene (either in just one layer or using multi-layering) and probably many more possibilities that are applicable in 3D ray-tracing rather than simple 2D.
So the answer is yes, more and more "recogniseable' images will appear, but just like in 2D the ones that stand out from the crowd will not be easy to reproduce.

> Something that can no longer surprise you?

That's a bit like asking the biologists the same question about life on earth - and the answer of course is provided by recent history in that case, I mean no-one expected the arsenic scenario !
There will always be images produced using *any* kind of fractal that will surprise even the most knowledgeable fractal artst and/or mathematician.
So far the biggest surprise that I discovered personally was the 'Magic Formula" (in mmf4.ufm if I remember correctly), for which certain parameters produce a Mandelbrot that is essentially similar to z^2+c except that there are disconnected (tiny) areas of inside in each iteration band - for the power 2 version the number of these increases by a factor of two in each iteration band - now I admit that's a surprise from a mathematical point of view really, but renders of this done artistically are obviously at least somewhat different from the visual point of view as well. (Note that experimentally it seems each of these tiny areas has a minibrot at its centre but I have no matehmatical proof for this).
Also from both a mathematical and aesthetic standpoint both the Mandelbox and Mandelbulb were surprises - in fact they are better described as shocks or even revolutions so again I expect such things to happen in the future - in fact I expect the rate that such occurrences happen to increase exponentially.

As far as finding decent fractal art on the internet is concerned, take for example the deviantArt galleries, if you just want to find "quality" then the best solution is to start with (or find) one member there whose art you like (or whose opinions you respect) and check out their favourites - and then check out the favourites of their favourites etc. This is massively more efficient than using general searches in that it's far more attuned to your personal taste and fairer in terms of priorities with respect to how long the art's been on the site etc.
I think the same idea basically applies to browsing around R'osity too.
Of course if you don't have much time to devote to this then at deviantArt just look through the Daily Deviations for the fractal art that has been given that privelege, this is still better than using the general searching on DA (and gives good starting point/s for searching using member's favourites).
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David Makin
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« Reply #8 on: December 05, 2010, 04:02:03 AM »

I'm a little concerned that 3-D fractals will never quite have the same diversity that the Mandelbrot set has because of the number of types of conformal transformations available in 3-dimensions is limited (rotations, scaling, inversion, and translation).

Even with this limitation, one of the things we haven't figured out how to visualize well, yet, is the aesthetically appealing escape time "coloring". Our current 3-D fractal images are just outlines of a single constant escape time, or ET function of a single resolution parameter.

From this point of view I would mention that at the moment IMHO the most "sophisticated" 3D factal rendering is being done by those using L-Systems (or at least L-Systems converted to a form of IFS) and that there are other fractal forms (DLA, plasma/fBm,attractors) that work in 3D+.
Also the 4D forms extending from the 3D Mandelbulb have yet to be fully investigated, I fully expect some of those to be more mathematically "consistent" than the 3D, maybe even finding a true "field" ?
At the moment all the differet fractal types are essentially being worked on by various individuals and groups mostly separately from each other (I mean the fractal types not the people necessarily) and I believe in the future "fractal algorithms" will be such that a user (artist or mathematician) can mix and match any/all of these essentially in the same way that a single fractal layer of say a Julia is controlled in fractal software now.
Consider this from the point of view of IFS - using the transforms one can essentially put *anything* anywhere even to the point of putting things at restricted levels within a hierarchy - so envisage say the L-system for a sunflower mixed with fBm and a Sierpinski with a Mandelbrot in the centre of the flowers just for good measure (one could even theoretically mix in strange attractors and say cellular automata).
Also all that could be done in a GPGPU friendly manner.
« Last Edit: December 05, 2010, 04:15:39 AM by David Makin » Logged

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miner49er
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« Reply #9 on: December 06, 2010, 03:07:27 PM »

How to you create this one Madman?

http://www.fractalforums.com/gallery/alien-dna/?topicseen

As it seems to be very new/unique-looking and pretty damn stunning if you ask me.
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bib
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« Reply #10 on: December 06, 2010, 03:10:52 PM »

I would bet on a rotated negative mandelbox. The point of view is very original though.
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KRAFTWERK
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« Reply #11 on: December 06, 2010, 03:21:41 PM »

After some exploring I share the feeling that a lot of 3D fractal are now similar, simply because each newcomer starts to explore the same parameters, more or less in the order they were discovered here. But I also think that there is still an enormous potential for the curious ones to explore new paths. After the first exploration phase of the boxes by hardcore enthusiasts we are soon to enter a new era of more diversification in terms of audience, that will result to a new wave of creativity, not only from a formula standpoint, but also from an artistic, material, mixed-media, printing, sculpture, why not architecture and design, street-art, event-driven standpoint.

Anybody hot for a 3D fractal animation projection bombing?

Well spoken bib.
I am convinced we will "stumble" over new interesting hybrids about once a month (joint effort)
I still think there are extremely many different shapes, just in the rotated negscaleboxes.
Just look at my images from the Rhino-version, and this is from ONE place, and I have lots of cavities left to explore wink

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4403


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4453


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4488


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4576
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Madman
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« Reply #12 on: December 06, 2010, 08:39:59 PM »

@miner49er: bib's right  grin. If you like, I can post the parameters

@bib: If you're also able to tell me the scale and the rotation, were getting close to my concerns in my 2nd post (or I'll need to take a hint and leave the rotated negative 'box). BTW, I like your idea about architecture, think we can find someone to try and built this: http://nocache-nocookies.digitalgott.com/gallery/4/3195_05_11_10_11_54_15.jpeg?   cheesy

@Kraftwerk: if this is all in one place in the same rotated negative 'box, I guess we'll have some time left before we run into repetitions! There are lots of nooks and crannies left in all those boxes!

@David: I think you have a very good point that there's tools and talent and if you haven't got the latter, it doesn't matter at all how good your tools are...

@all: It seems my worries are unfounded, for the time being at least. I'm convinced that there's still a whole lot of virtual world out there that's as yet unexplored! So it's time to create lots of broccoli and maybe once in a while find something interesting out there...
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bib
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« Reply #13 on: December 06, 2010, 09:03:43 PM »


@bib: If you're also able to tell me the scale and the rotation, were getting close to my concerns in my 2nd post (or I'll need to take a hint and leave the rotated negative 'box). BTW, I like your idea about architecture, think we can find someone to try and built this: http://nocache-nocookies.digitalgott.com/gallery/4/3195_05_11_10_11_54_15.jpeg?   cheesy

I'm not THAT crazy to guess the parameters themselves. But scale is probably between -1.2 and -1.7 and rotations are smallish (less than 15 degrees). Now let's continue to develop our skills and the reputation of this forum, and the architects will pass by sooner or later wink
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David Makin
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« Reply #14 on: December 07, 2010, 12:03:48 AM »

Just to expand a little on this:

**************
There are many, many images of 2D fractals that experienced fractallers will recognize as being say the standard Mandelbrot in elephant valley or some such *but* even then the *good ones* are likely to be such that reproducing a similar final result is very difficult simply because of the likelihood that the image is multi-layered with very particular combinations of colouring methods, palettes and merging and trying to reproduce something close would take as long or longer than the original did *even for the artist who created it* without actually starting from the parameters of the original !
**************

Fractal Art is still historically in its infancy even compared to say photography, let alone more traditional art forms and as time goes by the medium will inevitably mature such that the standard of the best work will improve and the number of people who recognise it as such will increase. At the same time the number of those first starting out will increase, obviously many essentially "going over old ground" as they learn.

Certainly at the moment I do not believe that the best fractal art gets the credit it deserves as far as the "art world" or "general public" are concerned but over time this is bound to change, especially given the sterling work of many such as those who organised the exhibitions in conjunction with the Mathematicians Congress and others such as the Fractal Foundation http://fractalfoundation.org/
Hopefully the education system here in the UK will eventually realise that for 11-16 year olds fractals, fractal art and fractal math represent easily the best way of getting children interested in mathematics generally.
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