True 3D mandelbrot fractal (search for the holy grail continues)

[KRAFTWERK]

this thread is going to be closed, or at least needs to be splitted up, please open up another thread for off topics

OK, I removed my "upload-help" posting, if that was what you meant Trifox...

[fracmonk]

Trifox-  Not clear whether the problem is size or subject, so if the problem is subject, does that mean this thread remains open, and we should just stay more on topic?

[fracmonk]

Jehovajah- Still lurking about by any chance?  You haven't weighed in for awhile in this thread, and I wonder if all is o.k. w you.  Posted some of my 4-d (in 2d) in "implementation" section, and if categorization is followed strictly, I should consult w those in "programming" thread at some point, if no other progress can be made.  Due to unexpected (+ otherwise unhappy) events in real life, had fairly unrestricted access to net lately, so I got the chance to put out what I promised. Look around for it, alright?  Silver lining...

See you later.

[jehovajah]

Yes thanks fracmonk. Just been busy elsewhere and tidying up other lines of thought. For me the search for the true mandelbrot is nearly over,as i have seen many wonderful exhibits of it and my real interest was to explore why polynomial numerals of the form a+ib+jc apparently were not able to produce good mandelbulbs. I now know that this is not true but it requires serendipity to find them and a willingness to be creative abut the squaring formula.

I would like to see one of your slices but i do not have the inclination to figure out how to do it with my current technology, as i am much more concerned with the exploration of the most fundamental conceptions underlyng thinking and algebraic thinking specifically as it derives from general language forms,structures, procedures and syntax,parsing etc.

I will keep looking here though especially if i find a new mandelbulb formula, and who knows what you are explaining may suddenly become clear to me how to represent it. I am still "figuring out" how to implement kujonai's mod 3 unary operator system, although he calls it signed.

[jehovajah]

Just seen trifox warning. Think he means that a new thread will make this topic more accessible to new searchers. It is very weighty now.
Fracmonk catch up with you in any new thread you start.

[fracmonk]

Jehovajah- Ha! That may be programming...got a raw idea about color-coding for depth.  Length, width, depth, 3-d...instead of the traditional use of black for nonescaping areas and color according to speed of escape, thought of escapees all being colored black (as a night sky) and then coloring the parts of the surface of a 3-d mass of non-escapers according to distance from the computer screen in the foreground, so to speak. Thought of a pretty efficient timesaver scheme for that approach.  Needs programming (like everything else). Follow your nose... 

[fracmonk]

J-  Thought I might add that if you get fractint (free) and put the pix and the .frm file together w it in the same subdirectory (folder), and call up fractint, u can view the pix, you can get info about them, zoom into them, save the palette, etc.  Not hard to use at all. The supplied palettes could be better in most cases...these pix use my "standard" one, though there is little hint of its range at the magnification shown.  I'd like to supply a collection of interesting coordinates for anyone interested, since in future I once again will not be able to send pix unless I get an opportunity on a "normal" machine again, and that's an unpredictable thing.  Only wish I had more time for this instead of other things I just must do.

[fracmonk]

Some coordinates for those who might want to use 2d to evaluate worthiness of conjuring up a 3d generator for full-field-property 4-d M-set:

I suggest, to know where you are, the famous "San Marco" J-set, (first line below):
        
a   (real)                             b     (i)                            c      (j)                         d     (ij)
-1                                      0                                    0                                  0              , and next to try:
0                                        0                                    0                                  .583679392
0                                      +.999                             -.047194378                  +.220344925 , ("zebra" J-set views)
-.048401254                       -.803391959                    -.810697561                      +.31742613

and try one where values for all (a,b,c,d) are: -.333361438

Enjoy!     (This is, of course, for hardcore enthusiasts only...no really, anyone...)

[fracmonk]

Thought i'd spend some time describing some things i'd found so far, without too much rehash:  I've noticed that many features found are very elongated, and if you've ever mixed paint or saw handmade papers, you'll notice that these cross-section pix have that feel to them.  If I did successive pix w small incremental changes, I get mostly the suggestion of some very complicated detail structurally, likely exceeding the ordinary 2-d M-set.  One set of coordinates i'm looking at now may have a bubble of escaping points fully enclosed within 3 dimensions, with an outlet, however, when one of those 3 is exchanged for the previously unused 4th.  That would mean that simple connectedness would require all 4 dimensions, in such cases.  The requirement of the origin point being non-escaping for the J-set to be connected (even in 4 dims) seems to still apply.  Trying to fully grasp the meaning of these concepts extended to 4-d is strange territory for me.  Don't want to sound like a broken record, (or a skipping cd), but this really needs a 3-d generator.  Terry Gintz, where are you?!!

[fracmonk]

Is everyone else on vacation?  I have yet to see see any fractal fans read the paper or try those formula files in FractInt and either concur with (confirm) or refute my results.  The grail belongs to anyone courageous and pure of heart AND knows where to look for it...in other words, anyone who wants it...

Gaston Julia would have been supremely interested, and then asked: "What's a computer?"

[fracmonk]

While working in totally unrelated biz, I've kept the machine busy w hi-res assignments.  They're a bit more time-consuming, but I've found another peculiar set of coordinates close to the surface of the 4-d object (but still within it).  In this one, there are vortices of non-escaping points surrounding eyes of escaping points.  Non-escapees get thinner and closer together as they spin in.  The structure is too similar in character to Hawaiian earrings, which is bad news for MLC conjecture (unless I misunderstand COMPLETELY).  There are in many j-sets some pretty incomprehensible structures, but the likelihood of simple connectedness in 3-d (as opposed to needing 4 for that) seems stronger generally the closer I study these things (in those cases when the point at the origin of a j-set does not escape, of course).  It would be really nice...and it doesn't escape me that the real in the M-set has an UNBROKEN AND CONTINUOUS set of non-escaping points -2<a<.25, so that if ever the capability shows itself, an animation of a 3-d, continuously shape-changing object;   b by c by d   , for microscopic incremental changes in:   a   , would make a very outstanding animation.  Dream fractal dreams...

[fracmonk]

To continue last reply: coordinates referred to are: -.790323258-.193903776i+.25j+0ij, and were easy (by cj+dij in a by bi) to come by.  2-d "mandelbread" slice studies suggest that connectedness (to standard a by b M-set) is robust, substantial, and persistent in 3-d. Tenuous (filament) connections in the standard M-set can be either MORE or LESS substantial in higher dimensional extension, depending on views chosen.  Good news for the MLC conjecture, unless I misunderstand it completely (see last msg) OK, I admit it was a provocation of timid academic lurkers, if present...I had this daydream about some Big U. math dept. prof. whacking me on the head w. a rolled-up math journal, shouting: "BAD topologist, BAAAAD topologist!!"

In the past, I had run into filled-in Julia-like features in 2-d index sets of other complex functions, sometimes neatly symmetrical and apparently simply connected.  When they weren't, I had assumed, in most cases rightly, that I did not get the critical point, and there was only local coincidence.  But this offers some insight into why j-sets take their characteristic shapes.

I have been able to include a zoom series of the aforementioned coords. for a by b in mags. 1 thru 100k in this & last post.  They should have 4 to 3 aspect ratio, but don't.  They look really good in 2048x1536 and proper proportion.  (Hint, hint...)

And now for something completely different:

This is as good a time as any to thank David Gilmour for an ancient (pre-fractal) favor, related to inspiration and empathy.  We're even now.  Stay connected.  Breathe...

One more thing: There's no reason why fractal artists wouldn't want to include kaleidoscopic effects from trigonometric manipulation of quaternion-like schemes in their repertoire.  "Hall of mirrors" comes to my mind inevitably when considering the search here, which is of a mathematically scientific nature more than an artful one, however.  Everything has its place.  

[fracmonk]

As a result of some dialogue in the "Programming" ("FPP") section of this site, there Schlega came up with a slight improvement (about 5% in overall running time, on average) for the .frm file.  I updated it, available below.

A further result of that discussion prompted two updates of my paper on this subject, also below. notquat9.doc has the latest, most tragic news:
The division scheme long discussed most likely has a FATAL FLAW, described within.  The rest is fine, however.

The best and hopefully final version of the paper is posted at the end of page 16 of this this thread.  The aforementioned flaw is subject to your point of view, and the situation is explained fully there.

BAD CODE NOTICE: See posts 251, 252

[fracmonk]

Here are views of the J-set for the coordinates given in reply 221.  They have the origin at center and take slices at various axis combinations.

[fracmonk]

And here is the rest of that set.  Notice the similarity between xy and zw.  Notice also that it is very difficult to describe this space without lots of pictures...