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

[fracmonk]

As nearly as I can tell, the poor behavior of division in this space would make it a commutative ring.  The 2d .frm file for cross-section views I made available only hints at the complexity of indexes and the expanded collection of Julias that exist in this space. A lifetime would not be long enough to do even a cursory survey of these.  One day it could become a favorite vacation destination of mathematical minds, and as such, never get too crowded.  But without input from others, it is little more than a message in a bottle.

These things are there, but do not become a part of our reality until they are discovered and revealed.  This job is far from done, and since this is more important than commercial considerations, I've provided what I could manage without restriction.  As an avowed enemy of orthodoxy, I've made some mistakes along the way and said some foolish things, but I feel this will emerge and flower on its own, if it gets enough water and sunlight.  To be clearer, the grail cannot be possessed with the intention of hoarding it selfishly.  The search is not over.  This is only one way of projecting the M-set outside the complex plane, even if it turns out to be the best-justified one.  Every picture NEEDS a story.  What have YOU got?

[fracmonk]

One result of (let's call it) hypercomplex analysis is:  (j+ij)/sqrt(2)=i.  ...which means that the M-set, a little compressed sideways, can be seen on the a by c=d parameter plane (45-degree angle).  By NOT plotting bi dimension, you get a simply-connected version of M besides the one we're so familiar with.  It IS a single 3-d object that it comes from, but will always require 4-d calculation to render.

See it below, along with a location zoomed into, along with a Julia view for the same angle on the complex plane.

[fracmonk]

The plane shown in the 1st pic above cuts the volume of the a by c by d object exactly in half, because of its own symmetric properties.

Coordinates for the 2nd & 3rd pix there are a=-.744381711, b=0i, c=.148284347j, d=.148284347ij.

The 1st pic below is of a by c for constant d. From the view of that plane, it would seem unlikely that a zoom into the picture's ctr. would yield a view indistinguishable from that seen in the second pic above, but it does.  2nd pic below shows the entire origin slice of the Julia set in x by z orientation for the same coords.  Obviously not connected, but the 3rd pic shows its ctr. magnified 1000x.  Close to the origin, connected, locally. Away, falls to pieces.  This is only a 2d effect, as it remains connected, as previously discussed, in 3d.  The location was very randomly chosen.

[fracmonk]

Here are a couple more views of the julia ctr detail.  Stretch and spin in these are inevitable because of the orientation of the first view shown.  Interestingly, xy looks like zw there, xw looks like yz, and xz looks like yw.  See the pattern in that?

I'm really not a monologuist, you know.  I feel a little like I'm wandering city streets after a nuclear blast.  Or maybe it's just MK...

Comments?

[fracmonk]

Has anyone else tried to image the Julia set for 1+i-sqrt(2)j, or similar coordinates?  It seems that though this point translates to zero, it escapes rapidly, because of its multiple components, analogous to something like 20+20-40=0 (if it was pertaining to a Euclidean situation).  It is sheer divergence, cloaking the ordinary complex behavior with another entirely different one, which strikes me in a way as entirely natural, while the complete conversion is quasi-euclidean, in some respects.  There is no reason to think that extension won't reveal unexpected behaviors.  A complete conversion from 4d to 2d is easy, and elucidates it nicely, but I found that an idea to do a sort of conversion which keeps 3 of 4 components in each iterative step, using the (j+ij)/sqrt(2)=i identity will preserve the 4d results you have already seen.  It is also time-consuming, however, and changes nothing.  Still pretty nice to see it in both incarnations.  Any takers?  Other thoughts?

[fracmonk]

Reflecting on things, I realize I probably could have been clearer in saying that there is no advantage to conversion to pure complex except understanding of a few algebraic questions.  As in a Euclidean plot, the dimensions must be kept separate to get any meaningful results.  In that way, 20 minus 20 is a position, not a sum, or problem to be solved, by an answer such as zero, as in the case following.  I must confess, for awhile I was having doubts about the veracity of my results because of the canceling effect of, for example, 1+i=sqrt(2)j.  This only applies to the purposeful collapse of 4-d to 2-d before iteration begins, when you have a position 1+i-sqrt(2)j.  If you want 4d, you just don't collapse it!  Some days, I just feel stupid, but I do come around again...

[fracmonk]

Whenever I can, I'm working on the "final" version of my paper, of which early drafts were posted here, which will have more elaborated info about M in this space.  Busy w. other things too, so it takes awhile.  I really need to get it right, including all the things recently found, in some coherent form.  Asimov's Foundation trilogy, if I recall correctly, ended with 2 chapters entitled: The Answer that Satisfied, and The Answer that was True.  Before I submit this for your consideration, I have to research some applications speculated about for M in the past.  A bit haunted by the notion that M is an electrostatic model as well.  Extra dimensions that behave the same should be of more than passing interest to those who study electrodynamics, even QM, where explanations could always make a little more sense!  Not expert on such things, that, for me, is really reaching a bit...  Sorry for the delay.

Thought I'd show you a pretty picture while you're waiting.  Funny, I always used the same palette and never even rotated it for everything I ever put on this site...

[fracmonk]

And how about some spelunking down some of the odder-shaped canyons of the 4-d M-set?  I was doing a coordinate on the a by c=d plane, and looked at other views parallel to, or at 90 degrees to, the complex param. plane.  Too many times, I would see something interesting, but never have the time to go there.  It's a zoom exploration here in *b by c*, starting at mag. 100.  *Oops!  Pix are mislabeled, sorry!*

(to be continued...)

[fracmonk]

Never know what you'll find in a cave, and in lo res, you can miss good stuff...but you can follow yer nose wherever it goes...

There seems to be a problem with connex today- tried to upload remaining pix in the series here.  I'll try again...it seems to only take text right now, if it does even that...

So now (under "modify"), I try again.  V. stubborn guy...

[fracmonk]

And finally, the last of them.  Nothing at all special about the location.  The journey offers good sightseeing, as these usually do.  FractInt only allows a 9-digit decimal assignment in input variables, so that limits the depth of Julia views, which always correspond w. index views.  So this last pic has a big black blob at its center, to all but insure that the coordinates are of a connected Julia set.  This is a problem when one does not see the trademark mini-M.  Its edges have that repetitious quality, however.  The series was done with 100,000 iterations.  Infinity is another story entirely.  I should find an interesting set of Julia views to go w. this series.

Also, on radio, heard an entertaining short story called "Chivalry", by Neil Gaiman, and na       by Jane Curtin in a show called "Selected Shorts" which had at its center the Holy Grail.  Absolutely nothing to do w. this, but v. funny.  If curious, maybe you can search it out.

Afraid posting this pic is also prohibited as w. last post, so once more, we'll have to wait for better conditions.

The line up above contains "and na       by", which refuses to be fixed.  The word is "na      ".

NA      !

N
A
 
 
 
 
 
 
!

[fracmonk]

Here is the last in the series, and some z by w Julia views, which are really equivalent of x by y ones, turned 45 degrees.  I've noticed that one does not have to go as deep into J-sets to get the roughly corresponding features in an index set, as in the relationships between the standard M-set & its Julias.  Anyone have an explanation for that?

Tried to fix last msg.  The word is DETARRAN spelled backwards.  This site doesn't like that word in some uncanny way.

[fracmonk]

The more I think of msg. 234 I laugh harder- it's like something out of Monty Python (& the Holy Grail?)
"We are the Knights of NA       !  We want your fractal pictures.
Yes, indeed, all of them will do.  Thank you very much.  Thank you very, very, very..."

[fracmonk]

From the left side of the first picture in this last zoom series are a couple more detail pix.  They continue to show how the complexity of M takes on a wholly different character in 4-d.  Currently working on hi-res, hi-iteration pix that take a long time and contain too much data to show here, according to the rules as I understand them.  Am proofreading when I can, too...

[twinbee]

Well.... what a long thread! -  keep up the good work. Although it'd take me a while to fully understand everything written in this thread (my math is by no means top notch), the search is still ongoing, and for all I know you're getting closer.

To toss ideas into the mix, I'm not sure if it's been mentioned before, but one might try alternation of rotation and scaling with the original Mandelbulb formula, rather than performing both rotations (phi & theta), and then one master scaling afterwards. My preliminary attempts weren't successful, though I haven't tried further alternating with the vector additions as of yet (splitting up x, y & z).

[fracmonk]

twinbee-  For awhile I thought I was being shunned or something!  I recently looked at mandelbulb for the first time, and can't see how the M-set there is not a flower pot from which trig projections are made to grow.  That's my first impression. Where can I find understandable info on what its geometry represents mathematically?

  Anyway, a couple things I'd planned for today may help you here:

Back to basics- The pix below are for the familiar 0+i+0j+0ij Julia set, various 2-d views.  I'd love to see a 3-d rendering of this from anyone with such a capable generator.  I still work only in 2-d for now, and don't have one yet.  Even w. max iterations, it would be fast, since its nature is not to pinpoint non-escaping points, but for escaping points to shadow them closely.  (Pleeeeez!)

Also, I've taken my paper about as far as I could, keeping it as simple as it can be, I hope.  It aims for those like myself who would not take my word THAT this space is a good, maybe best environment for extension of M, but need to convince themselves of HOW and WHY it works.  (Just wanted you to know-) You are not alone.

PAPER MAY NEED FURTHER CORREX: see posts 251,252