3 Main Mandelbox types (?) Illustration

[miqel]

I'm putting together a page on mandelboxes, --and without getting into hybrids yet--,
Am I semi-right in thinking there are 3 main 'categories' (or more accurately, fun places to explore for beginners) of the basic box?
Positive, Negative & Rotated.

Here's one of the illustrations. The top row, first 2 images are by bib, hgjf_radiolaria. All the others are ones I did in mbulber.

[bib]

Interesting analysis and illustration, and thanks for the reference.
Some mechanic/organic shapes can also be found in positive scale boxes:
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1956
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1957

[Madman]

Hi Miqel,

Great idea to try to categorize mandelboxes! I agree with your architectural positive boxes, but I don't think negative boxes are organic by nature. I guess the "crystaline" qualification is a good one, though. In my opinion, the rotation is the transformation that's responsible for the "organic" modification to the mandelbox, either positively scaled or negatively.

It may be interesting to compare your categories with Tglad's general fractal categories. Maybe there's a possibility for some symbiosis there...

Regards,

Marco

[miqel]

@Bib
Are you 100% sure those 2 are positive scale?
"Living rocks" looks exactly like the terrain found on the inside corner of a -1.5(ish) box. Do you know the scale for that pic?
Another category would be Interior/inverse - I haven't rendered or explored them yet, but I've seen a few here and they look quite different and interesting.

@madman
I agree ... i was struggling for words with that, but crystalline is closer to what i'm trying to articulate, because the rotated ones are absurdly organic & lifeform like. Still haven't found the right word though.
I was just putting this together as a general intro for non-math or casual fractal fans, so i didn't think too hard about a more scientific taxonomy. I'll compare with Tglad's, which are very cool ... like a periodic table of fractals.  

[bib]

Oops, well spotted miqel, these were negative scales indeed. 

[Jesse]

Nice idea, Miqel.  Dunno if these have to be another category, maybe they have more variety in shapes but this has to be dicovered more...

Some 3d cuts from 4d rotated boxes:

[reallybigname]

Some 3d cuts from 4d rotated boxes:

That hurts my brain!

[Jesse]

Hmm, just started an animation with a rotation in the 4th dimension, maybe i should soft-filtering it to prevent more damage?

[reallybigname]

Well, its not so much the 4th dimension that hurts my brain, its the idea that its cut in 3D!   I mean, am I cut in 3D?  And, if I was, how could I tell?

[cKleinhuis]

@jesse is 3d projection of 4d boxes included in v1.67 ?!

[miqel]

@Jessee

Hi thanks for posting those .. yeah that'd definitely be a new category.
Are those renderings the equivalent of something like a Tesseract, just a regular 3d 'shadow' of 4space?
Very cool, i'd love to see more! I'm fascinated with the 4d equivalents of the regular platonic solids, so seeing it w/ fractals is awesome.

I stayed up ALL NIGHT the past 2 nights experimenting with rotated boxes, testing if applying rotations in various combinations of the powers of PHI (Phi7 down to -5phi) yeilds any unusually aesthetic results. So addictive ...   :

My theory is that since Phi is a proportional ratio that is both aesthetic to the human eye and also 'functional' in some structural way (judging by it's frequent appearance in biology, physics, etc) - that maybe making the rotations various powers of Phi would yeild distinctively harmonious results.
So far I can't tell, i'll have to compare to other methods to see any aesthetic trends.
+ i'm still learning about the ABG axes and trying phi and number sequence combinations on the main rotation planes.

Here are a few that i'm tweaking for later exploration ....
For comparison purposes, first a pristine
-1.5 Mandelbox with A=0 B=0 G=0



-1.5 Mandelbox with A=Phi5 B=Phi4 G=Phi2



-1.5 Mandelbox with A=12 B=0 G=12



-1.5 Mandelbox with A=Phi6 B=Phi6 G=Phi6  (This one looks really wild -plenty of weird branches & knobs!)

[Jesse]

Reallybigname, that might be a more philosphical question.  Maybe 'cutted' is not the right phrase, you just can see 3 out of the whole 4 dimensions at a time, so you must decide what part from the whole bulb you will show by rotating the 4d bulb in the additional dimension and/or shifting it there.
So we can also see only 3 dimensions at a time, but time is the 4th dimension, so this is no surprise 

Miqel, i am no mathematician... using Phi as an angle multiplier looks somewhat strange but maybe there are also some rules for it, dunno.
The rotated box can have so many different shapes that you may never can tell if it is from a 4d box or not, for the common box you can tell (just re-exploring the scale 2 box in 4d).
And Mandelbulber offers a different method for rotations (at the folding planes), what seems to give even more interesting options!

So i can't rerender the rotated boxes, here is an example for 6*Phi at all angles (performed in a row at the origin, so this can't be equal):

[Jesse]

@jesse is 3d projection of 4d boxes included in v1.67 ?!

missed the question: no!

[Madman]

The next question is: Will it be implemented in the next version??? Looks like a fun way to go on experimenting! 

[Jesse]

sure (but a bit offtopic here!)