Is there anything novel left to do in M-like escape-time fractals in 2d?

[fracmonk]

(pix via bunny express)

I don't know what came over me in my last post, but sorry for that- it's sort of off-topic.  For years now, we've been expected to keep a sense of humor about such things.  Or else.

What?  That's not funny?

The 1st pic below shows the whole mapping for d=-i, with init. z=0, clearly not connected.  At a glance, the 2nd pic, showing the largest contiguous fragments, is an oddity, but upon closer examination, is not meant to be connected, probably, ever...and note the beginnings of a second tail as well as the period-3-looking antlers...but it is what it is, as they say...a jackelope.

The 3rd pic cycles the M8,4,2 nesting expression back into the function from the very beginning of this thread, so it appears in multiply connected form. There are many, many other approaches to generalization...

The last pic was a test to see how a hi-res 2048x1536 pic would show up in a post here.  In this view of the major antenna, the 'polarity' is reversed from prior renderings, so that all signs before d are consistent.

And TALK 2 me!

Problems, clarifications (if I can help), your own insights, criticisms, accolades (if applicable), (so-called) genius grants, your 2 cents, funny stories, explaining local connectedness in plain English somehow, musings on what Euler might have done had he had a computer, whatever...yes, all these are welcomed.

Later.

[miner49er]

I do like these images. I also wonder what 3D version of these would be like...how about even simple Quaternion versions of them?

[fracmonk]

miner49er-  If you understand the formulae well enough and can translate them into the particular syntax of the formula parser of your favorite generator confidently, knock yourself out!  Not sure if there's any advantage to quaternion renderings, though.

Lately, I've been following a "less is better" philosophy regarding dimensions.  The complex nos. are a closed set, so for a fixed d, there's no reason to depart from the complex plane.  In post 42, I was thinking of c=a+bi,  a by bi by d(real), for init. z=0, and particularly in the interval -2<d<.25.  I thought that would be ambitious enough for now.

You know that d=-1 and d=-2 offer connected results.  Lately I'm searching for a critical relation between c and d that yields connected sets for fractional d.  I'm probably chasing phantoms, at least, that's what I suspect as an outcome.  All the same, I'll try for a while, unless someone can show me that I'm wasting my time.

[fracmonk]

To continue last post, I don't think that the most important development in this thread should get lost:  that is, the multiple multibrot effect of d=-1 in 2d in its formula structure, within a simply connected one-piece set.

d=-2 turning out simply connected as well was a welcome up-to-the-minute bonus discovery.

The fact that both remain exact period 2 integers in the standard M-set is what informs me most with regard to comments in the last post about the unlikelihood of connected sets in the formula structure for fractional d.

When the going gets tough, the agile go elsewhere...

Later.

[fracmonk]

(pix via bunny express)

I turned my attention to odd powers, which had been in the back of my mind lately.  Prior to the posting of the pix of another new object below, only 2 sets of eyes have ever seen it before, most likely.  After posting yesterday, I went home and wrote the formula for it.

The 1st pic shows it whole.  It has M3, 9, & 27 multibrots, and is generated by the function: f(z)-> (((((z^3)c+i)^3)+i)^3)+i, with critical points z=0, z=i.  It is simply connected, one-piece, and follows nesting rules like those discussed in previous posts for successive squaring, only that in this case, it's cubed instead.  Theoretically, as degree 27, it should have (2d-2) 52 critical points in all in its locus of connectedness (and good luck w. that!).

Maybe then you will excuse the facile observation I ended my last post with.  Apologies for that.  It's NOT advice, but it was 'moving on' that produced these pix...

The 2nd pic shows the left end larger, where the 'main body' of the set is, dominated by M9 shapes, dwarfed by its largest bulb to right, which is dominated by M27 shapes.  The 3rd pic shows the "continental divide" these sets always have.

The last pic shows a smaller M3 mini than the 2 on the main antennae in pic 1, arrayed with both M9 & M27 minis.  The highest power shape always dominates in overall area and number. (maybe 'frequency' describes the situation better than 'number')

[fracmonk]

(via bunny express)

These pix below are given to show how the nesting of cubing expressions has similarity to squared ones previously shown.

The first is f(z)-> (z^3)c+i, which yields a standard M-like overall index structure, only with M3 bulbs, simply connected.  Pic ctr.:c=0.

The 2nd is f(z)->(((z^3)c+i)^3)+i, and yields a connected index set with julia-like features, as a sort of intermediate step, with M3 at its heart.  Pic ctr.: c=1.

The 3rd is the degree 243 nesting, where one can begin to see the resemblance (albeit rotated) to the Julia set for z^3+i, shown in the last pic.  i, of course, is an integer distance from the origin, as are d=-1 and d=-2 in the squared formula structuring you've seen here before.  Here too, if such nesting was taken to infinite degree, the 'index set' would be of identical shape to the Julia in the last pic.

So there are plenty of new toys to play with now.

Got any good ones yourself?

Later.

[fracmonk]

(via bunny express)

As long as I live, I will always be looking for more things like these.  For the sheer fascination they offer me, I've worked in isolation, but working that way has few advantages.  While I don't NEED support or feedback to do what I do, these things are appreciated.

What I think most miss about these objects is their cohesiveness.  These sets are determined to be in one piece, as if they possessed a will to do so, with each location clinging to an unbreakable whole.  And as in the original M-set, their detail is never exhausted at any scale.

At first, there was M, and its 'multibrot' generalizations, and most of us were left to believe or assume that those were the only M-like objects possible that were simply connected.

The class of objects containing multiple multibrots in this thread show clearly that this is not so.  In my view, they will very significantly complicate questions of connectedness applied to M-like objects, in both topological and number theory terms, and pose many new and worthy questions.

My original aim here was to ignite discussion of these issues, but discussion of theory requires dialogue.  So, any comments? (Those meant to be UNDERSTOOD are preferred...more about THAT later).

In the meantime, below begins a zoom series into the odd-power object first presented in post 49 here, into a valley dominated by M27 bulbs.  Once again, the destination coordinates are centered in each successive magnification.

[fracmonk]

(via bunny express)

I hope that the descriptions I've given here so far have been understandable!  If not, let me know...

In my experience, I've learned that it's smarter to ask when you don't know, and tragic when you feel you need to look like you know, when you don't know, and don't ask.

I've noticed a lot of people who SOUND like they they know what they're talking about.  Sometimes, they do.  But if no one can understand what they're saying, it's useless.  And then, there are some that just don't really WANT to be understood.  They just want people to think they're smart.  That's off-topic, but useful here (and everywhere).

Always ask questions...

The zoom continues in the pix below, in a dendrite, into the vicinity of an M3 mini.

[fracmonk]

(pix via bunny express)

There's a lot of info out on the web on M basics, most of it helpful.  Then there's "advanced" theory, coming from what amounts to a priesthood, which may not be terribly effective when it comes to helping beginners jump the huge rift we could call "intermediate".  Practically speaking, "intermediate" really isn't there.  I seldom understand what complex analysts say about M-like sets mostly due to notation with which I have no acquaintance.  I prefer more verbalistic explanations.  There's not a lot of help for us there, so I think we should help each other, having little other choice in that.

In the zoom continuation below, the 1st pic brings the focus close to an M27 mini above one of the canyons of one of the 2 largest bulbs of the M3 seen in at the end of the last post.  This vicinity influences the features found there surrounding the M27 as seen in the 2nd pic.  The last pics hints at the nature of the destination locality, to be seen in the next post.

Happy computing!  Later.

[fracmonk]

(pix via bunny express)

I think I need a website for this stuff, especially to show hi-res pix, and to make more extensive documentation available.  But then, I'd have to MAINTAIN it, as if I had the time...

But having only recently found all this new stuff, I have to catch up on that documentation as well, while my studies of it have only scratched the surface.

A special msg. to lurkers:  Much as you try, you won't suck it dry.  But keep sucking!

The zoom ends with the 1st pic below, inside an M3 mini.  The 2nd pic is of the whole Julia set for the point in the ctr. of the 1st pic.  The 3 largest contiguous groupings of non-escaping points in this 3-symmetric set are J27.  At the center is always found a J9 for this function.  In the last pic, another zoom begins with z=i at center, mag. 10x, for comparison with the index zoom just finished, to be continued...

Later.

[fracmonk]

(pix via bunny express)

I think that the subset of complex functions that, when iterated from critical points that allow it, can produce a one-piece connected index set of non-escaping points, should be classified topologically as such and given a name.  M is probably the 'simplest' in this category.

In fact, wouldn't it be helpful if ALL complex functions could be categorized topologically?  Then, one could say: "This function is known to belong to this topological category.", and know what to expect from it.  Maybe then too, patterns would emerge that could indicate WHY they would fall into the categories they do, once defined.

What do you think?

Right here in this thread is as good a place as any to speak your mind.

I believe there is a painful lag in the community's discovery of the rich significance of the material in this thread.  I'll be patient...

Again, what do you think?

As the Julia zoom into z=i begun in the last post continues, the 1st pic visits a dendrite of a J27.  The 2nd shows the neighborhood of the last pic, entering the fringes surrounding a J3.

[fracmonk]

(pix via bunny express)

The transformation of St.Valentine's Day into what it is today was only a contrivance of a greeting card company, to enhance its profitability, of course.  When you have no thoughts or feelings of your own (a growing trend), they'll sell you some, readymade.  How romantic...zombies in love...but my BEST advice, for now, is to throw away that fancy new cellphone you felt so compelled to get.  It'll make you still more stupid, if that's possible.

I've probably just pissed off nearly everybody, but I've done it from a safe distance, LOL. (Lots Of Loathing?)
And that's the point: "social networks" are only furthering alienation.  Still, send a card to all your "friends".  Maybe you'll get some strange.

Love is something else entirely.  May you find it.

Long since this holiday took on a life of its own, I do wish fractal fans cardioids on cardioids on cardioids on...(M3 does that, whether on its own or in the company of other M types, as in this function).

The Julia zoom continues below into a field of J27's each surrounded by dendritic forms suggesting its locality, as seen in the last pic.  Next time (if there is one), the destination of the series.

The sun still shines, the wind still blows, and I think I should be a bit less fatalistic, don't you?  

[Kali]

To be honest, I didn't read and study the contents of this thread yet, I'm about to do it later when I have some more free time. Anyway I'm a novice and I have to learn a lot from fractals and mathematics to fully understand most of the things publicated in the forum by researchers like you, but I encourage you and all the comunity to continue researching on mandelbrot-like fractals because my intuition says that there are a lot of things to be discovered yet, and maybe really revolutionary stuff. As for the "community lag" you mentioned, I want to say that I see the community of fractalforums more interested on the eye-candy side of fractals than any other stuff. I think fractal art is great, I'm also a little on it, and I have seen in this forum the most beatiful and amazing pictures and animations made out of fractals ever. But my motivation for being here is more transcendental, and I'd love to see more people also interested on researching and looking for answers because I'm truly convinced that there is A LOT of novel left to do.

[Kali]

Ok, I did a quick review of all of your posts, and I'm truly amazed! This is definitely something new and great!
Multiple powers shapes in connected sets, smoothly combined! and what about z=(z*a+b)+(1/(z*a+b))+c, no powers involved??
I'm at work now, I'll take a deeper look back at home, and try it in Ultrafractal.
I'm not sure why you didn't get almost any feedback...
Why don't you use imageshack.com or other site for uploading better resolution images and then link them in your posts?
I'll render some of them in a few hours and I'll post here to see if we can get more attention by using some "eye candy" like I mentioned before 

[fracmonk]

Kali-  Thank you, thank you, thank you.  You get it!