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

[Alef]

Second picture have inetersing bell shapes. Maybe with right colouring.
Just remembered.

Here's more of a philosophical question I've always wanted to bang around, but I'm not really sure if it belongs here:

Did these things already exist before they were found?

As far as they are part of numbers nature, and numbers are reflection of real world laws, I think yes.

Well, this could be very hard and old question. It is listed as "unsolved problems in philosophy" together with questions like "what constitutes an art?" or "should I drink bear or vodka?" dating back to ancient Greece and maybe even older indian and jewish tradition.
http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_philosophy#Mathematical_objects

What are numbers, sets, groups, points, etc.? Are they real objects or are they simply relationships that necessarily exist in all structures? Although many disparate views exist regarding what a mathematical object is, the discussion may be roughly partitioned into two opposing schools of thought: platonism, which asserts that mathematical objects are real, and formalism, which asserts that mathematical objects are merely formal constructions. This dispute may be better understood when considering specific examples, such as the "continuum hypothesis". The continuum hypothesis has been proven independent of the ZF axioms of set theory, so according to that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question. A platonist, however, would assert that there either does or does not exist a transfinite set with a cardinality less than the continuum but greater than any countable set. So, regardless of whether it has been proven unprovable, the platonist would argue that an answer nonetheless does exist.

[fracmonk]

Most people never encounter such questions, and blow them off when they do.  To consider them is a richer experience though often frustrating.  I'm very glad that you recalled this particular context in relation to these multipowerbrot sets.

Originally, the things I found were not randomly stumbled upon, since I was looking in the right place for them.  If there's an answer to whether they existed before that, waiting to be found, I'll never know.  I see very special things about them that I try to express, and wonder how good or bad I am at it.  I think that these sets are the most mathematically significant find in escape-time fractals since the discovery of the original M-set itself.  If I have my head in the sand when it comes to other kinds of fractals, (and I don't entirely) that is why.

The main thing is that they WERE found, for whoever is interested.

That said, you are confronted with a phenomenon that challenges you to deal with it.

What has really surprised me is how academics studying the local connectedness issues of "simple" M have not even acknowledged the existence of multipowerbrot sets yet.  I mentioned before that these sets can either help in that study, or enlarge the problem, but I've seen no evidence that they were even considered.  Do they know about them?  Do they FEAR them?  I've heard nothing of any of that.
But you won't find me personally chasing after them, trying to sell it. It shouldn't work that way, and I hope it hasn't come down to that yet, since it's about knowledge, not politics.

Maybe they think FF is not a legitimate publishing venue, and somehow, THAT's more important than the content...

Who knows?  Who cares?

What will be, will be.

Your thoughts?



 

[fracmonk]

Found something pretty strange, unexpected, and interesting about the real axis limits of the multiply connected object of post 193, (degree 2048) but I'll have to save it 'til I get some pix to illustrate.  I have them in the wrong size for display here, but will redo soon.  (and the flaky machine that made them is acting up...)

Have a good weekend!

Later.

[fracmonk]

The first 2 pix below are of the respective left and right real axis limits of the multiply connected (M2048) object I've been looking at lately.  They are at cusps of M2 minis in the centers of each of these pix, and are at 10 billion x magnification:

Left limit: (about)  >-.15450714937694...
Right limit: (about) < .65450714936977...

I thought that some of the digit places being identical was a pretty curious item.  The surrounding environment is pretty interesting, too, but there's a strict limit on the number of picture postings per day, I think...

The remaining pix are 10x larger than the previous ones, for more local detail.

[fracmonk]

Before I go and add anything more about the multiply connected set of high degree (M2048) I've been investigating lately, I thought I should clarify an important point about the simply connected Multipowerbrot sets, which are far easier to study:

That is, they are TOPOLOGICALLY IDENTICAL to the Mandelbrot set in 2d.  Anything that can be said of the M-set from a strictly topological viewpoint can also be said of Multipowerbrot sets, and for that matter, the simpler, older, and more well-known Multibrot sets that each have only a single power shape included in them.  That is the basis of what I said about whether the study of them helps or hurts the local connectedness issue in post 196.

Hope that helps...

Later!

[tit_toinou]

Hi.
I also consider your finding very important.
But when you say things like "the set generated by this iteration formula is connected" or "theses set are topologically identical", do you have a proof ?
I mean, it LOOKS like it is true. Don't say it is.

I'm saying this because I've been told that theses kind of proof were for the best mathematicians.

However i'm trying to compute a nice image of your M842, with a Distance Estimator and a Sobel filter... But the derivative grows so fast  ! I can only compute two more iteration after diverging (with double).

You should definitely make a summary of all your findings.

[fracmonk]

t-t,     Like BBM, I have enough experience with simple, reliable formulae to be able to conjecture those things with extreme confidence.  True, proofs are not my domain, but they weren't his, either.  Yet, he could be as reasonably SURE.  I would put money on the idea that it can't be DISproven.  Now, having said the magic freaking word, before the vultures descend, I must remind one and all that we are talking about the SIMPLY connected versions of Multipowerbrots only, and in 2-d, on the c parameter plane.  That's nothing different from what I said before, but maybe more succinct.  Anyone who had a good look would be foolishly throwing their money away to me, and to me, THAT would be a very new thing.

And especially if I annoy you!

I would suggest not using distance estimation, and I don't even know what a Sobel filter might be expected to do.  The formula is incredibly simple for the 8-4-2 version, and is most easily done with the fractint formula parser.  Everything you need to piece it together is in this thread (somewhere!).  I probably said that I don't like a lot of generating progs. because many were written with very specialized 3-d apps. in mind.  KISS, and you'll get more.

My machines are working on M2048 pix currently, and that formula requires the use of NO timesavers.  One recent pic took 46 hrs. with fractint in DOSBox.  8-4-2 should take seconds in low mag.

Later.

[tit_toinou]

We agree. Here is my image :
Applying this filter is like computing the discrete derivative. I used its norm to colour.
It is very efficient to show the border of the set.
This image is clearly conforting us in our belief that this set is connected.

[fracmonk]

t-t-   I see you turned it 90.

For those who'd like to do it as I do, in fractint:

M842(xaxis) {
  c=pixel, z=p1:;kcps=z0=-2,0,2 + more...
    s=z*z*c-1    ;M8,M4,M2 features
    t=s*s-1
    u=t*t-1
    z=u+(1/u)    ;for multiply connected version
    |z| < p3
}

Set p3 large for best results;

for the simply connected version, make line 5:  z=t*t-1  and leave out line 6, or set a semicolon before it, so it's ignored.
3-28-12: thought I should add that init z's that work would not then be as shown above, but -1, 0, and 1.

This is a cleaned-up version of my original formula, with one changed sign, that would reverse the orientation of the object.
It is untested, but *should* work...      ...!

Let me know how it goes, O.K.?  (Important)

Later.

[Alef]

Hi.
I also consider your finding very important.
...
You should definitely make a summary of all your findings.

If you were recieving education in something sphere, you could write master thesis on this;) That could count as publication.

[fracmonk]

I know-  sorry!  I've not been keeping up with my accustomed documentation.  I get a new notion, follow it, and then have to catch up with filing the pix so they aren't a mad jumble.  Still have to summarize, like you said, but there's other explorations in the objects I should do in more detail first, like locating significant values, as in post 198, or maybe to map occurrence of constants in the objects.  D842 has got a bunch, mentioned before.

I'll eventually get around to it, when I've got some time.  Helping Bunny right now...

Later.

[fracmonk]

"Playing away":  Again I've been in a spot beyond the reach of internet access most of the time, but I've returned to my remote backup, now the slowest of 3 machines at my disposal.  Still very busy with (survival in) the real world, BUT have that machine grinding away at higher-degree multiply-connected versions of Multipowerbrot formulae, in a way that I don't have to "service" it too often.  Most would say the pix are not that visually dazzling, as the higher the degree, the more numerous and larger minis clutter the visual field.  But I'm still very interested in how they structure themselves hierarchically as to powers.  Additionally, I think it important to discover the rules regarding each successive power doubling in multiples of 2.  For the simply connected ones, there's an alternation in the qualitative output (the overall structure of the connected index sets and associated Julias) for each doubling, so that resemblance only carries for every 2 doublings of the maximum degree that a formula is worth.  Degrees 4, 16, 64,... behave alike to each other, and degrees 8, 32, 128... behave alike to each other, in one formula structure.  One tiny change in the formula structure makes these 2 groups exchange their qualitative output.  THAT they do, I know.  WHY they do, I don't!

This phenomenon carries over to the multiply connected versions as the degree is doubled, but with different consequences in those.

I might have mentioned before that it behaves a bit like the periodic table of the elements, but it is purely an anatomy of numerical, not physical, behavior.  (Unless, as many have speculated, the physical world's most fundamental rules ARE in fact strictly mathematical.  Then, mathematics could no longer be considered only abstract, but very real...invisible as a magnetic field, but there, nonetheless.)

Idle musings?

Later.

[fracmonk]

Was up in the wee hours the night before last, and it occurred to me:

The biggest problem with the "real" world is the fake money.

Why is it so?

Later.

[fracmonk]

I just fixed an error in post 193 regarding behavior expected in M2048.  The formula structure outlined in post 203 leads to the following expections in the category of "different consequences" (from simply connected versions, as mentioned in post 206) found in the multiply connected ones, in the following scheme:

formula degree                      power shapes to be found
2                                           4,2
4                                           4,2
8                                           8,4,2
16                                         8,4,2
32                                         16,8,4,2
64                                         16,8,4,2
128                                       32,16,8,4,2
256                                       32,16,8,4,2
512                                       64,32,16,8,4,2
1024                                     64,32,16,8,4,2
2048                                     128,64,32,16,8,4,2
(etc.)                                     (etc.)

Earlier, I had bemoaned the notion of having to tediously count the lobes on the mini shapes that will appear on the plane with each formula permutation.  But if you don't, then you will miss the pattern revealed above, for instance, as I had done until recently.

Sorry about that, chief.

Later!

[fracmonk]

Now investigating a code error which happily resulted in a pretty wierd index set (degree 96, apparently), *connected* against ALL expectations, which will likely open up a whole new region of formula permutations (for each degree?) with their own peculiar rules.  Looking to bring the effect to the lowest degree possible (and see if it still "works") to know more about it before introducing any poorly-conceived theories about it!

(SEE NEXT POST about getting too far ahead of one's self...)

If only I could make ALL my mistakes privately, and find and correct them before letting them loose on anyone else!  (I try.  I DO, REALLY!  Honest!)

You must realize that it's not just about pretty pictures for me, though they come naturally with the territory.  I like the puzzle aspects as well.  I want to know what makes these things tick...

Notice in the last post that degree 2 yielded the M4 power shape in combination with the expected M2, seen very early on in this thread, because of the use of the inverse, so that it yielded a "quasi" degree 4 (in the distance between exponent 2 and -2).  I have since wondered whether the formula for determining the number of critical points (2d-2, where d is the highest exponent employed) is correct in all cases.

Can anyone help with that in particular?

Later...