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

[fracmonk]

My contest entry appears at the bottom of the Annual Fractal Art Contest 2011 page (where nobody goes) just beneath the categories, along with another pic, as if the 2 were highlighted.  It seems there are several avenues in which the contest thread can be navigated and therefore, it is easily bypassed.  I would have expected it to go in the list of other entries, and still can't figure why it didn't just go there.  So is it in or not?  If anyone can advise or assist...

Lately, I've been staying in a place where when nature bites, it bites HARD.  Eastern U.S. weather has been generally severe lately, enough to make the news.  Here, there have been high winds, torrential rains, minor (so far) flooding below, heat's been out for days now, and physically gaining net access from there is a hardship.  Hence the reference to the old TV show, where a really incredibly talented dog routinely rescues the rest of the cast from falling down wells, etc.  Lassie seemed to be able to bark in English too!

If you need to contact me, you can leave email @    fracmonk@mail.com   if it comes to that, and I will get back to you when I can.  There are reasons for the alternative, which I don't want to get into here and now.


makc-  Given my situation as described, I'm too tied up to give a fair look to your example, but I'm very interested in how it's different from what I said.  Please give details-  what YOU get from the citation you made, how it works, from your p.o.v....  You may want to take a leading role in this thread, if you care to, as I clearly can't right now.  It's foundation is good.  Build!


Thanx 4 your understanding.

Later!

[makc]

well the difference is obvious, what you got is four-leaf clover when you were supposed to get mandelbrot (on which border red points sit). so the application idea is this: user clicks along the "imaginary" oundary of fractal he wants, and we create the poly with roots in these points. nex (at this step is something I am not sure how to do) we find iteration rule corresponding to this poly in a way of that mandelbrot example. if this turns out to be impossible for random poly, I guess one could fnd some sort of "closest match". finally, you use iteration rule to render the image.

[fracmonk]

makc-  There was a notable absence of a constant in your original notation, so I hope you can see why I thought what I did.  Then it sounded as if you were proposing inverse iteration, or backwards iteration, which is also not new.  It requires randomly choosing between pos. and neg. roots in each step, and has the notorious weakness of not being able to reach into cusps, particularly.  It does demonstrate the same shapes to be available under iteration in either direction.  Forward is vastly superior in ease and accuracy, which is probably why the other method all but died years ago.  But you may still be talking about something else again, in which case, when it comes down to "Does it bother you that much?", it becomes a DIY proposition.

Lately, I've been wondering what bifurcation diagrams of real axes of the functions I've shown that produce multiple power shapes might look like, but it hasn't bothered me enough yet.  Anybody familiar with bifurcation diagrams can do that, if it bothers them that much.

Judging by history, I don't think it will.

Later.

[Kali]

I did a formula in UltraFractal for bifurcation maps some months ago, so I'm here to change history.

The formula I used was: z=(z*z*c+1)+1/(z*z*c+1) - 50 iterations

It's weird, as I expected... or a little more. It consist in two parts, one above and a lower one.
I did a full version, then a zoom on both parts.

Here is the full size to show you how the map is arranged:



link for better view:
http://img820.imageshack.us/img820/6504/bifurc1.jpg

And the zooms:

Upper:


Lower:



The black parts are actually very compacted vertical lines, as they show up with less iterations and zoom, but I forgot to upload and I'm too lazy now for uploading that to show you, sorry

[fracmonk]

Kali-  U R the best!  Any chance of changing the aspect ratio so it's less stretched vertically?  (These pains, they always want more...)

Thanx.

Later.

[Kali]

Oh, no more boring bifurcations... look at this beauty made by your formula, and colored by exponential smoothing. I'm very intrigued by results of this coloring method, which I use all the time. I don't fully understand how it works, however, as I'm very basic at math. With more iterations, the shapes becomes more precise and are much like the ones of escapetime coloring... but perhaps this technique, used for smoothing and forecasts of timeseries data, can tell us something more about this fractals made of iterative functions. As for now, it was for me a fun way to explore them:





Having read your post of your image for the contest, and the post 112 of this thread, among some other stuff you wrote here, It seems to me that you are a bit dissapointed about the path and protagonism that fractal art is taking lately, but correct me if I'm wrong. Also you said "we the fractalist", and I think you are mixing up things. Fractalists and Fractal Artists are not the same. I mean, they can be, or not. I'm myself a more a fractalist than a fractal artist (or at least I'm trying to be, I'm really new at this). I'm, like you, more interested in researching, understanding and in the quest of new types or methods of obtaining complex fractal images, always using the most simple formula possible. It's even a philosophical way of thinking, because I'm convinced the entire universe works much as this fractals, with all the inmense complexity coming out from the absolute simplicity. But I also like to admire fractals as art, looking for the aesthetically pleasing, but that's something not always related to fractalists. Many fractal artists, just play around with parameters, formula combining, colors, lights, etc. etc., without having much idea on what they are doing really, mathematical speaking. But, as artists, they have an special eye for visuals and makes stunning wonderful pieces of art without the need of understanding even what the hell is the meaning of the word "fractal". And is nothing wrong with that... I see this forum has become a bit more into Fractal Art than researching, but that's not bad at all... the fact is that the works are getting more and more amazing, and most people just like to see this spectacular fractals, without getting much involved into understanding or researching or whatever... I don't think there's a problem on that, do you? Anyway I see that there's some people doing research, sharing knowledge, making new softwares, colaborating with each other and so on... and the other people who are mostly "watchers", encouraging with their positive comments, so it's a really nice community... At first I was a little upset with some few things, like pictures uploaded here that are not even a fractal at all, or have very little fractal properties... i.e. an sphere with some wires dancing around... cool render, cool lights, cool colors, a lot of positive feedback... but maybe more suited for a 3D studio forum, don't you think? But is not big deal, each one has their own reasons for being here, and that's ok. I think the true reason for being upset was my lack of capacity for doing such cool 3D rendering 

Well, I have to say that I really enjoy this periodically updated thread of yours, it's very interesting, I'm always looking forward for new findings you make. I even had been exploring your formulas, and I based on one in my latest findings, as you can see in my latest post of the thread "Very simple formula for fractal patterns". Take a look at the whole thread if you didn't already, you have to like this, because is a really simple formula (and variations of it) that produces nice patterns. And they turned to be M-related, something I had no idea when I got the first version of the formula. This is the thread: http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/
I'd like to have your feedback on it... get out of this thread for a while!!!

Well, sorry for the long post, and I hope I was able to express my ideas using this poor english 

Regards,

[fracmonk]

Kali-  You're going to transform me into a man of few words!  Rarely does it happen that I would agree on all points with so many points made.  I take it that you prefer more monochromatic renderings than I.  I appreciate that, too, when I see it, but I don't do it, preferring CRAZY colors...most of the time.  As for exponential smoothing, I've been shy of such methods as continuous potential, true color, etc., because I'm more of a map freak.  I'm sure, however, you'll have many takers.  The rationale is usually to eliminate detail in general overview graphics, otherwise, where it can distract from a point being made.  There's room for all...

-except when there isn't, and then the guns usually come out...

And your thread is truly OUTSTANDING!

Thanx.  Later.

[fracmonk]

Kali-  Just back from your thread- It looks like a perky little formula I must find a way to play with.  Pretty sure it's not M-like, though.  Still, very impressed, and it looks like you've been v. busy w. it, too.

I still think that bifurcation diagrams of these functions, the ones that produce simply connected (as opposed to multiply connected) would be more interesting than you'd think at first consideration.  I'm going into serious fractal withdrawal, having nearly zero time for it.  Too involved w. low tech real world cave man stuff...

But GO, baby GO!

Later.

[Kali]

Fracmonk, thanks for the good feedback about my work, and I'm glad you agree with me on what I wrote in my last post here.

About my formula, keep in mind that you must use inner coloring for seeing the patterns, as the values doesn't diverge.
I don't know what coloring methods Fractint includes, but In my new thread I explain how 'exponential smoothing' works, so you can implement it as an add-on to the formula part: http://www.fractalforums.com/new-theories-and-research/a-new-look-into-escapetime-fractals-using-abs-and-inner-coloring/

Off course what I said about bifurcations diagrams was a joke, I'll try to get more decent ones and posting here. I wrote a vb.net application for plotting the regular M bifurcation, with an option to save the output to a sound file (yeah, a pretty strange feature), but I must add the possibility for going up and down the plot, in order to make it usable for the atipical diagram that your formula produces. I'm a bit busy now, but I promise I'll do it in the next days. Take a look at this thread to see what I'm talking about: http://www.fractalforums.com/new-theories-and-research/sound-wave-generation-from-1d-mandelbrot-formula/

Looking forward for your researchs, keep on the good work!

[fracmonk]

Kali,   My bunny reports that she has printed out the linx U supplied here, for me to study.  Thanx.

Fleshing out an earlier observation more fully, post 38 mentioned that in M, integer c is echoed by integer d in these multipower M-like sets,the simplest of them being: f(z)->((((zc-d)^2)-d)^2)-d.  The shape of d=1 is found in post 22, and the sword shape of d=2 can be seen in post 38.  The only other integer possibility for c within M is the origin, but that makes the entire plane non-escaping with the above formula.  But since any nonzero value for d results in a finite index set, I thought I'd test a d value ARBITRARILY CLOSE TO zero.  Pic 1 shows d=.0001, and curiously, changing its sign gives the same shape, with a 45 degree rotation.  As a non-integer value, it is NOT connected, and enormous, since there is an inverse proportional relationship between |d| and set size.  It does, however, begin to approach the shape of f(z)->(z^4)(c^4)-1 (pic 2), which IS connected and reasonably sized.  By simply squaring z in the first formula given above, one obtains the 8-4-2 formula, and, again using an arbitrarily small d, one similarly approaches the shape given by f(z)->(z^8)(c^4)-1 in pic 3.  These last last 2 pix both only yield the M-like power shapes of highest degree.  I thought some of you might find that interesting...


Irresistably off-topic:

A fool self-destructs, and is easily replaced.  We're the IMF, with global answers to local questions.  We don't want to run the whole world.  We do.  Things are never as they seem.  We make it so.  IMF.

The Wharton School will differ on this, but finance does not equal economy, but is the art of distorting it.

They would prefer that you keep believing that history is accidental...

Later!

[fracmonk]

Another thing I've been playing with lately is reordering execution of multipower formulae lines.  For certain critical points, one can give what amounts to very different longhand-expression equivalents that yield identical index sets.  It's a little like getting on a merry-go-round at different places, and still getting the same ride...

Elsewhere:

After 25 years of entraining the minds of women into a conformist (and perhaps vaguely anti-male) Borg, and being outrageously compensated for it, Oprah's show is over, finally, as promised so, so, long ago.  Sure it is...apparently, managing the many haystacks of money can wind up taking up all your time.

Some people say "Era" with a long e sound, some say it with a short e.  Anyway, one of her fans said it best, I'm sure without meaning to:  "It's the end of an Error."  Doesn't that say it all?

Maybe you had to be there, but that really made me laugh when it hit my ears.

Later!

[fracmonk]

Despite the unreliability of my ability to log on & make posts, I take every opportunity to check for new posts by others, when I can, especially if they contain inquiries.  Please don't feel forsaken!

Later...

[fracmonk]

Update from the real world, where the "war" rages on...that basically means I'm letting the computer sort of catch up.  I'm doing studies from sets I've found, to determine relationships between index sets and their Julias, but w. very time-consuming hi-res pix.  "Set it & forget it.", coming back much later, & maybe it's done, & I'll save it, since that's all I have time for lately.  Found out that FractInt does more than I thought it could, precision-wise.  The status screen (<tab>) will only show 9 digits of a fixed param, but it actually will do up to 14 in 320x200 and 12 in 2048x1536 accurately in its floating-point limit in formula mode.  That's its ceiling, for you flying types.  It can be checked by hitting <z> for the exact param info and then <esc> to restore calculation.  So going that much deeper and still being able to specify the Julias for the more precise locations is a blessing.

Well, it makes me just a little happier, anyway...

Later!

[David Makin]

Hi all, have been just keeping a scant eye on this thread and just realised one particular modification that I believe I've really only seen used by myself and Joe Maddry (though given that I only started in fractals in 1999 and never really had a proper look through the complete set of public formulas for Fractint I could be mistaken) and that is the use of the iteration count as part of the formula itself, I mean the following is tested by many:
z=z^iter +c
But I think few have investigated other ways of using the iteration count - in my case I came across the recursive algorithms for various polynomials at Wolfram Mathworld and incorporated them in my formulas for UF in as many ways as I had time for as my interested lasted (i.e. until I found something else to pursue) - you'll find these as the various "switch" formulas in mmfs.ufm at http://formulas.ultrafractal.com/ i.e. Switch Recursion, Switch Lucas, Switch Brahmagupta and Switch Morgan-Voyce. The Switch Gamma was just an offshoot of looking into such recursive relations as many involve factorials and the gamma function can be used for such evaluation as an alternative to simple recursion
Joe Maddry's formulas are also in the UF formula database as jam.*
I definitely think formulas of this type are worth further investigation - both artistically and mathematically.

[fracmonk]

Thanx David, I (for one) will try it, with some variation.  Straight iter would bring powers to fantastically huge proportions quickly, so I'd consider something to limit it, like iter/z, etc.

But I had planned, oddly enough, to talk about integer powers and second parameters today.  For the multi-power index sets I'd been discussing here mostly, they are apparently crucial to them if they are to share the same TOPOLOGY as M.  That would be the simplest way to describe the narrowest definition of "M-like" there is, I think.

For example, f(z)->((...(((z^n)c)-d)^n)-d)^...  would require that both n and d would be integer distance from 0, with d no larger than 2, and pure real or pure imaginary, give results that are simply connected in one piece.  That condition yields identical topology to M and may be useful in the study of both kinds, by comparison.  That is where I think the significance would be found when considering issues like local connectedness.

Fractional d or exponent on z does not offer this in only 2 dimensions, which, when one considers the complex numbers as closed, is where it counts.  In addition, the sign on d is crucial as well, as previous posts suggest.  Only within those constraints do the multiple-power sets have the narrow M-like properties described.

Topological identity in this sense, I believe, would be a valuable key for complex analysts.  Not to drop names, but if only they were still with us, Julia, Fatou, Mandelbrot, Douady, etc., would have been very interested, and many talented living enthusiasts SHOULD be, and my faith is that they will eventually discover this as I did.  Unfortunately, my sense is deeply 6th, as my own specific skills are wanting.  My discovery was partly accidental, while I was in fact looking for it specifically.

That should explain my particular obsession with it.

I'll add thoughts on that to elaborate as they come...

As now, 6-15, I thought I should clarify that d=i works with functions like the above when n=3.  I have not found any (and there are few) integer d values that aren't non-escaping in M as c values.  Better to update this post, as this thread's gone elsewhere in page 10.

Later...