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

[fracmonk]

(pix via bunny express)

makc-  Saw the Obama video you posted, but again, I get NO SOUND from here, so again, I'm at a loss.  I forego home internet by choice, however.  Cell phones, credit cards, GPS, OnStar, etc. can double as surveillance tools, and when I'm in a more paranoid mood, I think they were developed PRIMARILY for that.  If I was REALLY paranoid, you'd find me in a corner in a fetal position, sucking my thumb.  If nobody bothered, someone would eventually find the mummified version.

Thanks for the formula, though.  Using the same animation method in your post 101, it wouldn't be hard to apply my post 88 specs. to it, if you have command of the program.  Why not just do that?

In that object, there are always 2 complete M perimeters, and 1 or 2 distinct connected groupings of non-escaping points, for every d value, in every 2d frame.

The 1st pic below is for d=-.25+i in the same view as pix in the lambda series I showed.  The 2nd pic shows the center of that object, c=.75+i, slightly enlarged.  The 3rd shows a plot of c=d for the same formula, essentially then f(z)->(z+c)^2 using critical point (init.) z=-c, identical to M.

Later.

[fracmonk]

There are SO MANY MORE permutations available in the nesting formulae than I thought there would be that can yield one-piece connected sets, because I've found so many approaches to alter them that work that way.  The trick is to get an overview of what changes to make and what can be expected from them...in other words, I'm looking for and finding some interesting behavioral patterns.  There's a kind of recombinance of available power shapes.  It's gonna take awhile...a really long while...

(!)

Later!

[makc]

Cell phones, credit cards, GPS, OnStar, etc. can double as surveillance tools, and when I'm in a more paranoid mood, I think they were developed PRIMARILY for that.

I heared tinfoil hats help or wet towels, as in "total recall". or just towels, as in "hitchhicker's guide".

[Kali]

Hey, fracmonk, this image I posted for the Spring Competition, it's using a formula "inspired" by your posts. Should I share the prize with you if I won? 



One more thing: I really like your acid humor and criticism about our goverments and society. I don't think is a problem to have your comments inserted here from time to time, but you should open a blog for this, don't you think? I will be your first follower 

[fracmonk]

makc-  If you could actually benefit from potassium iodide pills, (or tin foil hats) you've got much bigger problems.  And you might...I bet you "know where your towel is" at all times!

Kali-  A very generous gesture on your part, and I'm honored.  But I think the contest prize is a cup with the winning pic printed on it.  While we debate over whether it's half empty or half full, some would get a few laughs from sawing it in half.  Still, I like to think old Solomon was just kidding about the baby and all...

I wouldn't blame anyone for getting angry at me for expressing myself beyond this topic here.  This is probably not its proper place, though I think it's the right time.  You could count the rules that really matter on one hand, and yet, we have so many more.  Governments are only subsystems of a more global effort of a very few to turn the rest of us into cannibalistic rats in a maze.  You can't let that happen and still call yourself human.

Keep the faith!

Later.

[makc]

I actually never listened to hitchhicker series beyond 1st 30 minuts. Much more verbose than the movie, and I didn't like the voices, too. So the towel joke is kinda over my head. In the movie, they use towels to slap vogons few times (in the end), but that's all.

[Kali]

Kali-  A very generous gesture on your part, and I'm honored.  But I think the contest prize is a cup with the winning pic printed on it.  While we debate over whether it's half empty or half full, some would get a few laughs from sawing it in half.  Still, I like to think old Solomon was just kidding about the baby and all...

 

There's another solution: instead of dividing space, dividing time. I'll use the cup for 6 months, then send it to you. But you pay the shipping 

I wouldn't blame anyone for getting angry at me for expressing myself beyond this topic here.  This is probably not its proper place, though I think it's the right time.  You could count the rules that really matter on one hand, and yet, we have so many more.  Governments are only subsystems of a more global effort of a very few to turn the rest of us into cannibalistic rats in a maze.  You can't let that happen and still call yourself human.

I don't think anybody can be such a fool to get angry for that. I was really encouraging you to ALSO write your stuff outside the forum!
As for "cannibalistic rats in a maze", I think we are that already, in a figurative sense, since a long time ago... But maybe we are turning into a more literal version of it!

"cannibalistic rats in a fractal maze" sounds better (so we are not off-topic anymore )

Keep the resistance!

Later.

[fracmonk]

makc- Try the Wikipedia entry about "Towel Day".  Explains all.

Kali-  Don't forget that you still have to win it first.  If you do, I'll be content if you posted a picture of a picture of...a picture of the cup, here.

I come from an art background, and once was Art Director of a mag. devoted to avant-garde art, until what they called "art" just offended me nonstop.  The same people who might think that gold-plating dog poo was somehow genius could never bring themselves to consider, for example, science as art.  In math, I would consider Euler's Identity as art, or the M formula, now responsible for millions of beautiful pix.  Not to obscure such beauty, I always present pix here with the same palette, never rotated.  And I'm not even a minimalist!  But much fractal art does obscure its own math.  To each's own, I suppose.

And thanx bunches for helping with the Rat Problem.  I was beginning to think it was just ME...



Returning to topic, so far, I've pictured over 100 connected sets using nesting formula variations, most of which were described previously in this thread.  As degree increases, more variations are available.  I see a sort of periodic table coming from relations between them.  What a puzzle...

Later.

[fracmonk]

(pix via bunny express)

Almost inadvertently, I found that there lives a programmer in Pennsylvania whose name is rare and yet identical to mine.  I figure he deserves no less than to be undisturbed concerning any matter I've discussed here.  Thank you for that.

The pix below are a few degree 8 variations of the nesting formulae I've been working on lately.  Each of these remains one-piece connected.  The 1st index set is notable for the absence of M2 features, the 2nd for the absence of M4 features, and the last for the presence of Julia-like features, with M4 shapes in it otherwise.  Figuring the rhyme and reason of them, so that their patterns can be understood, again, will take a while.

Later!

[fracmonk]

As you may have gathered, I've taken on the task of making sense of the nesting formulae, and also have some ambition to review this whole thread for a list of key posts, highlights, if you will.

The "real world" tugs harder at me lately, and that's why my posts have been intermittent.  My bunny is fond of occasionally chirping musically: "Nobody cares!", and there is much evidence to support that.

Are we to view the world as fallen and task ourselves to leave it behind?  Some think it can be saved, so I'm going off to do that singlehandedly.  (Sure I am...)  Either way, what will be will be.

For the foreseeable future, I'll be checking in here, looking for the random contributions of others, and, for the time being, only responding to specific questions, if any, when I can.

Also, I'm joining the "Save the Humans" campaign that the whales were kind enough to organize.

Later.

[makc]

it would indeed be helpful to summarize your findings. it iis nice experience to take random poly formula, tweak variables and see what kind of picture you get, but really useful outcome would be to scetch the picture of what you want to get and calculate formula for that. I dont think you're anywhere close to that  point, but feel free to correct me if you are.

[fracmonk]

makc-  Sorry for not getting back earlier.  Things are very hectic for me now.
You've got to crawl before you can walk sometimes.  Very early on in this thread, posts 3 & 4 addressed that question.  Not only am I nowhere close to doing that, I'm not currently pursuing it.  My ambition here is far more humble- only to discover functions producing M-like shapes that are connected.  If I could devote myself to this full time (and I can't), maybe then I would be able to anticipate outcomes better.  I'm still at a stage of amazement at the COHESIVENESS of non-escaping point sets produced by given functions, like the ones that produce the M-set, the lambda set, or some others I've given here.  I consider this "property" (or behavior, whatever) alone to be truly miraculous, and if I ever found out WHY it occurs in various functions where it does, and not in others, that alone would be a very satisfying personal accomplishment.

With that, I would greatly appreciate any help anyone could give me.

And yet, the cost of such things is often to lose a good portion of that sense of wonder one has while such things remain mysterious.  Sometimes too, we take seemingly obvious things for granted, and fail to wonder at things we should be looking at more closely.  New questions inevitably come up in both cases, and we only wish we could get better answers sooner.

Later!

[makc]

With that, I would greatly appreciate any help anyone could give me.

Some guy here posted interesting observation (result attached). Doing this in other direction might be the way - from graph of roots we could calculate the polygon in a form of (z-z0)*(z-z1)*...(z-zn) and if someone figures out how to rearrange that into "iterative" form, we got ourselves nice fractal designer app

[fracmonk]

Makc-  Got your msg. secondhand by telephone, and came up w. this, w. scant details:
I took the progression ( (((z^2)+z)^2)+...)as a longer way of saying: f(z)->((z^2)+z+c, where c=0.  Then you get the Julia below, the familiar "brain" fractal.  Its real interval is -1<z<0, so it centers at z=-.5.  It turns out all Julias for the function center there, it turns out to be the critical point for the index set, which, in turn, is like standard M, only shifted to left (negative) .25 real on the parameter plane.  I often find critical points by testing some values for Julia sets first.

Unless I completely misunderstood, which is always possible...

Have been looking to make an entry into the Spring 2d Contest, but, dunce that I am, it took a long while to figure the procedure.  Then it rejected my first pic submission as oversize, while its storage value is quite small. NOW, I'll have to drive all the way back to my place and make up a medium version one larger than the 320x200 puny one with annoyingly wrong aspect ratio, that I left to hold its place, and return to the library with it before it closes, as if it's got a Koch snowflake's chance in hell of winning...  User freaking friendly...

So I went back in a driving rain, made the 3-second picture, brought it back, had another exasperating time in this hall-of mirrors, and made a new submission only by wiping out the old one, cutting and pasting the text, etc., and then, I do not see it appear in the gallery.  So, maybe it's there, and maybe it isn't.  Now, I have to get back before the river crests and the road washes out, Lassie.  (No freaking kidding-  I JUST DON'T NEED THIS!!!)  But I felt I should let you know what was going on...

Have Floyd, will travel.  Seriously, it keeps me calm...

Later.

[makc]

I took the progression ( (((z^2)+z)^2)+...)as a longer way of saying: f(z)->((z^2)+z+c, where c=0.  Then you get the Julia below... Unless I completely misunderstood, which is always possible...

Yeah you did, just look at the picture of roots, they are all sitting on standard Mandelbrot border.