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

[lkmitch]

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.

Here's an idea that I've just started playing with:

init:
  iter=0
  z=0
loop:
  iter=iter+1
  rot=cos(iter)+i*sin(iter)
  z=z^power+rot*pixel
endloop

Modifications can be made to include a slider from 0 (regular Mandelbrot calculation) to 1 (full rotation effect) or to alter the amount of rotation each iteration.

[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.

Of course I should have added that although the recursive relations are normally applied to simply give a higher degree version of a given standard formula - such as a Chebyshev or whatever - the key thing is that my formulas for UF allow the recursion to be applied across the iterations rather than to a given degree on each iteration - essentially like z^iter+c but such that "z^iter" is replaced with the rising degree poly form. Applied in this manner the results are distinctively different from "normal" formulas.

[fracmonk]

David-  I PROMISE I'll try it, but I can't say WHEN!  Things are a little hectic for Bunny & me right now- the headlines are catching up to everyone, so we're struggling a bit to maintain life as we know it.  Hope things are more comfortable 4 U.

And...a...later (if there is one).

[fracmonk]

David- if you're still glancing in, there's a question I was meaning to ask you for a long time:  I started this thread partly because I had noticed that it was easier to get a simply connected object over more dimensions, and harder to do the same in less dimensions, generally.  Since you are very accomplished in 3d, I was wondering if over the course of your experience plotting fractals, you have found the same.  Your thoughts?

Also, I downloaded the complete UF formula zip from the site you linked, but it looks like it can only be read thru the UF program.  True?

Thanx, later...

[David Makin]

David- if you're still glancing in, there's a question I was meaning to ask you for a long time:  I started this thread partly because I had noticed that it was easier to get a simply connected object over more dimensoins, and harder to do the same in less dimensions, generally.  Since you are very accomplished in 3d, I was wondering if over the course of your experience plotting fractals, you have found the same.  Your thoughts?

Also, I downloaded the complete UF formula zip from the site you linked, but it looks like it can only be read thru the UF program.  True?

Thanx, later...

AFAIK anything that is disconnected in n dimensions can become connected in n+1 (or higher) dimensions - I don't know a stochastic proof for this but it seems "obvious" enough

The formulas in the zip are just text files in disguise - i.e. load them into any standard text editor and you should be able to read them OK.
For a quick and easy comprehension of how most UF formulas work just look one or two in mmf.ufm first, once you have the general idea then following them shouldn't be a big problem

[dwsel]

Also, I downloaded the complete UF formula zip from the site you linked, but it looks like it can only be read thru the UF program.  True?

Sorry for bumping into your talk.
Which software do you use for generating fractals fracmonk? You can download and unzip this whole formula pack, and in ChaosPro http://chaospro.de/ File->Import->Compiler Formulas and select all or only chosen ones inside the unzipped folder. 70-80% of them should work as expected. It looks that built in compiler has a bit different language from UF.

[David Makin]

Also, I downloaded the complete UF formula zip from the site you linked, but it looks like it can only be read thru the UF program.  True?

Sorry for bumping into your talk.
Which software do you use for generating fractals fracmonk? You can download and unzip this whole formula pack, and in ChaosPro http://chaospro.de/ File->Import->Compiler Formulas and select all or only chosen ones inside the unzipped folder. 70-80% of them should work as expected. It looks that built in compiler has a bit different language from UF.

That's essentially correct, but doesn't cover any of the UF5 class-based formulas (*.ulb) and of the others some functions introduced in UF4 and UF5 are not supported but I think any for UF3 or UF2 will work perfectly in ChaosPro.

[dwsel]

That's essentially correct, but doesn't cover any of the UF5 class-based formulas (*.ulb) and of the others some functions introduced in UF4 and UF5 are not supported but I think any for UF3 or UF2 will work perfectly in ChaosPro.

I was roughly aware of that, but after I've counted how many formulas were updated/uploaded to the database in last few days in *.ulb, then I see that there's missing much more than 30%

[fracmonk]

dwsel-  I'm a really old-fashioned guy, I guess: I use FractInt (and they all moved away from me on the bench...) and write formulae in Notepad (clean & simple).  Maybe other progs may offer more bells & whistles, but I try to keep it simple while I don't really have time to orient myself to their use.

David-  It's reassuring that you also view the greater chances of connectedness as more dimensions are available.  I felt it early on, and tended to observe it in practice later.  I'm comfortable enough to piece together a 3-d view from 2-d slices, but I'm sure some others may find it hard to visualize.  But I wonder if there is some object so hopelessly disconnected in 3d or less that it requires more than 6 or 8 dims to be connected in one piece, although we would not be able to visualize it in any way we're used to...and then, of course, there are String Theorists, (not to mention Theologists,) expert at finding questions that can't be proven one way or the other...I wouldn't know where to begin reading a proof, much less writing one!

'til next time,

Later...

[fracmonk]

Update- "theologists" is a word I coined for that subset of theologians who would condone killing over things they can't really prove, because (usually,) their one true God told them it was o.k.  My opinion is that things should never get so serious. Unfortunately, they do too often.

Otherwise, I've been trying to figure some rules of thumb for what power types to expect in combination for any given formula, but haven't had any time to do justice to it, and won't, I expect, for awhile.

Still otherwise, I've been looking at deforestation and its effects, and the sojurn of scarlet tanagers to our little island of woods after several seasons without them.  They make cardinals look drab in comparison, and cause my bunny and me to smile and be happier...

Little things, little things...later...

[fracmonk]

Not that I'm back exactly, but-

(This time of year, I'm usually building while not travelling to do building, most often with wood, less with stone, but it's all-consuming, & makes me a dull boy...)

Whenever I get a chance, however, I'm still exploring the many wierd new bugs with multiple multibrot-power-types that one can obtain by generalizing the method I've previously described in this thread.  My formula D32g yields an index set (pic 1 below) that contains M32, M8, & M2* power types, simply connected.  The function is f(z)->((((((z^2)c-1)^4)-1)^2)-1)^2**, for those who'd like to look for themselves.  The 2nd pic is of the whole julia set for a location ctrd. in pic 1, where there is a secondary mini of an M2.  The 3rd pic enlarges the center to show a J4 feature there, that ALWAYS appears about the origin in dendritic j-sets for this function, there at the origin, but nowhere else, nor in the index set.  Try to beat me to explaining why it's there...(it's easy, because I don't have a clue, nor any time to think about it)

But I thought I'd try to stay in touch...

Later!


_____________________
*  M2 is the designation for the "standard" M-set shape, as opposed to Multibrots of other powers.
** Oops!  Sorry, folks: I had originally misreported what was actually the formula for D32h (not shown): f(z)->(((((z^2)c-1)^8)-1)^2)-1, in which the index and Julias both contain 32, 4 & 2 power features, also not suggested by the function in any obvious way.  Also interesting to look at, however...

[fracmonk]

I met Irene inland- a bit less wind, as I figured, but the rain was biblical...very messy girl!

[David Makin]

dwsel-  I'm a really old-fashioned guy, I guess: I use FractInt (and they all moved away from me on the bench...) and write formulae in Notepad (clean & simple).  Maybe other progs may offer more bells & whistles, but I try to keep it simple while I don't really have time to orient myself to their use.

David-  It's reassuring that you also view the greater chances of connectedness as more dimensions are available.  I felt it early on, and tended to observe it in practice later.  I'm comfortable enough to piece together a 3-d view from 2-d slices, but I'm sure some others may find it hard to visualize.  But I wonder if there is some object so hopelessly disconnected in 3d or less that it requires more than 6 or 8 dims to be connected in one piece, although we would not be able to visualize it in any way we're used to...and then, of course, there are String Theorists, (not to mention Theologists,) expert at finding questions that can't be proven one way or the other...I wouldn't know where to begin reading a proof, much less writing one!

'til next time,

Later...

Intuitively I'd say anything disconnected in <n dimensions could be connected in n or more - I'm guessing there's a topological proof of this, but I've never studied topology.

[fracmonk]

David-  I've never studied topology (at least in any formal way) either, and at the risk of repeating an old comment to annoyance, I'm given to understand that Dr. M was never able to see the distinction between simple connectedness and local connectedness, either.  To clarify in my own case, however, I simply cannot understand the very CONCEPT of local connectedness itself, so my opinion on that is utterly useless.  The bunches of objects I've given that contain multiple power shapes in this thread are simply connected, but I still have not found the time to do more than list many of them taxonomically, with only the strong suggestion of a structural principle at work behind them, which has yet to be FORMALLY described.  Those who are expert at topology have not yet nailed down a determination of local connectedness for the simpler formula for M, as far as I know, (the big open question,) but I would think that if it were found to be so, it then might not be too hard to extend it to these as well.

[fracmonk]

Update-  I've been looking at multiple power shapes for odd powers again, with some work on combining both odd & even powers.  It seems that when it comes to powers, any combination can be made as long as only one of them is prime.  I'll try to show pix of other traits soon.

Later!