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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 42531 times)
Description: I think there may be. Your opinions are greatly desired.
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Ryan D
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« Reply #285 on: September 06, 2012, 01:34:30 PM »

Quote
That must have taken awhile to put together, no?

3000 frames, 2560x1440 resolution, render time was a bit over 20 hours.  I started with a low resolution animation (probably 256x192, that's what I usually use) that covered a longer trip along the Lissajous curve, found a patch that looked interesting and made a slow traverse along that portion of the path.

Of course, the time actually spent working on it is far less than 20 hours.  I've written a simple program to generate the batch file needed for the animation, and that's just a matter of some programming details to get the path where I want it.  Pick the number of frames, hit "Enter" and it spits out a massive batch file to make successive calls to Fractint.  It's not very romantic, it produces stuff that looks like this (3000 lines worth of it) -

Code:
...
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9653816388332706/0/1.003946543143455/0/-.8509944817947031/0/-1.003946543143455/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9659808786651696/0/1.00431701703897/0/-.8490195330148527/0/-1.00431701703897/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9665749764811259/0/1.004704089721826/0/-.847032650153877/0/-1.004704089721826/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9671639291186943/0/1.005107757972606/0/-.8450338611400208/0/-1.005107757972606/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9677477334428196/0/1.00552801843386/0/-.8430231940688784/0/-1.00552801843386/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9683263863458512/0/1.005964867610138/0/-.8410006772030078/0/-1.005964867610138/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9688998847475606/0/1.006418301868013/0/-.8389663389715289/0/-1.006418301868013/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9694682255951553/0/1.006888317436113/0/-.8369202079697344/0/-1.006888317436113/0/1/0
...

I don't keep the individual frames any longer than is necessary to render the final video, but while they're still on the hard drive, it's a good opportunity to check out some fine details.

Quote
I'm especially curious if a:
one of your dimensions was exponential (gamma?), and continuous thru fractional values,

The batch file snip above shows the parameter structure I used, as follows:
alpha real / complex
beta real / complex
gamma real / complex
delta real / complex
z0 real / complex

The rest of the batch file is the same, all parameters use real values only, and I have always used z0=1.  (There is some investigating to be done with this, using element90's method of calculating critical points and substituting that in to z0.)  This applies to element90's formula, z = c(alpha*z^beta + gamma*z^delta).

As mentioned, I'm using a 3D Lissajous curve to determine the parameters.  That is just a trace of 3 concurrent sine waves, one on each axis (or parameter, in my case).  As a sine wave, there are no discontinuities in the parameters, in fact the general structure is entirely repetitive.

http://demonstrations.wolfram.com/3DLissajousFigures/

Agreed, however, there are some definite discontinuities in the video.  The first one around 10 seconds in occurs when gamma crosses from the negative into the positive.

Code:
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9354191538966699/0/1.671626637850489/0/-1.849140382124068E-03/0/-1.671626637850489/0/1/0
FRACTINT @RYANANIM.PAR/e90monk BATCH=YES PARAMS=.9346009809914438/0/1.675944094356335/0/1.90003409364121E-03/0/-1.675944094356335/0/1/0

There are other surprising views, such as that around 1:00 in where 3 minibrots turn into 4 not by a split of 1 into 2 but rather by a merger of 2 into 1 while at the same time, 2 others appear out of nothing.  As I implied earlier, this probably isn't much for those into mathematical rigour but I like it nonetheless.

Ryan
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fracmonk
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« Reply #286 on: September 06, 2012, 05:09:50 PM »

Ryan D-

I gave myself one of those cringy feelings upon realizing that the exponents were beta and delta...sorry!
For that belief that I must crawl before I can walk, I've been avoiding situations where planes get bent, much less folded, spindled or mutilated...I need to keep things humiliatingly simple for myself!

In parallel, I was looking at the "time" dimension (the progression of the frames) as one assignment you used, but I'm still not clear on which it was, unless you decided to reach a point in the progression, make a previous variable fixed, and begin changing another.  You can never get done going down all available avenues in a multidimensional animal such as this.

It "looks" as if z0=1 remains critical most of the time, based on the M-shapes retaining their form.  If so, then the coefficients may not be continuous, either.  Are YOU sure of it?  Again, just curious, and I do not have a complete understanding of the journey, mind you...

Both in thinking of the how to, and the implementation, not to mention ridiculously long render times, I have spent longer on simpler things!  Labors of love these are.

Before getting into anything so complicated, I'm trying to nail down some rules or expected behaviors for much simpler things, and as usual, wish I had more time to devote to it.  I feel like Salieri to your Mozart, if you saw "Amadeus", but I harbor no jealousy as such.  Kudos on an interesting journey, and again, it's not clear to me the path it took.  Which reminds me of a favorite bumper sticker: Where are we going, and why am I in this handbasket?

Have you seen it?  Not that it applies here, but it came to mind anyway.  I'm a sucker for a good laugh, off-topic as it may be every time...

Later.
« Last Edit: September 06, 2012, 05:12:43 PM by fracmonk » Logged
Ryan D
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« Reply #287 on: September 08, 2012, 11:44:00 AM »

Quote
It "looks" as if z0=1 remains critical most of the time, based on the M-shapes retaining their form.  If so, then the coefficients may not be continuous, either.  Are YOU sure of it?  Again, just curious, and I do not have a complete understanding of the journey, mind you..

The parameters in my path simply follow sine waves.  Alpha and gamma are unmodified sine functions ranging from -1 to +1, beta is a sine function scaled and offset to range from +1 to +5, delta is the negative of beta.  Yes, the sine function is continuous.  That's intuitive to me, and as with many things, the proof causes me more confusion than just accepting what I already "know".

http://www.math10.com/en/algebra/functions/continuity-sine-cosine-function/continuity-sin-cos-function.html

Ryan
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fracmonk
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« Reply #288 on: September 11, 2012, 05:42:09 PM »

"I see!"  said the blind man as he picked up his hammer and saw.

The continuity I was looking for was in the object, in however many dimensions it has.  In just 2, it doesn't happen in every parallel plane, and probably never will, no matter what the function.  (Someone will prove me wrong about that, but please make it nontrivial, O.K.?)

Later.
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fracmonk
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« Reply #289 on: October 09, 2012, 05:05:51 PM »


Anyone up for a really good puzzle?

I have been very busy with OTHER THINGS (that...just won't let you be...), but I still find a
little time to knock this stuff around a bit, just barely.

I've been looking lately at functions that have no single numeric value as a critical point,
but ones with critical points that are algebraic expressions.  f(z)->(z+c)(z+(1/c)) is a
(relatively) simple example, as is one earlier in this thread in (my) posts in 88-97. 
z0=-(c+(1/c))/2 in THIS case, and already I was hard-pressed to figure that out, since I'm
not all that skilled at it.  I had originally looked at its Julia sets to determine the real limits
of the double M figure to the left in the first pic below, without realizing THEN that the 2
separate M figures to the right existed.  For each prisoner point c in this index set, there
seems to be a single connected Julia set, although the index set is not in one piece itself. 
The Julia sets resemble those of standard M without exception, although they are not
centered at the origin.  I would posit that the fact that the components of the index set
are obviously countable (4 M perimeters in 3 separate figures) is what makes this possible,
as opposed to being surrounded by infinite numbers of satellite islands.  That would then
mean that index sets need not be in one piece for this to be so, but the number of its
whole components on the parameter plane then still must be finite.  Then too, the argument
can be made that c and 1/c in each case of c is still a (split) constant, but a constant
single number when added up, all the same.  Here is an efficient fractint formula for the
function:

Mcinvc(xaxis) {;deg 2 index set
  c=pixel, d=1/c, z=-((c+d)/2):
    r=z+c       ;3 M2 figs. with       
    s=z+d       ;4 perimeters
    z=r*s       ;total
    |z| < p3
}

McinvcJ {;deg 2 Julia sets
  c=p1, d=1/c, z=pixel:
    r=z+c       ;3 M2 figs. with       
    s=z+d       ;4 perimeters
    z=r*s       ;total
    |z| < p3
}
 
Accordingly, I have found another function whose Julia sets are intriguing, and suggest the
existence of some similarly algebraically-determined critical points by virtue of the fact
that I have found one-piece Julia sets from it, f(z)->((2/(z^2)-z)c)-d.  This one may or may
not have an index set in more than one piece, I'm guessing it's in one piece, but I'm stumped
as to what expression involving c and d should be assigned as init z0 to obtain the index set.
Giving credit where it's due, this was inspired by the functional structures that element90
brought here recently.  Everything effects everything else. 
The Julia set formula is:

DMDC221J {;cp not found
  c=p1, d=p2, z=pixel:;quasi M2?
    s=z*z           ;d=1:z0=?
    t=((2/s)-z)*c-d
    z=t*t           
    |z| < p3
}

Pic 2 is of a one-piece Julia set using this formula, that appears to correspond to what would
be the approximate real right continental limit of the index set for d=1, c=.33881.  The others
show details of that pic in which it is easy to see that there are an infinite number of
bridges having zero width crossing the real axis.  At least one of them is critical...yeah, but
which one?

Anyway, can anyone please show me what I'm missing here (besides the index set)?

Later!



 


* MMcinvc.GIF (21.77 KB, 640x480 - viewed 76 times.)

* ContlimJ.GIF (57.31 KB, 640x480 - viewed 72 times.)

* FRACT006.GIF (84.26 KB, 640x480 - viewed 76 times.)

* FRACT005.GIF (84.23 KB, 640x480 - viewed 73 times.)
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fracmonk
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Posts: 356


« Reply #290 on: October 23, 2012, 05:37:05 PM »


The function f(z)->((z^2)/c)-(c/(z^2)) appears to create an infinite set (detail pic 1), but many of (if not ALL) its
(specifiable) Julia sets are finite sized.  The problem with the index set is not in zooming IN, but zooming OUT, when the limitations of a given generator determine how much of the set you can zoom OUT to.  The 2nd pic is
of fractint's limit that way, and the 3rd pic is an inversion, showing how the same index set looks on
the 1/c plane.  Limits again show themselves when zooming in to the origin of the inversion, as in the
last pic, where limits to the bailout value blank that local region.  You won't in any event get to see
all of this set, but there's still a lot of it you can.  Try it:

Zdivcinv(xyaxis) {;deg 2
  c=pixel, d=1/c, z=p1:;z0=2
    r=z*z     ;infinite set?
    s=r*d     ;mostly foamy
    t=c/r
    z=s-t
    |z| < p3
}

ZdivcinvJ {;deg 2
  c=p1, d=1/c, z=pixel:
    r=z*z       
    s=r*d
    t=c/r
    z=s-t
    |z| < p3
}

I picked a spot that made an interesting Julia set, and did a zoom to it, and then of the critical point
z=2 in the Julia, in many formats using FFW, trying to determine correct x mag factors to yield
correctly proportioned pix, including monstrous, time-consuming 2048 sq. pix (xmag=1.3333...) that would
never ever fit here.  When I don't have time to give the work attention, the machine still grinds
away...like inflation grinds away at buying power when they make so much fake money.  That's off-topic,
of course, but always in my thoughts as time marches on and entropy unfolds as it does.

I might add that distortions in the proportions of M shapes are often a matter of what the formula will
correctly yield, as in this case and the set shown in post 261, differing from MALFORMATIONS of the M
shapes, as can be found in the last 3 pix in the set in post 210, indicating an incorrect (in that case UNFOUND) critical value.

Later.


* Smn2.GIF (66.4 KB, 320x200 - viewed 348 times.)

* Slimmn5x4.GIF (79.55 KB, 320x200 - viewed 345 times.)

* Sinvplw.GIF (22.44 KB, 320x200 - viewed 346 times.)

* Sm5.GIF (76.26 KB, 320x200 - viewed 341 times.)
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fracmonk
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« Reply #291 on: October 24, 2012, 08:52:15 PM »

Note:
Please don't mind me if I get snarky sometimes.  I've been long in the dark heart of the Land of Disenchantment with the Arrow of Time poking out my lower back, and yet, it's work or wither, and winter's coming...

Later.
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Ryan D
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« Reply #292 on: October 25, 2012, 02:47:55 PM »

Quote
Zdivcinv(xyaxis) {;deg 2
  c=pixel, d=1/c, z=p1:;z0=2
    r=z*z     ;infinite set?
    s=r*d     ;mostly foamy
    t=c/r
    z=s-t
    |z| < p3
}

ZdivcinvJ {;deg 2
  c=p1, d=1/c, z=pixel:
    r=z*z       
    s=r*d
    t=c/r
    z=s-t
    |z| < p3
}

A question - why does your M-set equivalent above equate z0 to a fixed value (p1) rather than the pixel value?  Obviously the J-set must use the pixel value, otherwise you'll just get a single colour over the entire screen since every pixel would start at the same value and follow the same escape path.  But if you're trying to create an M-set that serves as an index to the J-sets, it seems to me you'd need to have an equivalent initial value for z.

I implement it as follows - I use the ismand variable to allow toggling with the space bar, and through force of habit I always force a non-zero default bailout value.  As a direct toggle, the M-set now has z0=pixel, the same as the J-set.  I've also generalized it by adding p2, allowing for exploration of varying power terms of z.  Your hard-coded value is p2=2.  P2=sqrt(2)=1.4142134 is interesting.  (Virtually all the formulas I fiddle with will be generalized in some way so I can vary parameters to create animations.)  Symmetry was removed because this formula now looks at both the symmetrical M-set and the non-symmetrical J-sets - if I were to create an M-set animation, the formula would be rewritten to remove the ismand toggle and the appropriate explicit symmetry would be restored.

rdfmonkZdivcinv {;original hardcoded with r=z*z, z0=p1(suggested=2)
  if (ismand)
    c = pixel
  else
    c = p1
  endif
  d=1/c, z=pixel:
    r=z^p2     ;infinite set?
    s=r*d     ;mostly foamy
    t=c/r
    z=s-t
    |z| < 4 + p3
}

Following are a couple of M-set images for p2=sqrt(2), one with a considerably zoomed-out view (magnification about 0.05) and one of a zoomed-in mini-blob.  There are also two J-set images, one located at the Elephant Valley equivalent to the zoomed-in blob and one along the chain that disappears off the edge of the zoomed-out image.  Fun stuff, I need to take some time and go through a bunch of the previous formulas to see if there are worthwhile animation candidates.

Ryan


* frac0031.gif (24.67 KB, 400x300 - viewed 325 times.)

* frac0032.gif (46.93 KB, 400x300 - viewed 327 times.)

* frac0033.gif (50.37 KB, 400x300 - viewed 360 times.)

* frac0034.gif (13.59 KB, 400x300 - viewed 325 times.)
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fracmonk
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Posts: 356


« Reply #293 on: October 25, 2012, 10:54:59 PM »

Ryan D-

I used z0=2 in the index set because 2 seems to be critical in the connected Julias. 
From the discontinuites and separate islands in your pix, I can tell you're not starting with the critical point for whatever other values you may be assigning.  I see these kinds of qualities in my own initial tests often enough, and then know I don't have the right c.p.  Not that you can't have interesting fun otherwise, but I'm kinda obsessed with connected sets, and why the critical points are critical, and sometimes I THINK I know all the patterns that produce the results I get, and find there's usually something new, some exception that keeps me looking...

It would be great, for instance, if you had a reliable formula for adjusting the init z correctly for every change of, say, the exponent of z, that may in fact be pixel-dependent in some cases of fractional exponent.  I don't know about that myself.

I'm just trying to digest the "simple"...it's complicated enough for me...and besides, I think of integers as the most basic constants, anyway, followed closely by rationals.  By that thinking, 2 is no less magical than its (trancendental?) square root, but every number has its unique identity, plain or ambitious...

Later.
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fracmonk
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Posts: 356


« Reply #294 on: November 01, 2012, 04:18:04 PM »

Hey!

We were lucky enough to be in a decent spot in NYC for the storm.  We had little rain and very high winds, but most everyone we know did pretty poorly in the tidal surge.  Most we know are w.o. power & phone, one burned out @ Breezy Point, others flooded catastrophically in Long Beach.  Been busy...

Machines doing long pix...

Later.
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fracmonk
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« Reply #295 on: November 08, 2012, 05:27:51 PM »

Also off-topic for here, but still fractal...

The second storm would have been almost inconsiderable, if not for the damage from the the first, not unlike rubbing salt in a wound.  No wound, salt's not a big deal.  Saw a fuel tanker stuck in a line for gas today...

Snowcover, not well tracked, reflects sunlight and heat away from earth, so cold gets colder.  On the other hand, lack of reflection makes warm warmer.  Logistical equation strikes again?  (That is, due to sensitivity to initial conditions, by the law of averages, on average, there's a wide departure from the average.  Let's call it the Law of Average Irony.  Any subscribers?)

Later.



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Alef
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« Reply #296 on: December 10, 2012, 05:37:33 PM »

Here i have buddhabrot render of one of your early formulas: http://www.fractalforums.com/fractal-programs/problems-with-implementing-budhabrot-in-uf/msg55319/
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fracmonk
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« Reply #297 on: December 18, 2012, 08:27:44 PM »

Alef- Not me, anyway.  I don't recall ever having had anything to do with that.

Sorry, I've been either too busy or not well or both.(Like Now)
Otherwise, I'd be paying more attn.
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fracmonk
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Posts: 356


« Reply #298 on: January 17, 2013, 05:34:14 PM »

Started working on an illustrated summary of Multipowerbrot funcs and how they work.  I will draft it in Word, but eventually, maybe with Bunny's help, put it in html form.  Many pix in it will probably be same as those seen here, but either way, it won't fit in FF's size constraints, except the text portion, which I will be happy to post once finished.  Some of that should, for most, be easy to follow, making reference to things most fractalists know.  Others, like zoom tours, will need the pix.  Meantime, the thread here remains a a good resource for those who would like to study them in depth.
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fracmonk
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Posts: 356


« Reply #299 on: January 31, 2013, 05:39:34 PM »

Pretty sure that ALL points in a Fatou dust (for f(z)=z^2+c) escape eventually, but has anyone done a proof for it?  They say if z0=0 escapes, it's not connected, but then, there's nothing to connect, right?

If anyone can shine a light on that, I'd greatly appreciate it.

Later.
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