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

[fracmonk]

...so let's start this post over.  I got an error message saying that it exceeded 256K.  I don't know if that meant 1 pic, or all.  So if you got less than intended, don't complain to me.

The first 2 were explained in the last post, and the next two are views of J4 minis, for which the same observations about nonstandard features apply as well.

(After trying this and that, I found it would not take the last pic, a larger view of the center of the 3rd one.  Such is life.  I have decided to try not to fight chaos anymore, if I can help it.  It's a natural force.  Altruism is not altruism, by the way, if one feels forced to do it, whether by coersion, compulsion, or an externally imposed sense of duty.  It must be a free choice.)

There's some really neat stuff to be found in these objects beyond 16-digit floating point.  I feel like I'm always only scratching the surface.  Wouldn't it be great if someone felt charitable enough to allow us to go beyond that limit for "formula" in Fractint?

Later.

[fracmonk]

Here's the pic that was rejected yesterday.

But there's MORE!

Finally, I tried something I used to do routinely in formulae.  Yes, squaring or cubing the constant c also doubles or triples the symmetries in these sets as well.  Great way to get snowflakes, for instance, if you like snowflakes.  If you don't then never mind...

Anyway, the second pic shows a degree 2 with c cubed, and the last 2 are degree 4 variations.

I've been exploring near the origin in the very last one.  Most prisoner points on the real axis are in 4th power shapes, while all (?-looks that way) on the imag. axis are M2.

Later!

[fracmonk]

As suggested in last post, here's some pix from the last index set posted there.

The 1st one has the coordinates of the index set dead ctr.  The second is a rather lacy whole Julia for that spot.  In this one, all the minis are found at narrow spots in the filigree.  The 3rd greatly magnifies z=2, a critical point, where a J4 is found (corresponding to the feature local to the index set coords).  The last shows the "barbell" J2. (This would have been the case, but again, the server/program rejected the pic, which was well within stated criteria.)

These phenomena show up again and again, and are characteristic; the entire behavior, I'm starting to grasp, is that of how lower powers are locally mimicked within higher power behaviors.  It's easier to see than to describe...

Have a good weekend and then some.

Later.

[fracmonk]

Quite a long time ago, I became curious about the theory that all the minis and dendritic structures in the M-set were composed of miniatures of M.  With the minis, this is obvious, but the theory goes on to say that assuming one could zoom in as deeply as one desires, any dendritic structure would be shown to be composed of minis also, just really tiny ones.

If this is so, then what about structures including multiple powers?

Any thoughts on that?

Later.

[fracmonk]

Hey, I've been away from the Internet since last post here, practically, and I bet I'm happier than you!
I don't imagine anyone missed me all that much...if you are interested at all in this thread, please let me know, so that way, I can either pay more attention myself, or forget about it...

Playing with the formula structure for multiply connected forms of the Multipowerbrot type,
I came by an elegant spinoff.  I cut it out of a now enormous formula file that I use, and
it contains my sort-of-shorthanded notes, which help me keep the many permutations, how to
use them, and what they do straight...

M4La(xyaxis) {;deg 4,M2 only MC
  c=pixel, z=p1, d=p2:
    s=z*z-d     ;d=1:z0=-2,2:2sym
    t=(s*s-d)*c ;d=2:z0=-2,0,2:8sym
    z=t+(1/t)
    |z| < p3
}

Notice its brevity.  It yields the 1st 2 pix below.  The last 2 pix are subsequent doublings
of degree, to 8 and 16, respectively, obtained by repeating the squaring and subtraction of
d as in line 3.  Alas, they have only the classic M2 shape in them, and so, they are not in
the Multipowerbrot family.  For degree 2, not shown, the calculating portion would be simply:

    t=(z*z-d)*c
    z=t+(1/t)

For each degree, the number of equally largest M2 figures is double that degree.

Also, I had mentioned before that I always use the same palette without rotation.  It creates
a standard for comparison for everything I show here.  For theoretical consideration, I think
it creates less confusion.  I hope that's not tiring!

Later.   

[Alef]

MM4Lad2.GIF is pretty interesting.

[fracmonk]

Asdam-   Yeah, and what I was trying to point out is what you get from the formula structure, no matter what the degree doubling.  Nice, symmetrical rings...

Looking at Im d in some of the formula structures, where it works.  You can get these odd blobs, with Julia-like shapes, assymmetrical, linked with little M2's, but also apparently connected (when you're lucky!)

Maybe I can get some pix together later.

And

Later.

[Alef]

Now all folks went to colour method based formulas;)

[fracmonk]

Asdam-    Your last message implies what exactly?  Is it that a consequence is horrendously (and unnecessarily) complicated coding that's no fun at all?  Just guessing...

Here's the fractint formula for the degree 4 version of what I spoke of in my last post.  If d=i or -i and your init z=2 or -2, you will get an index set object like pic 1 below.

M4L {;deg 4
  c=pixel, z=p1, d=p2:
    s=z*z-d   ;d=-i,i:z0=-2,2,more?
    t=s*s*c-d ;M2+J-like features,MC
    z=t+(1/t) ;d=1:z0=-2,-1,0,1,2 NC
    |z| < p3  ;highly distorted
}

By repeating the 3rd line (1st line of calculation), that squares and subtracts d, the degree is doubled, resulting in the 2nd and 3rd pix, degree 8 and 16, respectively.  Notice how an off-origin julia-like symmetry begins to emerge in higher degree.

Leaving that line out altogether gives a degree 2 version, which actually then can have y-axis symmetry, as shown in the last pic.

This formula structure variation will yield another totally different shape that is most likely *not* connected (NC), with very distorted sausage-like lobes and broken minis, when d=1 is used.  See formula note above.  The effect becomes deceptively more regular in higher degrees.  I include that note for those who like such Frankenstein's monsters that require minimal effort.  (Generate & post yourself if you like...)





In the U.S., Independence Day is celebrated tomorrow.  Enjoy!

Translation from modern Orwellian:

DO NOT USE THOSE ILLEGAL FIREWORKS ON DEPENDENCE DAY.  Turn them in, along with your neighbors and family members, with pictures taken with your cell phone.  Use an online substitute fireworks show instead.  Don't drink and drive.  Have another barbequed soyburger.  They're healthier for our profit margins.  Take a pill (prescription only, it's O.K. then).
Obey all rules of the kinder, friendlier international totalitarian police state. 
Become whatever your TV is showing you.  Never ask why we show or tell you anything.  If you don't think there are already too many laws, imagine new ones, and obey those, just in case. 
Only with your outstanding cooperation can we watch you so closely.  Buy our latest iSuck product.
Things are getting better all the time.  We won't eventually murder you.  We promise.
 :)Have a nice day.


Later.

[weavers]

_________________________________________________________________________________________________________________


Diagnosis Criteria : Compositionality : regarding Orwellian eye to brain to subject text civilization satire perfect : Sound of your verbal articulated voices are more than good, excellent choice of wordings, clear and so in tuned with the prevailing farceties : tok = Theory of knowledge mood achieved excellent : verbal descriptions constructs : Un reserved : Un-restrained achieve goal of telling it as it is! : Dynamic style coming from the hippy dual perspectives juxtaposed realistically : professional seconds into [ text at  police state ] Excellent pause as we enjoy reading what you have to say poignantly to they max : the truth hurts, your word are biting truthful : Thanks do more!

[Alef]

I meant, UF colour methods have hard time with some of these formulas;) Multiwave coloring for Mandelbrot by Pauldelbrot works well, but some orbit trap based methods not so well.

[fracmonk]

Asdam-  I suspected something like that-  -it's one reason why I stick to fractint.  I can communicate the basics as easily as I can think up a formula and test it.  No muss, no fuss.  Gets ideas out quick & easy.

weavers-  My first thought was: "Uh huh..."  -a critique on delivery?  How about where YOU stand?  You know, on the subject matter...



and hey, how come no one ever tries MY questions on for size?  No seriously, check the history...

Later.

[fracmonk]

I had a nice, moderately-sized post, with a formula, some small pix, and good explanation of it all, went to post it, and was told the upload box was full.  I left out the pix then, and tried to post it, and then was told I already did.  If I did, then it should be there.  Do YOU see it?  Didn't think so.  Maybe next time...

I've been looking at AI and the techno singularity lately.  There's nothing to worry about, apparently...

Later.

[fracmonk]

Just kidding-  human deception, corruption, arrogance, and incompetence in
the name of control, and the pathological competitiveness it breeds and
excuses, will create a technological singularity with a character that
will make the Unabomber look like a saint, in retrospect...there's a bunch
to worry about!

What follows here is what I tried to put up last time, with a little
updating:


Here's an adjustable (by d) function I've been looking at since last week
that exhibits a *very strong tendency* toward connectedness in the c
parameter plane.  Only the M2 type (standard M shape) is yielded by it,
everywhere in it, in the examples below.  Adding the multiplicative inverse
in this (any many other) function(s) makes the connectedness (when
available, as here) multiple.  I've wanted to further explore that
(previously mentioned) idea of *cohesiveness* in certain functions not
found in others.  I have this feeling that that distinction will one day
become very important...

M4F(xyaxis) {;deg 4 for Re d only!
  c=pixel, z=p1, d=p2:;M2 features,MC
    s=z+(1/z)   ;d=1:z0=-1,1:2sym
    t=s*s-d     ;d=2:z0=-1,1,-i,i:4sym
    z=t*t*c     ;d=-2:z0=-1,1:2sym
    |z| < p3    ;d=-1:z0=-1,1:2sym
}               ;d=0:z0=-1,1:2sym

M4Fi(origin) {;deg 4 for Im + some cmplx d
  c=pixel, z=p1, d=p2:;only; M2 features,MC
    s=z+(1/z)   ;d=-i,i:z0=-1,1
    t=s*s-d     ;d=-2i,2i:z0=-1,1
    z=t*t*c     ;d=-3i,3i:z0=-1,1
    |z| < p3    ;d=1+i,2+2i,etc:z0=-1,1
}

;The only difference between these is the timesaver for symmetry type.
;The second version will work for all d listed in the comments in both. 
;Shown below as examples are maps of sets for Re d for now, as labeled. 
;(A negative d value subtracted in the code is actually positive then.) 
;Others next time.

;For Julia sets, the following *should* work in all cases, but to be sure,
;remove the (origin) spec:

M4FJ(origin) {;deg 4
  c=p1, z=pixel, d=p2:;M2 features,MC
    s=z+(1/z)   ;d=1:z0=1:2sym
    t=s*s-d     ;d=2:z0=-1,1,-i,i:4sym
    z=t*t*c     ;d=-2:z0=-1,1:2sym
    |z| < p3    ;d=-1:z0=-1,1:2sym
}               ;d=0:z0=-1,1:2sym (etc)

Notice that all examples offered so far involve whole numbers, and no
values exceed 2.  With d values >2, and fractional d values, index sets
like those we've seen before here in this thread, that have fractured and
distorted M-like fragments and localized julia-like features appear. 
Often, as in other functions shown here before, they appear as if they
might be connected, but one would have to look too deep trying to prove
instances that are obviously not connected.  In other instances, lack of
connectedness on the specified 2d plane is obvious.

In my experience, it is easier to find connectedness in an object shown in
more dimensions rather than less.  There is no question that a 4d index
set of this function for the c by d complex parameters would be connected,
but probably not finite.  I have not yet looked far enough into that
property.  Fractional d values that are real (and pure Im as well) should
yield a connected 3d object that is also probably not finite, from what I
have seen.  In that, after stacking parallel 2d slices at tiny incremental
distances from each other, observation could reasonably suggest continuity.
In other words, various combinations of c and d complex parameters could be
connected in 3d, but would require that many dimensions to be so.



Judging by recent comments, I would renew my call for a version of fractint,
or another workalike program with its basic map design concepts, that can
display true 3d in the formula parser without resorting to trig. 
Ridiculously fast processor speeds now take away the (what? 20-year-old)
argument that it would take too long to calculate.  Currently, it can
*calculate* in as many dimensions as you can keep straight mathematically,
but can only display 2 of these dimensions at once.  An assignable 3d pixel,
along with a simple scheme to color-code proximity in a depth dimension,
would really be a great thing...

Ambitious showoff programmers Wanted.  Inquire Within.

Later. 

[fracmonk]

I don't know if it's just me or the equipment I've been using, but I have fidelity issues with the way the pix appear here.  Rather than a faithful pixel-by-pixel display, they look like they are being "interpreted", as by the (really crummy) schemes that MS uses in the picture display programs they incorporate in their recent OS versions.  Some spots look smeary or out-of-focus, not sharp.  I truly miss MS Photo Editor, which once came with XP.

Is there a reason for that?  A fix?

Comments from others?

Later.