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

[fracmonk]

Here are more pix from that adjustable function.  The first is for d=-2,
significantly different from d=-1 in the first report.

The next 2 are for pure imaginary values for d: d=i, d=2i.

I find the last most interesting, since it has a 4sym, but turned 45 degrees.
That's d=2+2i, and worth comparison with d=2 in the last report.  Its
traceries have a certain grace to them, and the entire figure seems to be
made exclusively of the M2 form. 

It can be argued that for each d value, you are really dealing with a
different function.  I was concerned here with the similarity of the
structure of the formula from one to the next, and how there can be both
profound similarities and profound differences simultaneously when comparing
the figures that result.

What are the numbers trying to tell us?

Bear in mind that we are dealing with on-off, ones-and-zeroes, dichotometric
Boolean math at a more fundamental level here.  That most of the functions in
this thread, where the most interesting things seem to happen, are dependent
on zero, one, two, and powers of two seems poignant to me.

Dream big, dream small, but dream!
 

Later.

[fracmonk]

Spent more time driving than sleeping in last 3 days.  Saw Furthur @ Bethel Woods.  Almost Dead.

Later.

[Apophyster]

Spent more time driving than sleeping in last 3 days.  Saw Furthur @ Bethel Woods.  Almost Dead.

Later.

hmmm... not much time to wonder about tapestries behind bands, but... Dead Heads on the Bus?
whew!!

[fracmonk]

Apophyster-

Sadly, the stage background had little to do with fractals.  Sorry, it's off-topic, them voices tell me so...

Don't worry 'bout me...

[Apophyster]

My apology that I go OT.  You and so many others are far advanced in math beyond me.
So I gawk at all of your experiments, yet oh... I'm such a numb skull.  :

[fracmonk]

Apophyster-  [Nah, nah,] you misunderstood!  It was ME going off topic, and making a little fun of myself at the same time.  The stuff I do here is simpler than you think, [when you look at it right].  Anything I can make clearer somehow, just let me know.  Does that skull have one eye, two, or three?

(OK, enough references to Dead culture and lyrics...I get carried away sometimes...)

Later.

[Apophyster]

Two and a bolt.  Been a while since I checked the bolt for new eyes.
There are continual new representations.. 
Personally, IMHO you may feel free to carry sometimes on, anytime on then.  I'm on Sugar Magnolia just now. 
Or just ending.
Since '67 iym dead to the bone, through the breast, and all 'round m' heart!
  Friendly Fredley

[fracmonk]

A-

Then you understand, and got it all.  See?  No numb skull after all.

Funny, I considered a defense of GD, if one was needed, but left it out of here days ago.  
They have origins in CIA LSD experiments, but look at this:
You have to make your way in a violent, desperate, and often merciless society, and you come to realize deep down in your heart that you're not gonna change it much in any meaningful way.  Information control limits influence.  What do you do?
As poets, they do music.  Another kind of poet does math.

You keep your fingers crossed that it won't wind up hurting anyone.

Or you make movies that make the rest of us more careful about what we wish for.

Later.

[Apophyster]

As far as the CIA category is concerned, it helped get Larry Ellison and Oracle going.
So however the spy guys interjected themselves into the band and other facilities and people in the BA and elsewhere, may not be so unsettling as what they're doing in the IT field.

I read very little about that type of thing though.  People may have dug up more info since I've read aything about the subject.

If there's another time around I hope the band and all the asscoiated troupes will be there to collectively turn on the love light again.

[fracmonk]

A-

Exactly.  Here came Oracle, there went your freedom.  Everything needs a critical eye.  Even the GD.  I have no major complaints, though.  They've gotten me through a lot, even if I came out of it a little crispy.

As far as turning on the love light, that rests with each of us, and it's in what we do.

ALL of it together, not just this or that part-time cause.  It's a lifetime of research to figure out how to avoid working for the bad guys and also avoid winding up...well...dead!



Sorry, no pix at the present time.  Working on variations on the formula that gave squaring-effects-without-squaring, seen very early in this thread, the first bunch of pix, in fact.  Designated it M1La, the variation that gives me all the current puzzlement.  I should add that they are remarkably slow, or they reveal nothing.

Later.

[fracmonk]

The formula variation for squaring effects without squaring I
mentioned last time:

M1La(xaxis) {;deg 1, Re d M2 features
  c=pixel, z=p1, d=p2:;z0=-2,2 seem
    t=(z-d)*c   ;to be critical
    z=t+(1/t)   ;d=2:z0=-2 notable
    |z| < p3
}

The very first pic I posted in page 1 of this thread can be
duplicated with this variation of its original formula by using
init z=2 with d=0, and gives a double M-set inside the unit
circle, arguably a lambda form, mapped there.

The first pic is yielded by d=2 when init z=-2.  I have not seen
a SINGLE M-set shape packed inside the unit circle like this
before.  In both cases, the main bulb edge maps to the unit
circle, the first one twice, and this one once.



YOU DO NOT SEE THE PIX BELOW, since they were rejected because of size, all having lesser storage requirements than the one you do see.  Rules are rules...nothing over 256K(?) allowed, but they weren't, and the site said they were.  Who am I to argue?  I'm only human...

...and I have better things to do than to drive miles to a computer to have this awesome dude bot treat me this way!  I'm thinking about having nothing to do with it any more.  (That may make you HAPPY!)

Different values for d cause an emerging M-set shape outside the
unit circle.  This one (pic 2) for z=2, d=3 is pretty elegant.

Another example is seen in the next pic, z=2,d=2.5, in which
there is complete separation, but there are still residual
dendrites that are not a part of it outside the unit circle.

If you study the formula, there are at least 2 combinations of
init z and d values that will result in division by zero: z=2,
d=2, and z=-2,d=-2.  The closer you get to d=2, the further away
the exterior M-set gets away from the unit circle.  The last
pic, z=2, d=2.05, shows the emergent M well distant from the
unit circle using a d value arbitrarily close to 2, which makes
one rethink pic 1 as a whole set, which may have its "nemesis"
at infinity.

(...but there's More!)

Later.

[Alef]

Why don't you share Fractint formula file.

[fracmonk]

Asdam-  Nice to see you here again.  It's right there in my last post, at top.  Don't know how you could miss it.  It's for fractint formula.



This is my Olympic Semiathlon event where half is done on one day,
the other half now, and I care half as much and try half as hard.
Since it is not an official Olympic category yet, it is outside
the jurisdiction of the IOC.  The IOC should know I am not trying
in any way to exploit or damage their "brand", which they might
want to burn into EVERYONE's backside, apparently, without
exception.  I am not criticizing them, which is also probably
illegal by now.  In fact, I want nothing to do with them, and
let's see how that works out.

Here are (with any luck) the pix that didn't stick last time, 2,
3, and 4.  For explanation, see my last post notes (after the
complaint).

And again, there's more, but it will have to wait until next time.

Later.

[Alef]

I was thinking about formula file in attachment. Most of Fractint formula files are pretty simple text files with lots of formulas.

[element90]

This thread is inspiring, there follows a picture using



For version 3.0 (eventually ...) the formula will be altered to

z = alpha(beta*z^gamma + delta*z^epsilon)

when any of the complex parameters alpha to epsilon can be substituted with c i.e. the location in the complex plane.



This is from the blunt end of one of the Mandelbrots in the fractal generated using that formula. The "chains" and "beads" attached to the more normal Mandelbrot structures are of course missing from the standard Mandelbrot in a similar relative location.

Keep up the good work it is appreciated.