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

[Apophyster]

Off Topic: now I'm...old...
...and maybe in the way...
(For Fred)

Not to hijack the this very constructive thread heheh, but, no one's an obstruction (in the way). The rest of the herd is just in too much of a hurry!

More on topic, I've been picking around in Fractal Explorer trying to use some of the basic formulas from this discussion (as best as I can understand them).
I don't get the critical point stuff really.  (Fractal math flunkie.)  But I've been able to produce the ringed sets of mandelbrots.
Kinda darn cool really.  Lots of fun to zoom on. 
I'll try to post some images if I can figure out how to do that here.
If not I'll try to put some up on DA.  For many of the images I zoomed to the limit of FE, about e-15.
I'm not able to get close to any of the beauties I see around FF.  But in FE I see myriads of little mini brot spots all over the place.  I've never witnessed that kind of thing before. 
I will include parameters when I get some of these uploaded.
Fred

[element90]

fracmonk - I discovered that for one particular form of 'my' formula (that you've picked and run with) is sensitive to its bailout value. The form is where one power is set to zero and the other power is negative.

So for

z = c(z + 1/z)

the following are produced by changing the bailout value:

norm(z) > 2



norm(z) > 4



norm(z) > 8



norm(z) > 16



The number of the main buds or bulbs is determined by the sum of the powers, ignoring signs. When there are only two buds they merge into each over as seen above. As the number of buds increases the bailout value can reduced such that the buds are not cut off.

z = c(9z + 1/(z^9))

the result has ten buds and a bailout of norm(z) > 2 doesn't truncate any buds or Mandelbrots.



I've been playing with the variation of the formula you mentioned in your last post. The formula isn't available in Saturn and Titan and there is no mechanism to add custom formulae, so I'm using Gnofract4D to try out this variation with extra parameters.

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

What I've found is that if I adjust alpha and beta so that the critical value is 1 I get completely different pictures to you. If alpha and delta are set to 1 I get the same results as you. With the earlier formula I could use any values for alpha and gamma work out the appropriate critical value and the resulting picture was the same in all but size and location. If I try the same thing with the formula above I get different pictures for each critical value. A good check to see whether the critical value is correct is the presence of well formed Mandelbrots and each picture does indeed contain well formed Mandelbrots.

No pictures using the above formula yet, Gnofract4D has some bugs such that it doesn't correctly generate high resolution images, it has other annoyances like automatically changing the number of iterations. I'll post some pictures once I they can can be correctly or nearly correctly generated. I've just installed the newest version to see whether that's any better.

[element90]

The following pictures are produced using Gnofract4d.

z = (c(2z - z^2) + 1)^2

z0 = 1 and c is the location in the complex plane.



z = (c(z - z^2) + 1)^2

z0 = 0.5



z = (c(z - z^2) + 1)^3

z0 = 0.5



z = (c(z - z^2) + 1)^4

z0 = 0.5



z = (c(z^3 - z^4) - 1)^5

z0 = 0.75



This shows the limitation of Gnofract4D there are a least three areas where calculation wasn't completed, at the centre of the fan and in the spirals above and below the main body. For some of the images Gnofract4d also refused to increase the number of iterations above 256.

z = (c(2z - z^2) - 1)^3

z0 = 1

[fracmonk]

element90-  
Go really large on your bailout value if you have any control over it, and you'll get better results.  Look more closely at the edge of the unit circle in the last pic in post 271. you see the buds get notched there too.  They're just smaller bulbs.  I get the same problems myself sometimes, no matter what.

You last post suggests all this stuff is getting knit together, and entering its baroque period.  I'm not sure, but is your prog. picking out critical points for you?  And I didn't know that some of this stuff has has been worked into type lists.  I guess that's good, but the puzzles of working out the formulae and critical points yourself are more rewarding.  Do you agree?

As promised, I wanted to talk about the loop dendrite
emerging from the seahorse valley regions of objects
like those in post 264, actually, that particular
object in this case.  

Pic 1 has a julia-shaped formation common to those on
the main dendrite pretty deep in that valley, that
one would expect to find there.  It's a bit distorted,
so that an endless spiral appears to be at its center.  
This pic has its center one of the larger minis on
either side, probably its actual structural center,
but it's hard to tell, since the components are
somewhat repetitious, as one would expect.  
Pic 2 zooms into the center of the 1st pic, to a
location close to the complex equivalent of the
Feigenbaum point of that mini there.            >>>but you won't see it, since the posting bot
                                                                 >>>rejected it for size.  C'mon, were doing FRACTALS here!
                                                                 >>> Get a REAL server, with some CAPACITY!
The 3rd pic
shows the whole Julia for that location, which was
the main thing I wanted to show here.  It is also a
loop shape.  The last zooms into the critical point in
it, but not too much.  See note about pic 2.

Have a decent weekend.

Later!

  

[fracmonk]

Apophyster-  

You have to go to the very basics of what the M-set is to understand critical points.
Originally, Dr. M was looking at Julia sets for f(z)->z^2 + c, and noticed that if they were connected at the origin, they were connected everywhere.  He came up with a way to index them, by plotting out one picture with that critical point, zero, as the starting z for every c value, mapped as pixels in the picture.  The pattern of it turned out to be pretty remarkable.  The M-set.

Here, we're doing the same thing.  Once you know the critical point for a function, then you can make up one picture, an index set, that summarizes that behavioral aspect of all its Julia sets, if only one constant is involved.  More complex constants would then require more dimensions to map, but here, we're only dealing with 2 at a time.  At least, we're supposed to...

Later.

 

[fracmonk]

element90-  I think we might both have been looking in here at the same time, but now I think you've left.  It would have been an interesting dialog, maybe.  Regarding the start of your post 271, that variant of "your" formula is in Beauty of Fractals, you know, and rather ancient.  It was a good starting point for me at the beginning of this thread.  I might have mentioned that then.  I would imagine that things both you and I have done have grown from that, not the other way around, no?

Later.

[element90]

I work out the critical value in the same way as before using pencil and paper, the introduction of the extra constant and powers had me going for a bit as the differentiation gets more complicated. However,

g(f(z)) = g'(f(z))f'(z)

Now for g(f(z)) = 0 can be met by f'(z) = 0 so I only have to deal with inner function and I can forget about the powers. For some formula I often get an expression to solve that has a c in it, that is the location in the complex plane which stumped for a while  until I realised that 1 can be provided that the first iteration results in c.

For formulae of the form

h(z) = f(z)/g(z)

h'(z) = (f'(z)g(z) - f(z)g'(z))/(g(z)*g(z))

to solve h'(z) = 0

it's more difficult, g(z) can not be zero, so f(z) and f'(z) must equal 0 for at least one of the formulae I use this is impossible, so I've come to the conclusion that there is no critical value for some formulae. This thread has been very useful to me as I hadn't really understood how critical values were determined and I can't remember where I came across the rule that the critical value can be found by solving f'(z) = 0, working out critical values for this thread has helped enormously. I think I'm right in say that if the there a "well formed" Mandelbrots in the resulting image the critical value is most likely to be correct.

If I gave the impression that this stuff is knitting together please disregard such a suggestion as I think the surface is just be scratched.

The reason for using smaller bailout values is because using very large values for the examples shown the result is an incoherent mess and in my opinion even with clipped bulbs the results can be pleasing to eye.

fracmonk - I don't know what your statement "I didn't know that some of this stuff has has been worked into type lists" means.

fracmonk - I've stopped uploading any image files to fractal forums I just insert links to images elsewhere on the net, you can upload them to any free "cloud" storage on the net and then you can around the problem of losing you images from your posts.

Here's a dendrite spiral coloured using a more exotic colouring method that just colour by number of iterations:



These things crop up all along the thin threads and if you delve deeply enough into them you'll find unsurprisingly a Mandelbrot.

The formula I called Cczcpaczcp i.e.

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

is the subject of a multi-part guide on my blog, it's currently up to 5 parts and I think there'll be at least an other 3 parts before I'm finished.

Fracmonk - re post 275, I put quotes around 'my' formula only to indicate the implementation in my software I pretty sure that anything I've come up with has already been thought of before. I've got a copy of the Beauty of Fractals and I've implemented the Magnet Model 1 and Model 2 from there, I don't remember seeing a variant the formula above, I'll have to have a look as I haven't looked at it for some time.

[cKleinhuis]

stunning last image! you created an art piece

[fracmonk]

element90-

My own copy of BoF is hundreds of miles away right now, but the object I was referring to had the formula f(z)->z(c+(1/c)) I think, so I was wrong about that.  I think the results are exactly the same, though.  I have my machine grinding away at a hi-iteration, no periodicity, hi-bailout version with the formula you gave.  The difference in expressions between the one above and the one you gave is the z/c in the above case, and c/z in yours, I think.

It was I who said this stuff seemed to be knitting together, but you're right about merely scratching the surface.  I had a preprepared thought, shown below, that I had put together over the weekend which would in part point out spots where our thinking is in parallel.

By "type lists", I meant the formula collections for various generators, that these things are being absorbed into them.  Also, I believe much of what we've been playing with here has NOT been done before, AFAIK.

Remember, I wrote the following with no knowledge of anything beyond post 275:


A (somewhat self-) critical note:

I doubt that more than a small handful of people care
all that much about what we do here in this thread.  
I think I've already made the case for its origins,
and told the story of how my own work evolved.  I saw
things beautiful and profound, and wanted to share my
amazement at them.  The biggest problem I have seen is
that newly discovered objects appear with such rapidity
that there is no time to study or explain them
adequately.  I don't think it's enough to say: "Oh look,
here's another new shiny thing!" and not explain it
clearly before moving on to something else, also
inadequately explained.  I'm sure I've done that myself
on enough occasions.  We simply happen into new forms
that quickly.  But there are patterns to these things
which ought to be understood.  I think it's important
to bring these things to light.  "New Theories and
Research" meant to me that we should be talking more
about HOW and WHY these things work, and not wasting
everyone's time trying to prove that we're freaking
geniuses or anything.  Leave the showing off part to the
Mensa crowd who have only that much to prove.  I prefer
it when people make themselves understood, and are not
going out of their way to appear mysteriously awesome.  
It has to be be simple enough to understand, or it's
useless.  I TRY to make it so, and apologize if ever I
fail in that way.  Many so-called smart people fail
miserably that way.  Worse, some will go out of their
way NOT to be understood.  Then it's doubtful, at least
to me, that their intentions are wholesome.  Have you
noticed that too?

(Kinda wandered off-topic...)

If I haven't made it clear before:
One of the things I've been trying to do is to avoid
coeffients to deal with these things in their most
irreducible forms, and explain patterns I've found in
them as families in the simplest terms available.  The
fact that their formulae are all related either in form
or algebraically is a profound thing not to be
neglected.  So that's why I avoid using coefficients
myself, though I see nothing at all wrong with them.  

A while back, I communicated the idea that some
functions are connected in complex space while others
are not, and how it would be nice to find some quality
that would make one able to recognize the difference
in advance.  I may be closer to that now than at that
time...or some fact will turn up that makes it a still
more complicated problem than I originally thought.  I
still can't tell about that.  And remember, no one
seems to care anyway...  

That's just one worthy Holy Grail, and that's the kind
of thing that comes to my mind when I think of the idea
of what 'research' means to me.  There are as many
Grails as there are seekers for them, and more we
probably SHOULD pursue that still have yet to reveal
themselves.  I think there are enough of them to pass
around when company comes over.

"Here!  Have a Grail!"

"Thanks!  Don't mind if I do."

-or-

"Just had one.  I'm full.  But thanks anyway."


Also, I've seen enough blank rectangles, each with a
little red square with an "X" in the middle of it.  I
think this is called link rot.  It is why I post my
necessarily miniscule pix directly to FF, so you can
see what I'm talking about as long as FF is there, when
some other linked sources perish inevitably.  (You have
likely seen my complaints regarding the limitations.  I
do not trust "The Cloud" in any way, shape, or form,
and you shouldn't either.  See posts 248,249.  You have
to be out of your mind to trust your data to unknown
others, but then, look around, plenty crazy people...)  
Anyway, I THINK that the metadata for each pic goes
with it, so there's more about it available in it if
you take it to fractint together with its formula.  
There, you SHOULD then be able to play with these pix
in any way you like.

As an example, here's a formula and a pic that goes with
it below.  You SHOULD be able to call up this pic in
fractint with the accompanying formula inserted into a
.frm file and zoom into it, etc.

DM98(xaxis) {;deg 18
  c=pixel, z=p1, d=p2:;quasi M2
    p=z*z       ;d=1:z0=8/9
    q=p*p       ;only!
    r=q*q       ;no per. ckg.
    s=r*z       ;nonstandard
    t=(s-r)*c-d ;dendritic
    z=t*t       ;structure
    |z| < p3
}

The pic is one of few in a series involving a zoom into a
"furry" spiral with small enough storage requirements to
be accepted by the FF posting bot.  You should be able to
find the zoom destination.  By switching the z and c
assignments, you will have the formula for the Julia set
then, too.

I looked at real axis limits for patterns in the values
between 21, 32, 43, and 54 exponent combinations.   If
there's anything there I can recognize, I'll let you
know, unless you can't wait and have to look
yourselves...

Many thanks are due to element90 for bringing this
formula structure into the mix.  



Later.

[fracmonk]

element90-

I must apologize that my personal circumstances are
not leaving me enough time to do your contributions
any justice at all right now, but they are very
powerful in my estimation.  Without my having to
more than "scratch the surface" of them, they have
helped focus some of my own observations.  I will
try to explain how.

If you took the simplest formula, for Multibrots,
f(z)->(z^n)+c, you will find that n must be integer
or you will get discontinuities in the object, AFAIK. 
I have noticed that many Multipowerbrots that have
taken up the bulk of my own contributions to this
thread require that any second constant used must
also be integer, for the same or similar reasons. 
This leads inevitably to, and here's the big idea, a
quantum effect. 

While we know that in the case of exponents, the
discontinuity already infects any parallel 2d slice
for any fractional n in the c plane, the second
constant d in the Multipowerbrots I've shown here
SHOULD have continuity over (a MINIMUM of) 3
dimensions, in some, not all cases, as a solid
object.  So that quantum effect would then be
expressed only in exponents.  I'm not clear on WHY
this is so about exponents, but it is so.

That's where my thinking has been going lately.  Your
work brings out new forms consistent with this notion.

Here's where I go into COMPLETE AND UTTER speculation:

If this kind of math does in fact say anything at all
about numbers governing the design of our universe,
we may be mapping actual elementary particles, and
there may not be only one, when everything is said
and done.  While there may, in all eventuality, be
many such possible forms, the patterns that govern
these forms should be painfully finite.  If the
objects aren't the particles themselves, then they
might dictate the laws that make the forms.  The
latter is more likely.

As in string theory, luckily for physicists living on
government funding, it's all too small to test!

How's THAT for mad scientists overreaching with
physics fairy stories?

Obviously, it could use plenty of analysis, merely as
mathematics and geometry that is beautiful in itself. 
It's only a big maybe right now, although the
experimental results we get here are indicative of
SOMETHING.  But what?  The habit of these numbers
staying in one piece under the right circumstances
cannot be ignored.

Given time, some conclusion will be drawn about that
(besides "fracmonk is NUTS!").

But those personal circumstances are always a
hindrance.  Everything happens in its own time, I
guess...

Have a great weekend, unless, as G. Carlin once noted,
you insist on having a crappy one.

Later.

[fracmonk]

No comments at all?

To be maybe a little clearer:

Variables and constants can be considered continuous in as many dimensions as their number require.  The same can be applied to coefficients, which usually only govern scale and proportion.  The inherent behavior of fractional exponents, like quantum behavior, is not continuous when mapped onto the complex plane.  Is there a relationship there?

Later!

[fracmonk]

So, here's the thing:   If you think it's totally preposterous, say so, and say why.  That's what we're here for, no?

Later.

[fracmonk]

element90-

I checked out your post 272 a little better, but my online time at the library is severely limited, and it still won't print right so I could study it right later.  But I think your approach to cp's is practical, and I still think that while some formulae won't calculate right under any circumstances, some others might under ridiculous conditions.

Very busy with other things, I let the machine grind away with the hope of making a clean version of the barebones formula from post 271.  set to 100000 bailout, 20 million iterations, no periodicity checking, it took (only) 140 hrs. 21 mins. ...but it did come out right.  You know, I hate it when machines and their programs of any kind think they know better when it comes to what you want to do.  Just leave the driving to us, they say, and when they didn't think of everything, you might just make the evening news...

Life is just a bowl of...

Later!

[Ryan D]

Following up on element90's reply #265, it looked like a good opportunity to try feeding some values into the many parameters used in his generalized formula.  I tried a couple of different methods and found some interesting views when the base terms alpha & gamma range between -1 and +1 and the power term beta ranges from 1 to 5, with the power term delta = -1*beta.  I then set up a path following a Lissajous curve that travels within these boundaries and made a rough animation over a long path.  I excerpted an interesting portion and made a proper animation.  Nothing to look at if you're interested strictly in the integer values, but there's a lot here that's interesting to me at least.

Ryan

<a href="http://vimeo.com/moogaloop.swf?clip_id=48652524&amp;amp;server=vimeo.com&amp;amp;fullscreen=1" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=48652524&amp;amp;server=vimeo.com&amp;amp;fullscreen=1</a>

[fracmonk]

Ryan D-

That must have taken awhile to put together, no?  I'm especially curious if a:
one of your dimensions was exponential (gamma?), and continuous thru fractional values,

and b:

If the the discontinuities I mentioned are reflecting that, in your view.  I seem to see them clearly as stills.

Really action-packed illustration of the power of that formula structure that element90 brought us
anyway, I think.  I really appreciate it, for one...

Later.