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

[fracmonk]

So almost immediately, considering what I said in the last post, I find that I was likelier to be wrong than right.  (It figures...)  The M96 object I spoke of (1st pic below) had a certain misshapenness to the largest formation of non-escaping points, which experience has shown me always indicates that the object is *NOT* connected.  I had applied the same original code error to what amounted to a degree 12 multiply connected object (2nd pic), since experience also suggests that lower degree will reveal such problems more obviously.  There, the object had a clear discontinuity (last 2 pix), as when one uses a fractional exponent, which was in some ways pretty and interesting, but leaves one in a position of having an index set whose "critical" points can't tell you for sure about the nature of the julia sets it indexes.  The critical points would not then be very critical, would they?

I had put pix on a disk to show, then brought the wrong disk...next time, hopefully.  (5-2-12: This was what I thought yesterday- Erased a disk w. XP and burned the stuff I wanted to it, but the laptop I was using w. 7 actually found what I thought was erased, and ignored the existence of the new contents.  Weird, or what?)

Here's the thing:  Usually, one does not have to look far for evidence to show that an object is not connected.  In this case, there are features that signal that that is so, and yet, it might not be possible to obtain that evidence (ie., a separate "island" from the rest) beyond the reach of floating point precision, when that's your zoom limit.  And, of course, experimental evidence in no way constitutes proof, though I have never seen it to mislead if properly interpreted.

But I'm pretty satisfied that this little side road will turn out to be another of many dead ends encountered in this kind of experimental math.

There are always other things appearing to show more promise that might not be apparent if such dead ends were not scouted out.

I often wonder whether new approaches could be exhausted entirely, but then I think it's more like stage fright or writer's block, and I think that works the same way.  Just because you reach and it's not there, only means that you didn't reach into the right place yet...

Later.
  

[fracmonk]

Apparently most of my pix just barely exceed the max size of 256k, so I'm experimenting with the program and its "input procedure requirements" (my term), to get to show and say anything at all...

I wouldn't mind half as much if it let a posting go though and left out whatever it objected to, but all your carefully articulated notes are lost as well when you are forced to abandon the effort and start over to get anywhere with it at all...

The pix it seems to be willing to accept show a location just next to a spot where one would expect an M2 mini to be found, where surroundings are symmetrical, where instead, there is merely a hole.  The above restriction doesn't let me show the view that shows it best.  The next shows a slightly deeper view, but not the one that shows the M2 mini that actually can be found at the centers of both, upon deeper zooming.  Apparently, you'll just have to take my word for it that it's there.

More flexibility please!

One of the things I was trying to say in these attempts was that it is one thing to have this or that multiple of a power locally dominate some part of the plane, as well-negotiated subsets, which seems to happen with Multipowerbrots, and another thing entirely to have two conflicting systems interfering with one another in the same space, as in the discontinuities just mentioned in the last post.  The two have an entirely different visual character.

Later!  

[Alef]

p.s.
Legitimate publication place is arxiv.

[fracmonk]

Asdam-  Thanx.  I'll look into it, but you know, I'm one of those people who questions everything, like what makes one venue any more "legitimate" than another.  I'm not hacking out a career in this, but looking around and sharing what I've found, so I don't need to care too much about formality.  What I'm looking for is others who find these things somewhere near as fascinating as I do, and want to help flesh out and further develop the concepts involved.  Maybe it would trigger someone else's wildly different take on it, or that some pretty cool applications might occur to others as a result of exposure to it.

I've returned to concentrating on successive squaring resulting in power shapes associated with multiples of 2, and lately, as applied to adding multiplicative inverses of expressions to get multiply connected effects.  Below are an assortment of lower-degree transformation variations, each resulting in another curious index bug.

[fracmonk]

Today I intended to begin showing a nice zoom series from the last pic in the last post, but the laptop I borrowed for this, with its trusty Windows 7 OS refuses to read the cd that was made with XP.  This has happened before.  In fact, last time, I erased a disk that was written with 7 to put stuff on it from an XP machine, and the laptop acted as if the disk was unchanged from its original condition, reading the erased data as if it wasn't erased.  When I put the disk back in the xp machine, it read it as intended.  I bet it'll turn out to be the same deal this time.  You'd think I'd have learned...

Sorry, folks, maybe next time.  But let's call the program MS Typical.

I'll bet that I'd have to transfer such data to my 7 desktop, then reburn the disk with that machine to get the laptop to recognize it.  I don't know why anyone puts up with MS, other than the fact that everyone else does, and we're all just SHEEP.  They just SUCK.

I feel so much better now...

Later. 

[fracmonk]

The situation I described in the last post was true.  I put the stuff on the disk into the Win 7 desktop, erased the disk, and burned it back from there.  It should read right on the laptop now.  A curious thing happened when I tried to shut it down after making yesterday's post, however.  It literally never got done logging off.  I had to put it in sleep mode, and brought it home, and opened it, and it was still trying to log off.  Slept it again, and took out the battery.  Started it up on wire in safe mode, and looked at bizarre messages (like one "critical", as opposed to "error", or "warning") in event viewer before doing a system restore. That got rid of 2 MS updates, and nothing else, and then it seems now to be working O.K.

Hmmmmm...

So the pix below begin from the last pic in the last post, M16a.  It has a ring shape more pronounced due to higher degree and a slight tweak in the usual formula (for formula structure, see post 203), leaving the last subtraction of d (=1, in this case) out before the addition of the multiplicative inverse, which gives the multiply connected topology.

The journey begins near the very large M4 mini, in one of its valleys.  Notice there the dendrite system that daisy-chains smaller minis together, following the chasm all the way in.

Further next time (if there IS a next time...)

Later!

 

[element90]

I've been following this thread for a while, it's been interesting, especially with regard to different power Mandelbrots appearing in the same fractal. I've noticed that the standard Mandelbrot shape keeps turning up sometimes misshapen sometimes not, lately I've been finding multibrots along with the Mandelbrots. Here is an example using a formula I've called Double 3F Onion M, onion refers to the fact that the function calls are nested so the brackets resemble the layers of an onion, 3F for 3 functions in each onion, double for two onions and M to signify that the onions are multiplied together (I have an other where D signifies division).

z = alpha*f1(f2(f3(beta*z + gamma)))*f4(f5(f6(delta*z + epsilon))) + c

alpha to epsilon are all complex parameters and are all set to 1, c is the location in the complex plane and the initial value of z is set to c
f1 = sqr
f2 = tan
f3 = cube
f4 = sqr
f5 = sin
f6 = tanh



Obviously, the set is not connected, at the edge of the large island at the compass points multibrots are found, this one is at the eastern edge edge.

[tit_toinou]

@fracmonk:
I think you have done a lot. Now, it is our responsibility to use your whole bunch of formulas so that they (and you) get noticed.
I think a big video in a M842 connected set would be SO awesome............

[fracmonk]

element90-  You're realllly deep into assigning functions, I see.  I'm curious whether what you found was accidental or on purpose.  Sometimes you find yourself in a situation that can be hard to describe to the uninitiated.  Either case may have implications about (let's call it) "natural occurrence".

t-t-  Thanks, you're very kind.  I too have a fondness for the degree 8 inclusive, since it marks the extent that many of the most popular constants are operative, but ANY doubling can have interesting things in it, and I just happen to be looking at one of the M16's at the moment.  You can see that they have behaviors in common that I'd love to see generalized coherently.  It's the "coherently" part that's tough...

The zoom series I began last time continues below.  We go between 2 M4 lobes into another "daisy chain" dendrite, and into an edgy spot on a M8 mini...

('til next time)

Later.

[element90]

Generally speaking I've no idea what I'll get from any new formula. I'll take an idea regarding a formula and extend it and add extra parameters just to see what will happen. I'm currently developing version 3 of my software which will change the way it deals with formulae, a formula may have a number of parameters that have values that are fixed and a value that is the location in the complex plane (I call this value c), for version 3 a formula will have a number of parameters any of which can be substituted with c, so the basic Mandelbrot formula will be used for both the Mandelbrot and its Julia. This will reduce the number of formulae defined but will increase the number of possibilities, taking Double 3F Onion M formula as an example the number of variations due to parameter substitution will be 64.

[fracmonk]

element90-  Keep us posted, o.k.?

[element90]

fracmonk

Will do. I announce releases here in the "announcements & news" section. If I find anything fruitful I might even start a new thread.

One of the main features or Saturn and Titan is the ability to orbit plot all the fractal types except Lyapunov. The orbits can be plotted as escape orbits only, captive orbit only and all orbits with the option to omit a number of points at the start of an orbit. So far I've discovered that Julias produce poor orbit plot images and fractals using the Mandelbrot algorithm (both divergent or convergent) produce interest images, I'm mainly concentrating on captive orbit plots and all orbit plots as the escape plots (Buddhabrots) haven't been that successful. There is an exception to this general rule however and that is that Pickover Popcorn (I extended the formula with an extra parameter, the seed for the Julia or location in the complex plane for Mandebrot) produces good orbit plots for Julias but poor plots for the Mandelbrot form, both produce forms nice gnarls when treated as ordinary escape time fractals (although often the bailout is set to none).

I've drifted off topic some what.

I'll leave you with a variation that produces a seriously altered "Mandelbrot". I have posted this before elsewhere on this forum but you may not have seen it. Since the the image is held on a different site I'm not wasting the forums valuable disc space.



Parameters

Formula: Combination 4

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

alpha = 0.25 + 0i
beta = 2 + 0i
gamma = 1 + 0i
delta = 1 + 0i
epsilon = 2 + 0i
zeta = -2 + 0i

Programs: saturn and titan
Number of A transforms: 0
Number of B transforms: 0
Transform sequence: A
Number of Complex Plane transforms: 0
Initial values of z: 0 + 0i
Image centre: -2.208 + 0.128i
Image width: 15.008
Rotation about image centre: 0 degrees
Maximum iterations: 600
Bailout: norm(z) > 160
Colour selection: outer for bailout otherwise inner
Outer colouring: iteration
Inner colouring: fixed colour

The set is clearly disconnected, on the needle we get this:

[fracmonk]

element90-  Your pix are along the lines of what I spoke about in post 210, but maybe more esoterically (dis)organized.  Excuse my very subjective, purely personally perceptive take on the visual field there.

Clearly, your choices are not entirely random, and you've developed what looks like a fun toy.  What I said before relates to users understanding how it works enough to get pleasing results from it.  I vaguely recall doing pix a bit like the second one in fractint, again using multiplicative inverses.

Keep going!


My own offering for this time reaches the destination coordinates and shows the Julia set for them.  One more shows the center of the Julia magnified.  Notice the smaller J4 minis in the periphery surrounding the center J8.

I plan to show a few more views of this Julia to show correspondence between it and the index set, but I don't know when I'll manage it, exactly.

Life is often, well, um, chaotic...

Later.

[fracmonk]

The nonzero critical point z0=2 is at the centers of all the pix below, and the next bunch to come, most likely.  They start with mag 1 and increase by 10x each, just as the index set had.  It's not hard to see the usual similarities in local detail as magnification increases in both.  Notice the abundance here of small j4's, and also the general shape of the daisy-chains of minis in general.

Later.

[fracmonk]

The zoom series into the Julia is completed below.  Since the coordinates in the index set were within an M8 feature, this winds up in a J8 feature at the critical point z=2.
Next time, a look at the hybrid J2, if I can find it...if you recall, when the coords are in a higher-power feature, then all Julia minis of lower power than the coords one are hybrid, that is, never as found in a normal multibrot, but heavier at their extremities.  A J2 then would be generally barbell-like, unlike those found in Julias for coords in standard M.

To be honest, I think the Julias for standard M are very pleasingly proportioned in comparison.  I don't know whether I perceive them that way due to familiarity (that I have come to expect them the way they are) or if it's something completely unconscious that governs that sense.

Comments on that?

Later.