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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 32685 times)
Description: I think there may be. Your opinions are greatly desired.
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Fractal Supremo
Posts: 1138

« Reply #60 on: February 15, 2011, 02:26:48 AM »

I rendered some pics and uploaded to imageshack, for the formula (z*z*a+b)+(1/(z*z*a+b)) drawing the plane a
The first is for b=0 and then the rest for b=1

There are only two showing the multiple powers minibrots, because you did already a lot pics of that. I'm pretty sure the "multipowerbrots" are the more important aspect of your discovery, but I find the sets also very beautiful and with a great complexity of shapes.

I'm VERY interested on what you've done, and I will be researching more on your ideas when I have more time.

Thanks for sharing your findings.

<OFFTOPIC>I liked your thoughts about St. Valentine's Day, I totally agree with that.</OFFTOPIC>

« Last Edit: February 15, 2011, 02:34:22 AM by Kali » Logged


Fractal Fertilizer
Posts: 356

« Reply #61 on: February 15, 2011, 02:36:52 PM »

(pix via bunny express)

To hopefully wrap up yesterday's theme, I'm obviously not a complete Luddite, as this fractal research requires super-fast calculation.  My concern is that we're overautomated, making us weaker, less able and more dependent, and that's very unappealing to me.  We must carry too much crud as it is, to get through a day in a practical way.  Again, sometimes, less is better.  I'd rather travel lighter, and maybe stave off early dementia...

At the end of the Julia zoom series, the J3 is dwarfed by 3 nearby J27's that surround it.  The critical point z=i is found well within it in the last pic.  Next, same object, different spot.

Kali- Those pix are very familiar to me, and I can't begin to tell you how reassuring it is to have you showing up here when you did.  Did you notice that certain parts will get cut off unless the bailout's really high?

Here's more of a philosophical question I've always wanted to bang around, but I'm not really sure if it belongs here:

Did these things already exist before they were found?

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* s3jim9.GIF (29.82 KB, 320x200 - viewed 406 times.)
« Last Edit: February 16, 2011, 04:21:10 PM by fracmonk, Reason: t.b.t.r.n. » Logged
Fractal Fertilizer
Posts: 356

« Reply #62 on: February 16, 2011, 02:39:08 PM »

(pix via bunny express)

A new journey into the M27-9-3 object begins in the pix below, in the left cusp, where M9 continental features predominate.  As usual, each view is 10x larger magnification than the previous one, unless otherwise noted.

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« Last Edit: February 16, 2011, 04:22:23 PM by fracmonk, Reason: t.b.t.r.n. » Logged
Fractal Fertilizer
Posts: 356

« Reply #63 on: February 17, 2011, 02:54:02 PM »

(pix via bunny express)

In the pix below, the zoom continues deeply into a spiral...

I'm constantly thinking of ways to get more eyes on this stuff.  I thought of becoming a celebrity, since they're the only ones people pay any mind to these days.  But, you don't get the microphone unless you have little to say worth saying, a catch-22.


Oh well, back to the drawing board...

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« Last Edit: February 17, 2011, 07:12:16 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #64 on: February 18, 2011, 03:03:59 PM »

(pix via bunny express)

At the hairy edge of the spiral in the pix in the last post, the zoom continues in the 1st pic below.  In the 2nd, the features that surround an M3 mini are seen.  In the last, the destination will be in one of the 3 M27's there that dwarf it.  Notice the 9 arms that emanate from the M3.

For technocrats:

Very generally, I seldom encounter trouble using periodicity checking in index sets, but it often wreaks havoc with Julias, especially at lower magnifications.  Can anyone explain why that's so?

Some like it hot:

Solar flare activity is high lately, but I don't need to know what's going on at the surface of the sun right now.  I'm content to wait the customary 8 minutes or so...aren't you?

So much for instant gratification...does away=ago?

Have a cool weekend!

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« Last Edit: February 18, 2011, 04:19:50 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #65 on: February 22, 2011, 02:35:02 PM »

(pix via bunny express)

In the pix below, the zoom ends inside, but near the edge of the M27 mini.  For this location, the whole Julia set is seen in the last pic.  The 3 largest groups of non-escaping points are J27, and surround the J9 at center, which should be considered hybrid.

Earth is more accurately 8 mins. 19 secs. at light speed from the sun.  No real human has ever departed from that distance from this hot fractal by more than a couple seconds.  By the weak anthropic principle, it's the perfect place for us to evolve as we had.

Too bad it's run by ruthless parasites.

IMPORTANT:  We are to believe that wealth is the only measure of success, and obtainable only by our degree of collaboration with them.  Often for ridiculously little, most collaborate, and their hold gets stronger.  The first rule is not to anger them, and having your own thoughts does that more than anything else.

You must save yourselves from such subversion!  Take a pill and apply to some large, famous computer graphics company.  There must be some texture sub-subroutine you could be working on, as part of the Borg.  When you find out whether their main objective is to cartoonize humanity or to humanize cartoons, let me know.

Too alarmist?  Want more?  Just keep those parking tickets coming.  It takes that little to set me off lately.

For today, at least,

"That's all, folks!"

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« Last Edit: February 22, 2011, 07:34:08 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #66 on: February 23, 2011, 02:37:38 PM »

(pix via bunny express)

The 1st pic below is a detail of the whole Julia set in my last post.  In it, the center is z=i and the origin is at the midpoint of the bottom edge of the pic, and about half the central J9 is seen there, mag. 1.   As coordinates near to the origin in the index set, the Julia sets for them tend larger.  Zooming in on z=i, the scenery in the 2nd pic includes spirals, and J27 and J9 minis large enough to identify.  V. pretty in hi-res, not shown*.  In the last, the zoom into the nonzero critical point reflects things seen in the index zoom previously shown.  In these functions, as you may already have noticed, the nonzero critical point in the Julia set is more informative about the local features in the index set than others.

* I am forced to GUESS, by number of views indicated on pix posted, that some of you may be having problems viewing higher-res. pix for reasons I can only speculate on.  I have therefore avoided posting such.  Again, if you had difficulties with any of the few I had, let me know.  Some take a lot of space & presumably slow things down, an effect unwanted by the moderators, I imagine.

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* s4jim2.GIF (24.28 KB, 320x200 - viewed 327 times.)
« Last Edit: February 23, 2011, 04:25:28 PM by fracmonk » Logged
Strange Attractor
Posts: 272

« Reply #67 on: February 23, 2011, 05:45:13 PM »

Did these things already exist before they were found?
As far as they are part of numbers nature, and numbers are reflection of real world laws, I think yes.
Fractal Fertilizer
Posts: 356

« Reply #68 on: February 24, 2011, 02:42:23 PM »

(pix via bunny express)

As the Julia zoom continues in the pix below, so does the correspondence with the earlier index zoom.

makc-    I agree wholeheartedly.  My theory is that if numbers govern nature, we should understand better the nature of the numbers themselves.

I've never been very good at application, a condition which has drawn me into pure study, for pure love of it.  Many criticize "solutions in search of a problem", as so much in pure mathematics can be, but I think everything there finds usefulness eventually.

I truly delight in those who CAN easily find applications for new results.  "It takes all kinds.", my grandmother said.

But I myself am restricted to an intuition, at best, that the things you see here in this thread will relate somehow* to how nature does things, as I think Dr. M. had, with regard to the M-set.  On a spiritual level, it is an article of faith for me.  So am I preaching to the choir?  I would like to think that we fractalists are midwives for a new way of looking at the world, a necessary one, given our current state as a species, which I too often view with pessimism, and Rays Of Hope detectable only with state-of-the-art sensors**.

(I cringe, awaiting the arrival of high-velocity stones!)

* The nastier question is exactly how.
** Available from ROH Technologies, $29.99+tax, each.  HA!  Made you laugh!

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« Last Edit: February 24, 2011, 07:27:10 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #69 on: February 25, 2011, 04:15:14 PM »

No stones?

To my great sadness, Bunny Express is out of commision temporarily, and pix planned for today will be postponed until Monday at the earliest.

Those will wrap up that series, and a new perspective involving an earlier object, previously neglected, will be explored later.

Have another cool weekend...
Posts: 45

« Reply #70 on: February 26, 2011, 08:44:27 PM »

Is it possible to have a simply connected set that has both z^2 and z^3 msets?
Fractal Fertilizer
Posts: 356

« Reply #71 on: February 28, 2011, 03:09:23 PM »

(pix via bunny express)

kjknohw-  I haven't managed it yet, and I'm thinking that maybe odd+even power shapes may remain incompatible.  But I've been wrong before...you know, I've tried to make that happen quite a lot- no success.  It may be subset condition involving multiples. 

The 1st pic below shows clearly the surroundings of a J3, seen more clearly to right in the 2nd pic.  It must be presumed to be a hybrid, since the coordinates for it are found inside an M27 in the index zoom.  Here, the nonzero critical point at the centers of these pix is within one of 3 J27's that surround the J3, as seen in the last pic here.  In Julia sets of this class, all prisoner point groupings of lesser degree shapes than that of the coordinates locality are hybrid.

Something a little different next time...later.

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« Last Edit: February 28, 2011, 07:26:20 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #72 on: March 01, 2011, 02:44:38 PM »

(pix via bunny express)

The pix below show an object first seen in post 22 in this thread.  It was a trial-and-error precursor to M4-2, also first seen there.  The formula for this object is the same, except for a change of sign (for consistency, reversing its orientation), and in leaving off the last (-1), so that f(z)->(((zc-1)^2)-1)^2.  Since it wasn't what I sought when I first saw it, I never gave it much attention until very lately.  The more I consider it, the less able I am to explain it, but I'll try...

The largest contiguous area in it is Julia-like, and there is a hierarchical distribution of M2-like miniatures bristling its edges.  The largest of them is obvious in pic 1.  The M2 minis in turn bristle with more Julia-like blobs, which in turn bristle with...
Left (c=1-sqrt(2)) and right (c=1+sqrt(2)) real axis limits are greatly magnified in the other pix.

More next time...

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* JLrl.GIF (3.24 KB, 320x200 - viewed 261 times.)
« Last Edit: March 01, 2011, 07:22:02 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #73 on: March 02, 2011, 02:44:18 PM »

(pix via bunny express)

The 1st pic below shows the antenna of the object in the last post, centered at c=2.  The 2nd pic is of the Julia set for c=2, shown centered at z=1/c, as all Julia sets for this function are.

The point c=.25 in stardard M, called the "cusp", I prefer to think of as "navel" instead.  In M, it's the only one without an "umbilical cord", as I like to think of dendritic connections to all the miniatures.

The equivalent point for the largest M2 in the set currently discussed here is c=>.84375... (approx.) and not shown.  If you want to kill your machine for a fairly uninteresting pic, hunting for the exact location is a good choice.  The last pic shows the Julia set for the value I gave.  The shape makes sense, right?

More later...


Behavioral psychologists now believe they can control the entire universe, upon learning that it revolves around the actor Charlie Sheen.

Who knew?

Can't you just FEEL that manipulation?




Charlie Sheen OK, found clinging to large white rock in Bahamas

(I got a million of 'em...)

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« Last Edit: March 02, 2011, 04:51:47 PM by fracmonk » Logged
Fractal Fertilizer
Posts: 356

« Reply #74 on: March 03, 2011, 02:37:19 PM »

(pix via bunny express)

Does no one get my jokes?

The 1st pic below is of the Julia set for the real axis left limit c=1-sqrt(2).  The 2nd is for the approximate "zero point" of the largest M2, c=.91, and the last is for the center of its main bulb, which appears to be c=sqrt(2).  Not shown is c=1, which is identical in shape to the San Marco Julia for c=-1 in standard M.  Here, the location c=1 is found within the cardioid body of the largest M2 in this index set, oddly.  As to why, your guess is probably better than any I might come up with.  I just find 'em and show 'em...

* JLjll.GIF (5.86 KB, 320x200 - viewed 228 times.)

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* JLjmbc.GIF (5.57 KB, 320x200 - viewed 228 times.)
« Last Edit: March 03, 2011, 07:17:51 PM by fracmonk » Logged
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