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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 32342 times)
Description: I think there may be. Your opinions are greatly desired.
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fracmonk
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« Reply #30 on: January 11, 2011, 02:43:41 PM »

(via bunny mail)

I should point out here that no matter what the degree, functions of this kind have Julia sets w. overall 2-symmetry, and so far, I haven't found critical points for any of them that don't turn out to be preimages of z=-1.

Generally, lower-power minis are more elaborately framed by dendritic structures than the higher-power ones in proportion to their size.  The framing reflects the local symmetry, which usually improves as one zooms in.

The first pic below shows a hybrid J4, and zooming out from it in the 2nd pic shows how surrounding 4-symmetric dendritic arrangements help to locate it, echoing its focus.  The 3rd pic magnifies the left tip of pic 2, showing how the J2 in the last pic was found, by its surrounding 2-symmetric echoes.

For this class, all minis of powers less than that local to the index set coordinates will be hybridized shapes in the Julia set, as are the ones shown here.  Not shown in this bunch is J8, but the same rule applies.

From here, I would rather return to the 8-4-2 combination, which I am somehow fondest of, as was partially explained previously, and for aesthetic reasons.  Next time.  Later.


* 512Jm4m5.GIF (51.7 KB, 320x200 - viewed 546 times.)

* 512Jm4m3.GIF (10.71 KB, 320x200 - viewed 535 times.)

* 512Jm2m4.GIF (6.32 KB, 320x200 - viewed 539 times.)

* 512Jm2m8.GIF (50.44 KB, 320x200 - viewed 543 times.)
« Last Edit: January 11, 2011, 07:23:19 PM by fracmonk » Logged
fracmonk
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« Reply #31 on: January 12, 2011, 02:57:05 PM »

(via bunny mail)

The centers of the 1st 3 pics below are for coordinates inside a canyon that bores its way toward the origin in the index set.  The left side of the canyon has M8 features, and the right has M4 ones.  I like to call it the "continental divide". Index sets of all degrees in this class have them.

The last pic is of the Julia set for these coordinates, for a bulb on a mini there.  Values for c close to the origin naturally produce large Julia sets.  c=0 would cover the entire plane. (!) 


* M8Is2m0.GIF (6.18 KB, 320x200 - viewed 519 times.)

* M8Is2m2.GIF (3.51 KB, 320x200 - viewed 525 times.)

* M8Is2m9.GIF (32.71 KB, 320x200 - viewed 523 times.)

* J8Iw.GIF (25.67 KB, 320x200 - viewed 526 times.)
« Last Edit: January 12, 2011, 04:42:28 PM by fracmonk » Logged
jehovajah
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« Reply #32 on: January 12, 2011, 07:13:46 PM »

Nice renders fracmonk! Way to go!
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fracmonk
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« Reply #33 on: January 13, 2011, 02:34:56 PM »

(via bunny mail)

Wonderful to hear from you again, J, & THANX for the encouragement.  Have you read this whole thing?  I've many questions, but for decidedly vain reasons, I wonder if this stuff is truly *new*.  Only lately, I saw a paper in PDF by Wolf Jung, published sometime in 2009, that had a picture that somewhat resembled the M4-M2 combination seen here in post 22 here, but with a more complicated, less obvious formula.  Who got there first?  I know at least SOME of the things I put up here are original, I far as I can tell.  If you know, let me know...spent so much time doing this stuff, I haven't had a decent chance to look around. (hence the screen name...)

The reason why I've concentrated on on minis here (& their edges, at that) should now be obvious.  "Continental" coordinates yield Julia sets that are monolithic, & resist revealing their inner structure without something like a BoF60 treatment.

The 1st pic below shows a detail of the Julia set in the last post, with the same center as the more magnified 2nd pic, that shows the J2 feature, @ 300x.  The 3rd, M4 @ a mere 7.5, from the origin, & the last, M8 @ a mere 2.5x, also visible in the 1st pic, lower left.  M8 in the 8-4-2 combination is found largest @ z=1/sqrt(c).


* J8J2s2m0.GIF (29.12 KB, 320x200 - viewed 507 times.)

* J8J2s2m2x3.GIF (55.03 KB, 320x200 - viewed 516 times.)

* J8J4s2cm03q.GIF (47.23 KB, 320x200 - viewed 523 times.)

* J8J8s2m0xq.GIF (50.65 KB, 320x200 - viewed 517 times.)
« Last Edit: January 13, 2011, 07:25:00 PM by fracmonk » Logged
fracmonk
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« Reply #34 on: January 14, 2011, 02:38:19 PM »

(via bunny express)

Another instance in which B0F60-type interior studies would be useful is in verifying "zero points" of minis.  For lack of better terminology, these are counterparts of c=0 in standard M, only in minis.  The Julia sets below, from the major antenna, *appear* to have the following zero points:  1. M2, c=-2.  2. M4, c=-1-sqrt(2).  3. M8 c=-e+1.  Other estimations distort the circles more, while they already exhibit "tidal" eccentricities.  Circles are pulled both horizontally and vertically within each set.

But these values demonstrate how 8-4-2 seems to be packed with the most well-known constants interacting.  That's why I find it the most interesting of the bunch.  As a single object, it seems to unify them.

At the time I found these objects, I liked to try to nail down such points, a habit which hid the non-self-similar structural aspects (shown in previous posts) within many of the Julias for a while.

The last pic is an artifact, a common error for FractInt when it calculates Julia details.  While the index set is relatively well-behaved, some of its coordinates will defy study.  This one is found in the same antenna, so it is assumed to be non-escaping.  Any width is beyond floating point's precision, but calc rebels at relatively low magnification.

Happy Computing (anyway!).


* J8Im2z.gif (4.48 KB, 320x200 - viewed 519 times.)

* J8Im4z.gif (3.9 KB, 320x200 - viewed 519 times.)

* J8Im8z.gif (4.83 KB, 320x200 - viewed 520 times.)

* FRACT147.GIF (65.22 KB, 320x200 - viewed 513 times.)
« Last Edit: January 18, 2011, 07:19:24 PM by fracmonk, Reason: 2 fix typos » Logged
jehovajah
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« Reply #35 on: January 15, 2011, 02:13:26 PM »

No fracmonk, not yet.

I can see that your labours though hard have been rewarding, mine also.

I have not done research on your choice of roots of unity, but have a general idea now of how they fit in the general picture. Your choice of roots of unity are a natural extension which i could not evaluate before, but now can make some sense of, in comparison to quaternions.

I think Hamilton found only one of the combination of solutions to the roots of unity for this level of dynamic equilibrium. He may have found others but i have not determined that. Your set of solutions are another variant. I think that there should be a solution that consists of triples with a mod( 8 ) ring structure.

Although this sounds complex it is only the terminology, it is better explained by a geometric model, 8 cubes stacked in a cube form.

Hamilton was a whizz at spherical trigonometry, so he may have visualised the roots, but not tested them all! Your set of roots would lie on two orthogonal great circles, even though you choose one at π/4 in the so called complex plane.

We are very fortunate to be able to animate these magnitudes and their relations , but i note you are old school and hold pictures in your head of what they should look like. That makes you an expert in my book!

Keep up the good work.
« Last Edit: February 09, 2011, 03:04:26 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
fracmonk
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« Reply #36 on: January 18, 2011, 02:31:22 PM »

(via bunny express)

Jehovajah- I would be very glad if this stuff can help you in your researches.  It's exactly the kind of thing I'd hoped would happen.

When I first encountered the figure shown in post 15 here, I was then only trying to get lucky finding a simply connected counterpart for it and did.  (!)  I thought that that might offer more considerations to those trying to prove MLC.  In later posts, I will affirm that there's something unique to unity (no pun intended...) in yielding multiple power shapes.

The pix below continue the last post.  The 1st is a fairly small M8 mini whose ctr. is well approximated in 9 digits only.  The whole 2-symmetric Julia set follows, along w. the J4 ctr. detail in the next, and the a J2 in the last.  Since the local features are roughly circular, one can only tell the types by the arrangement of secondary features surrounding them.

Later.


* M8II.GIF (60.51 KB, 320x200 - viewed 484 times.)

* M8J0.GIF (10.26 KB, 320x200 - viewed 482 times.)

* M8J4c.GIF (53.74 KB, 320x200 - viewed 485 times.)

* M8Jn1.GIF (51.04 KB, 320x200 - viewed 486 times.)
« Last Edit: January 18, 2011, 07:21:21 PM by fracmonk » Logged
fracmonk
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« Reply #37 on: January 19, 2011, 04:29:14 PM »

To clarify the formula structure I've outlined, an alterative function yielding the standard M shape is:

f(z)->(z^2)c+1.  Changing the + sign to - only reverses its left-right orientation, so let p=(z^2)c-1.

f(z)->(((p^2)-1)^2)-1  then yields degree 8, by nesting p thusly.  Let the expression after the arrow = q, and

f(z)->(((q^2)-1)^2)-1  then yields degree 32, (etc).

And the calculation is as efficient as it gets, as well.  Is that cool, or what?

Standard M, for all its intricacy, stays simply connected, in one piece, tenaciously so, at any scale, a property of its function on the complex plane, and nothing about that is short of miraculous.  These sets I've shown do exactly the same thing, only for MULTIPLE "multibrots".  This would have been very unexpected if it weren't for the encounter with the object in post 15.

It is now for those who do complex analysis to explain why this formula structure yields the results it does, especially in light of post 23 here.  I like to believe that such things help rather than hurt!

Next time, substitutions for -1 in the formulae.

...& new pix, a job for...BUNNY EXPRESS!

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fracmonk
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« Reply #38 on: January 20, 2011, 02:39:05 PM »

(via bunny express)

The 1st pic below completes the series from post 36, & shows the J8, with its immediate surroundings.

For squared z in this formula structure  f(z)->(((((z^2)c+d)^2)+d)^2)+d, there is apparently ONLY one other value besides d=-1 that yields a one-piece simply connected index set.  d=-2 is shown in pic 2.  A while back, I came up w. a formula that elongated the M-set, nicknaming it "Mandelsword"(not shown).  This object might deserve such a name more than that one.  A look at the 3rd pic shows the leftmost cardioid, the only one with no "tail", which then must be considered the "main" one in the set therefore.  Its dendritic "hairiness" proportionally exceeds that of the main cardioid of standard M, so it is M-like, but not a distorted version of M itself.  In this object, all shapes are uniformly M2, however, not uniformly M8, as the degree might suggest, when uniformity would have been expected.  This also happens in the simply connected sets of higher degree where d=-2, as in the degree 32 nesting, shown in the last pic.  I noticed that as degree increased, the set shape looks more and more like the Julia set for standard M for c=-2.  The "line" often gets too thin to even show, but it seems to be there.

With z squared in the formula, all Julia sets center at the origin.  Non-integer d yields non-connected index sets.  More about that later.

More generally, it seems that an index set must be connected in one piece itself, to reliably indicate the same property in its Julia sets.  That's been my experience.  Yours, too?

Does anyone know about this for sure?

For some reason, only one-piece index sets attract me.  There's a certain perfection to them...


* M8J8.GIF (50.47 KB, 320x200 - viewed 455 times.)

* D8w.GIF (3.75 KB, 320x200 - viewed 450 times.)

* D8ld1.GIF (9.08 KB, 320x200 - viewed 448 times.)

* D32w.GIF (2.62 KB, 320x200 - viewed 453 times.)
« Last Edit: January 20, 2011, 09:16:50 PM by fracmonk, Reason: formula fix-sorry! » Logged
fracmonk
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« Reply #39 on: January 21, 2011, 02:29:29 PM »

(pix via bunny express)

To detail the left end of the object in the last pic of my last post, the 1st pic below has a 35x mag. & the 2nd is at 100x.  Notice the even fernlike quality of the dendrites when the degree increases.  The 3rd pic is only mag 10x, from further right, and shows a sort of evenness in the spacing between the larger minis in an object the comes off as "all antenna" overall.

For extremists, the last pic is again of the left end, this time in a degree 2048 nesting of the formula, near c=0



Now may be a good time for free advice, and you know what they say about free advice...

The objects in these last 2 posts have never been seen by human eyes before about a week ago, so far as I know.
If anyone asks you what experimental mathematics is all about, tell them that.

That's why I do what I do.  Usually, you don't find new things so frequently, and that's made things pretty hectic for me lately, trying not to make mistakes, etc.  If you do this stuff, it's for the love of it, and, as the saying goes, you "don't quit your day job".

Does that make me "old school"?  If so, I'm honored by the description.

'Nuff said, I think...

Happy computing!


* D32ld1.GIF (6.23 KB, 320x200 - viewed 431 times.)

* D32ld2.GIF (11.66 KB, 320x200 - viewed 429 times.)

* D32ld3.GIF (2.68 KB, 320x200 - viewed 432 times.)

* D2048ld4.gif (20.79 KB, 320x200 - viewed 442 times.)
« Last Edit: January 21, 2011, 04:25:33 PM by fracmonk » Logged
fracmonk
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« Reply #40 on: January 24, 2011, 02:38:06 PM »

(Via bunny express)

For the formula in post 38, initial z=0 is critical for d=-1 and d=-2, and in those cases, will yield connected index sets.

The 1st pic below is an index map of non-escaping points for init. z=0 for d=-1.5, and is NOT connected.  The second shows a Julia set from it, for c=.25+.029i, d=-1.5, which is PROBABLY connected.

The 3rd pic is the index for d=.25 for init. z=0, also NOT connected.  The last pic shows the Julia set for c=6+6i, in which the origin point also does NOT escape, but the set is NOT connected.

In these cases, z=0 apparently can ONLY indicate the presence of non-escaping points in a Julia set for fractional d, but NOT its connectedness.

My skills give me no innate ability to find critical points, so I'm a little lost there...in degree 8, there are supposedly 14 of them.

Any suggestions?*

Obviously, I've been looking at correspondences between values for d in this formula and those for c in f(z)->z^2+c.
Next time, a look at d=-1.75 (a strange case, so you'll find me there...).


__________________________
* In a worst-case scenario, it could be that an intersection of ALL index sets resulting from ALL 14 critical points for a given fractional d may be needed to determine simple connectedness in their Julia sets.


* 1inc.GIF (3.83 KB, 320x200 - viewed 407 times.)

* 2jpc.GIF (7.62 KB, 320x200 - viewed 409 times.)

* 3inc.GIF (5.43 KB, 320x200 - viewed 406 times.)

* 4jnc.GIF (8.71 KB, 320x200 - viewed 415 times.)
« Last Edit: January 24, 2011, 07:25:48 PM by fracmonk » Logged
fracmonk
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« Reply #41 on: January 25, 2011, 02:46:03 PM »

(pix via bunny express)

In the M function, c=-1.75 is @ the cusp of its largest mini.  W. the function I've been discussing here, staying w. init z=0 the index set for d=-1.75 is notable for these properties:  1. Its relatively large size, given its distance from the origin, compared to the index sets for many lesser d distances from the origin.  2. Its largest M2 shape points left, as opposed to the "polarity" of the index set for d=-2.  3. It perfectly preserves the shape of standard M in every detail.  4. It has an infinite collection of smaller separate satellites that themselves share these properties 3, 4, 5, & 6.  5. Its escape set is one-piece multiply connected.  6. Julia sets for non-escaping z=0 have their own corresponding set of similar properties.

(3,4,5,&6 are a stab at a kind of set theory poetry...)

The 1st pic below shows the whole index set at .03x magnification.  The next 3 sample a zoom series, keeping the destination coordinates at the center of each, that being (just barely) inside a (distant) satellite M2.  Mags. are .1x, 100x, & 1000000x, respectively.

Next time, a look at the Julia set for the coordinates.  


* 1n175w.GIF (3.06 KB, 320x200 - viewed 391 times.)

* 2zisn1.GIF (4.81 KB, 320x200 - viewed 403 times.)

* 3zism2.GIF (3.68 KB, 320x200 - viewed 394 times.)

* 4zism6.GIF (24.09 KB, 320x200 - viewed 402 times.)
« Last Edit: January 25, 2011, 07:26:04 PM by fracmonk » Logged
fracmonk
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« Reply #42 on: January 26, 2011, 02:32:32 PM »

(pix via bunny express)

The 1st 3 pix in this Julia series zoom in at the origin from the whole set in pic 1.  Notice how locally, the shape is specific to the coordinate location shown in the last pic in my last post.  The last pic below shows this again in a randomly chosen satellite away from the origin.

For 3d fans:
Considering d as strictly real, stacking its incremental 2d index sets for complex c with initial z=0 constructs a 3d object whose limits are not finite:  d=0 covers the entire plane; and any large d still has a prisoner set, however tiny.  This set should be connected as a solid only over those 3 dimensions. In other words, for it to be connected would require all 3.  Similarly, complex d would then yield a 4d index set.

Next time, just one more look at another satellite, this time, closer to the "continent".  See ya...


* jcm0x2.GIF (2.36 KB, 320x200 - viewed 377 times.)

* jcm4x2.GIF (9.92 KB, 320x200 - viewed 381 times.)

* jcm6.GIF (72.3 KB, 320x200 - viewed 385 times.)

* jr1.GIF (23.81 KB, 320x200 - viewed 385 times.)
« Last Edit: January 26, 2011, 04:20:08 PM by fracmonk » Logged
fracmonk
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« Reply #43 on: January 27, 2011, 04:24:44 PM »

(pix via bunny express)

On the date of the post, yesterday, the library was closed due to a storm, but bunny express got thru.  The pix were right, but the labeling was wrong, my fault.  The pix were corrected.

The 1st is mag. 1x, showing the large size of the continent for d=-1.75 again.  Mag. 10000x in the second pic shows a 'cloud' of satellites that betrays its location in the 1st pic in the way embedded Julias might in other functions.  The 3rd pic, mag. 100000000x, is from the approximate middle of the formation, as pic 4 attests.  Note the clouds at every scale.


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* n175s6I4.GIF (7.69 KB, 320x200 - viewed 341 times.)

* n175s6I8.GIF (19.3 KB, 320x200 - viewed 339 times.)

* n175s6I9.GIF (29.28 KB, 320x200 - viewed 336 times.)
« Last Edit: January 28, 2011, 07:25:45 PM by fracmonk » Logged
fracmonk
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« Reply #44 on: January 28, 2011, 02:46:42 PM »

(pix via bunny express)

In the Julia set pix below, the 1st shows the whole Julia for the coords in my previous post.  Not surprising is the cloud shape of J satellites surrounding the origin in the 2nd pic, mag. 20x.  In the 3rd pic, mag. 1000000x, the local feature at the origin is an intact Julia one would expect from the last pic of the previous post.  The center of the last pic, mag. 100, of the cloud in the 2nd pic here, contains a local feature similar to the one shown in the 3rd here.

Though I prefer connected sets, as stated before, there is still one more weirdie you should see.  It's kind of a "jackelope".  For those unfamiliar, a rabbit with antlers, a favorite old joke among taxidermists...and maybe a new one for a certain very large, very litigious genetic engineering company that shall remain nameless.  Hint: they steal DNA, splice it into things they probably shouldn't, patent it as their own, and sue, for instance, farmers when their fields get polluted by the pollen from it, when it shows up in THEIR crops, as if they WANTED that somehow.  Nice guys...

...got good genes?  Maybe you'll need to pay them a licensing fee if you decide to have children...

Anyway, jackelope, next time, be good...



* n175s6Jc0.GIF (2.58 KB, 320x200 - viewed 362 times.)

* n175s6Jc1x2.GIF (5.87 KB, 320x200 - viewed 355 times.)

* n175s6Jc6.GIF (19 KB, 320x200 - viewed 361 times.)

* n175s6Jr2.GIF (15.56 KB, 320x200 - viewed 362 times.)
« Last Edit: January 31, 2011, 07:32:41 PM by fracmonk » Logged
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