fracmonk


« Reply #15 on: December 17, 2010, 02:37:29 PM » 

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The 1st pic below squares the original function seen in the very first pic in this thread. You may have seen things like it in McMullen's work.
The 2nd is *new*, I think, and significant. I couldn't believe my eyes when I first saw it. Nowhere in the literature have I found a suggestion that multiple power shapes could coexist on the same plane. There was only various multibrot generalizations by themselves. Take the function used in the first pic, square it, as in pic 1, and make b=1 real. So simple, and the essence of fractals is (simplest formula)+(iteration)=(complicated results). Kind of a philosophy of mine, so that my part is no more complicated than it needs to be. This example is multiply connected. Others are not, and instead simply connected, also as single objects.
Over a year ago, I first began developing these expressions and doing some decent documentation, along with many hires pix, that unfortunately won't fit under the rules about posting on this site.
For now, I'll just let it soak in a little, and show more later.
Happy Computing!


« Last Edit: December 20, 2010, 07:24:39 PM by fracmonk »

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matsoljare
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Posts: 197


« Reply #16 on: December 17, 2010, 08:07:04 PM » 

I still want to see the Newton formula with power varying by one of the axis, as well as Supermandelbrot and Superjulia versions of the Nova and Phoenix formulas...



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fracmonk


« Reply #17 on: December 20, 2010, 02:44:05 PM » 

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I'm reposting b1d1M.gif in 320x200 below just in case some of you have trouble w. the 640x480 (true aspect ratio) format in my last post. If you do get such problems, please let me know, ok?
The last pic in my last post began a zoom series that contines with the last 3 below, keeping the target coordinates in the ctr. of each. Minis in the main object are dominated in size and quantity by 4th power shapes (M4). These are usually in the ctrs. of embedded julia shapes that also suggest 4th power (J4). An infinite amount of the more familar 2nd power minis (M2), generally smaller, are found similarly at the centers of J2 embedded shapes. All are found in a netlike structure, whose character, as a background for them, reflects its locality in the set.
If anyone knows this object to be previously discovered, or any that do some similar things, please let me know. Structurally, this set is of a class more complicated than standard M. I, for one, would like to see it more widely known.
With some parens., I've clarified the formula in previous posts. FractInt is tempermental enough doing division as it is...


« Last Edit: December 20, 2010, 07:23:34 PM by fracmonk »

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fracmonk


« Reply #18 on: December 21, 2010, 02:36:34 PM » 

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The first 3 pix below more or less finish the zoom series. The last picture is an almost whole version of the Julia set for the center of the zoom, with z=0 at the ctr. of the pic. Julias under this function are symmetrical but hollow and offcenter on the plane. There is another critical point, z=4, but I'm getting ahead of myself. More pix to come...


« Last Edit: December 21, 2010, 07:11:29 PM by fracmonk »

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fracmonk


« Reply #19 on: December 22, 2010, 02:39:58 PM » 

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A closer look at z=0 is shown in pic 1 below, mag. 4x. Notice the shadow embedment in it, due to its proximity to a larger mini (see b1m3.gif) shown early in the previous zoom series. Next is mag 1000x. The 3rd shows z=4 at its ctr., mag. 10000000x. Notice that its central feature is a bit offcenter. Also consider that z=0 is always a preimage of z=4. There are also many J4 features within it, as in the last pic.
An alternative function, f(z)>((z^2)c+1)+(1/((z^2)c+1)) yields an IDENTICAL index set to this one, but the Julia sets for IT are arranged differently. More on that next time.
(Your input on this is very welcome. Feel free at all times!)
Later...


« Last Edit: December 22, 2010, 04:20:37 PM by fracmonk »

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fracmonk


« Reply #20 on: December 23, 2010, 02:36:09 PM » 

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For the alternative function mentioned in my last post, the whole Julia set looks like the 1st pic below. A closer look at the J4 feature @ ctr. is shown in the next one. The last pic shows z=2 @ ctr., greatly magnified. In both functions, nonzero critical points do not center the features there.
I see things we can do with complex numbers iterated on the plane as infinite geographies to explore, and this one is quite rich. Next time, there are a few more details to cover before moving on. This story is still nearer its beginning than its end, so hold on tight...there are surprises.
Don't miss the formula files available below, from which all the pix in this thread were produced. If you're an old hand with FractInt, you'll know what to do to look around at this object. Helpful comments precede the list. To read, use any primitive text editor.
Gotta take time out to thank my Bunny for so graciously uploading for me. It never hurts to have someone who believes in you!
Our Best to you for the holidays! Later.


« Last Edit: December 23, 2010, 07:24:55 PM by fracmonk »

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fracmonk


« Reply #21 on: December 28, 2010, 03:52:33 PM » 

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The 1st pic below is of a greatly magnified M2 mini, and the 2nd is a detail of the corresponding Julia set for the coordinates at the ctr. of the 1st pic, using the type 1 formula. Also magnified, it has at its ctr. z=0, which shows the quadratic (J2) shape that can always be found there. So what happens when the ctr. coordinates are within a quartic (M4) form, as in the 3rd pic (same mag as 1st pic)? In the 4th pic, you can see what I call a "hybrid" quadratic, for lack of better terminology (same mag as 2nd pic). Note that such shapes do not exist in Julia sets of standard M, notated for our purposes here as M2.
For easier study, I was determined to find a formula for the same effect (quadratic and quartic M shapes in the same object on the parameter plane) in a simply connected form, but I was totally unsure if I had sufficient skill to do so. I fooled with it algebraically, and will show the results of that in the next episode of this continuing saga...


« Last Edit: December 28, 2010, 07:43:19 PM by fracmonk »

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fracmonk


« Reply #22 on: December 29, 2010, 03:01:45 PM » 

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The oddity of M4 shapes in an M2 general structure (1st pic below) is obtained with the formula f(z)>((z^4)c)+1. For any M generalization in this form, f(z)>((z^n)c)+1, I like to stick w. integer n.
The 2nd pic was found in my algebraic searching, getting close, and worth showing: f(z)>(((zc+1)^2)1)^2. Notable for Julialike features in an index set.
The 3rd pic has the formula f(z)>((((zc+1)^2)1)^2)1, and was exactly what I was searching for! M4 & M2 in a single simply connected object! I was enormously satisfied to have found it, in March of 2009. Was I first to do so? (Please advise!) Notice the repetitious nature of the formula, which aids in further inclusive generalization. More about that next time...
DO NOT MISS part 5 of the documentation I was making of it some months later, also below. It's an easy read, but contains references to hires pix that go w. it, but will not fit in this site's format.


« Last Edit: December 29, 2010, 04:31:29 PM by fracmonk »

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fracmonk


« Reply #23 on: December 30, 2010, 02:38:28 PM » 

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The 1st pic below contains M8, M4, & M2 shapes and is obtained w. the formula f(z)>((((((z^2)c)+1)^2)1)^2)1. The shapes may be discerned better in the left (major) antenna detail in pic 2. Zooming in on the antenna ends revealed an interesting number x=2.55377397403003... which satisfies x(x2)=sqrt(2). In fact, practically ALL the most wellknown mathematical constants can be somehow found at work in this particular set, including pi (the M2 bulbs), e, and the golden mean.
I had long wondered why these functions that inclusively generalized powers of 2 in these sets all shared w. standard M the exact same (famous "San Marco") Julia set for c=1, as in pic 3 for the 842 formula.
But then, JUST YESTERDAY, while gathering pix for this entry today, I reexamined the nesting of the formula within itself that produces these inclusive generalizations. I had long thought of the formula structure itself as a fractal. I then wrote more "iterations" into it, enough to generate a 512degree index set shown in the last pic.
Consider the calculation of pi to billions and trillions of digits for world records, when so few digits of precision are required to describe a circle as wide as the known universe to the accuracy of a human hair. Here, it is equally absurd to claim "the most complicated object in mathematics" when the record could be broken daily.
But WHOAAA!!! I realized, as you can see in the last 2 pics, that as these index sets increase in degree, they more and more resemble San Marco!
So I could confidently conjecture that San Marco is the equivalent of an index set with this formula structure taken to infinite degree.
You never know what you might find...so keep looking!
Happy New Year!!!


« Last Edit: January 15, 2012, 11:00:54 PM by fracmonk, Reason: 1152012! (the \"c\" in the formula) pretty crucial! ...my bad, as they say... »

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fracmonk


« Reply #24 on: January 03, 2011, 02:30:42 PM » 

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Shown below are minis from the antennae of the 512degree inclusive generalization function at various magnifications: M2, M4, M8, &M16. If you have already explored Mgeneralizations, you already know that the higher the degree, the larger and more crowded the minis appear. Here, higherpower minis seem to dominate lowerpower minis in both size and number in the space. Still, there is an infinitude of every power shape, from 2^1 to 2^9 powers, and their arrangement is complicatedly hierarchical.


« Last Edit: January 03, 2011, 07:13:14 PM by fracmonk »

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fracmonk


« Reply #25 on: January 04, 2011, 02:35:00 PM » 

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The progression of minis I began in the last post can be continued up to M512, but it's pointlessly tedious (for me, anyway) to count all those tiny lobes...
Below instead begins a zoom series into it that keeps the destination coordinates at the center of each picture. Currently, my machine is grinding away at a hires version of the last of them.
Does anyone know of a formula parser that goes beyond floatingpoint precision? Too often, I'd like to go a bunch deeper, and hit the wall. Have you ever run into formulae that you feel deserve a deeper look, but you just can't go there?
I speculate that it would be a programming nightmare to try to provide such a thing. Does anyone know, with any specifics, why I haven't heard of it yet?


« Last Edit: January 04, 2011, 07:17:20 PM by fracmonk »

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fracmonk


« Reply #26 on: January 05, 2011, 02:39:44 PM » 

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The zoom series continues below, augering into a spiral that doesn't end...but not for long, since it uses up a lot of that precision I spoke of in the last post. I found a destination point that had 9digit precision, since fixed variables in the FractInt formula parser cannot be specified beyond that, and later I will show the Julia for the center coordinates there as well.
I'm belaboring the study of these (pretty randomly picked) coordinates because as an example, it will help show the dynamic structural details of this entire class of inclusive generaliations. I hope to demonstrate their importance as a new find. Hope you like things even more intricately complicated than M...


« Last Edit: January 05, 2011, 04:19:54 PM by fracmonk »

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fracmonk


« Reply #27 on: January 06, 2011, 02:42:10 PM » 

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The zoom series reaches its end with the 3rd picture below, in a lobe of an M16 mini viewable whole in the 1st picture. The coordinates remain at center, and are nonescaping.
The whole Julia set for those coordinates is seen in the last pic, w. a J32 feature at its center, z=0. I'm looking for a pattern that can predict the Jn at the center of each respective set of maximum degree, and haven't worked it out yet. (As of 111:) For any of these functions of degree 2^n, Julia set center features are of power shape 2^(n+1)/2.
I use an older machine that might be slow by current standards. The hires version of the 3rd pic here, mentioned while in progress in a previous post, took over 30 hours! The detail is rewarding, however. Though I forget the exact procedure, so long ago, I had to retrieve a 4line file from the XP OS collection and install it to get 32bit DOS emulation. I notice that "improvements" imposed by MS only make it harder to do good fractal pix as time goes on...lo res pix like these at least get done faster.
Any ambitious viewers doing any of this stuff on their own now?


« Last Edit: January 12, 2011, 04:48:29 PM by fracmonk »

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fracmonk


« Reply #28 on: January 07, 2011, 02:42:42 PM » 

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The series below begins a zoom into z=1 in the Julia set, a critical point for which z=0 is always a preimage. Once again, we see that nonzero critical points are in no way visually obvious. Scaledependent features seen in the index zoom are again encountered here in the Julia set at appropriate depths, as will be seen in the continuation, next.


« Last Edit: January 07, 2011, 04:11:50 PM by fracmonk »

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fracmonk


« Reply #29 on: January 10, 2011, 02:42:21 PM » 

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At the end of this series, we find that z=1 is inside a lobe in a J16 feature, not unexpectedly. The feature is offcenter, as are all @ nonzero critical points.
When the degree goes very high, those higherpower minis tend to eclipse the lowerpower ones both in number and size. Next time, ways to track the lowerpower ones down "from afar".


« Last Edit: January 10, 2011, 07:14:18 PM by fracmonk »

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