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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 25165 times)
Description: I think there may be. Your opinions are greatly desired.
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fracmonk
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« Reply #180 on: February 15, 2012, 07:38:16 PM »

That one (end of last page) was from post 12, with my formula for it.  I almost didn't recognize my own shot!

The torch lights are actually inversions of infinity (I call them "infinity holes", and I believe they are centered on Misiurewicz [sp?] points, but am not sure) and the stalks are the usual dendrites, although they seem to have more regularity of shape than on standard M, oddly enough.
« Last Edit: February 16, 2012, 07:44:16 PM by fracmonk » Logged
Alef
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« Reply #181 on: February 20, 2012, 04:38:55 PM »

I just tried, how these super advanced 2D formulas would look in 3D.
Even power formula as expected produced just another quaternionic rotation surface.
z=(((((z^2*c)+1)^2)-1)^2)-1; (another proof of i=j aka y=z generating rotation thing)

But the result of odd power formula is pretty interesting one.
z= (((((z^3)*c+1i)^3)+1i)^3)+1i;
For an easyness and lack of quaternionic ^3 operator I rewritten it as chain of the formulas.

z=z*z*z*C+1i;
z=z*z*z+1i;
z=z*z*z+1i;

Vuala, here it is, a bitt strange horned quaternionic Brotosaurus.


* FM_brot_1.jpg (34.82 KB, 333x327 - viewed 231 times.)

* FM_brot_2.jpg (38.8 KB, 339x311 - viewed 233 times.)

* FM_brot_6.jpg (44.17 KB, 355x359 - viewed 245 times.)

* FM_brot_7.jpg (49.78 KB, 366x330 - viewed 238 times.)
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fracmonk
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« Reply #182 on: February 22, 2012, 04:48:52 PM »

Asdam-   I recognized the silhouette of your first pic immediately, since the connected 2d of it extends in probably all ways to the furthest distance from the origin on that plane, so none of the rest that is off that plane is in the way of seeing it.

I also program with the basic formula structure you showed.  You can see that it's most efficient, and I might hazard that that fact has to be related to whatever makes it appealing...
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fracmonk
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« Reply #183 on: February 23, 2012, 07:26:11 PM »

Asdam inspired me to go back to that formula, which goes like this in my file:

(initializing info)...
s=z*z*z*c-d
t=s*s*s-d
z=t*t*t-d
|z| < p3
}

The pix below show various details of the ci by di (2d) parameter plane slice subset of the 4d animal it actually is, where real c and d are 0, in this case.  In each closer view, the details are not surprising, given the degree 27 formula.  The last one betrays the fact that that particular 2d slice is NOT connected.  
I would suspect that the 4d hyperblob is connected over its 4 dimensions in one single piece, and most likely any 3 of those 4 in 3d, while the remaining one is set to 0.  By hyperblob, I mean the 4 axes that cannot be displayed at 90 degrees from each other all at once: a by bi by g by hi, where c=a+bi, and d=g+hi. (Not to be confused with any 4 dimensional single number of any kind, but just the combined parameters of a pair of complex numbers as the axes)

(Hope it's clear...!)

It was just out of curiosity-

Later...


* D27cidi1.GIF (3.74 KB, 640x480 - viewed 30 times.)

* D27cidi2.GIF (7.82 KB, 640x480 - viewed 25 times.)

* D27cidi3.GIF (25.54 KB, 640x480 - viewed 26 times.)

* D27cidi4.GIF (64.99 KB, 640x480 - viewed 37 times.)
« Last Edit: February 24, 2012, 12:16:13 AM by fracmonk, Reason: greater clarity » Logged
Alef
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« Reply #184 on: February 28, 2012, 05:37:16 PM »

I think in 2D now most in are interested in (low max iteration and exponent smoothing revealed) kaliset like pattern formulas.

First were simple formulas, then came long formulas, then all switched back to short universal formulas but with
sophisticated colouring techniques and layering. Now at least here in ff complex geometrical pattern formulas grabs the most atention. Entangled trees and somewhat arabic patterns, maybe it have something to do with middle east being at the centre of world politics;)

In UF mailinglists unit vector thing (thanks to Toby Marshall implementation with many functions) become quite an used, but mailinglist seems to be a bitt behind;)
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fracmonk
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« Reply #185 on: February 29, 2012, 05:33:14 PM »

I have few opportunities to post pix, but here's some of current interest, just done:

The first shows the whole "D27" index set, with the coords of the center of the second pic in it as well.
That spot is at the edge of the tip of an M27 bulb, and can be considered continental or non-dendritic.  The object has M27, M9, and M3 features in it.

The 3rd pic displays the whole julia, which has a J9 feature at its center at the origin.  It also has 3 large J27's near the ends of its major arms.

In a dendrite near its furthest extremity, the hybrid J3 shape in the middle of the last pic can be found.  Note that it is hard to pick it out of the field of shapes, since they sort of overlap as feature parts.

Later!


* FRACT029.GIF (14.32 KB, 640x480 - viewed 27 times.)

* FRACT028.GIF (113.49 KB, 640x480 - viewed 37 times.)

* FRACT027.GIF (46.24 KB, 640x480 - viewed 32 times.)

* FRACT030.GIF (158.92 KB, 640x480 - viewed 34 times.)
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Alef
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« Reply #186 on: March 02, 2012, 03:05:31 PM »

I was thinking that pattern sets being in current main 2D interest at FF, so this could be area of (2D) research.
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fracmonk
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« Reply #187 on: March 02, 2012, 04:27:49 PM »

Asdam,  "Pattern sets" meaning?

Later!
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Alef
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« Reply #188 on: March 06, 2012, 05:58:12 PM »

Like this http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/90/
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Alef
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« Reply #189 on: March 06, 2012, 06:07:14 PM »

I just implemented your formulas (as single formula) in Ultra Fractal database as Walbrot. http://formulas.ultrafractal.com/cgi/formuladb?view;file=em.ufm;type=.txt

Sorry that didn't asked a permision, just couldn't wait to share it;) (If you are strongly against I wll delate.) But credits and links to this thread is included and you name would be seen in every time one uses the formula (placed info just below formula switch). But woun't be able to change the name of formula to something more pleasing sounding. embarrass

Added few extra modifications (unit vector) and power 3 soap like fractal (z=1/(z^3*c+realD)+z^3*c+realD), coz it generates soap buble julias with david stars.

Found nice julia spot, throught extreme complexity of geometry makes a bitt hard to find right colour method. All good mathematical names alredy are taken, so just remembered that mandel is almonds, but walnuts have much more complicated shape. At least there is no walbrots.

It looks that division placed first, then multiplication and just then addition calculates a bitt faster.
So z=z/cabs(z) * @vfactor+z is faster by about 1/20 than z=z+ @vfactor*z/cabs(z)  
The same goes for z=1/z^2*c+d+z^2*c+d




z=((((z^2*c+realD)^2)-realD)^2)-realD and z= ((((z^3*c+imagD)^3)+imagD)^3)+imagD
with
z=z/cabs(z)  *  (-2)+z
Produced this. It have small tricorn fractal inside.

And two small power 3 tricorn fractals (crosses)
« Last Edit: March 06, 2012, 06:16:29 PM by Asdam » Logged

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fracmonk
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« Reply #190 on: March 06, 2012, 07:30:59 PM »

Then there must be univerality for the tricorn, too, under the right circumstances.

Sorry, stuck for time,

Later!
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Alef
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« Reply #191 on: March 06, 2012, 08:22:37 PM »

Well, tricorn is derived from mandelbrot (there is no aviable description why it generates that shape), so it could appear (noone knows why). tongue stuck out
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fracmonk
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« Reply #192 on: March 07, 2012, 09:26:38 PM »

Asdam,   I'm pretty sure it requires the conjugate version of z, or, at least, a "virtual conjugate", that would come about due to a programming detail that creates the effect by accident.


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fracmonk
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« Reply #193 on: March 07, 2012, 10:05:26 PM »

Let me take you back to nearly the beginning of this thread, where multiple power shapes first appeared in multiply connected form, that is, where the prisoner set is connected in one piece, but the escape set is not.  (See post 17)

Later, in post 23, the "San Marco" effect came about when the degree was really high, so that the iterations in the formula made it look like that particular Julia set (for c=-1), only it was a second constant, d=-1, that was taking its place, that was repeated more than the parameter being mapped as an index set.

In post 45, one picture, MC842, showed how multiply connected sets could be made by using a z=z+(1/z) instruction after so many z=z*z-d instructions and just before test.

In the first pic below, a long string of just such programming gives a degree 2048 index set with 7 (corrected 4-23, originally reported as 11, see post 208 for explanation) different power shapes in it.  It is multiply connected in one piece, and takes on the San Marco shape overall, with traceries of the mixtures of different powers in daisy chains. To merely identify some of them (for instance, M128 against M64), one would have to make multiple enlargements to tediously count how many lobes each has.  These preliminary pix zoom in a little for an idea of its details.

The 3rd pic shows M8, M4, and (a tiny) M2 in a succession from right to left, while the last shows the tiny M2 a bit larger.

More later, but not sure when...


* M2048d1.GIF (18.82 KB, 1024x768 - viewed 24 times.)

* det2.GIF (36.56 KB, 1024x768 - viewed 22 times.)

* det3.GIF (37.29 KB, 1024x768 - viewed 25 times.)

* det4.GIF (34.83 KB, 640x480 - viewed 27 times.)
« Last Edit: April 27, 2012, 07:00:24 PM by fracmonk, Reason: specific correx » Logged
fracmonk
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« Reply #194 on: March 08, 2012, 07:32:13 PM »

I've noticed that when doing Julias for this family, especially edgy ones, false periodicity detection makes errors that fill in otherwise interestingly detailed areas.  The <G> command line fix is periodicity=no.  In these latest multiply connected INDEX sets, it is necessary for the first time to do it there, too, at some magnifications, so that errors don't give low quality pix.  That and the added division step realllllly ssssslowwwwwww things dowwwwwwnnnnnnn!

But I think the results are worth it.

Later!

« Last Edit: March 08, 2012, 07:50:58 PM by fracmonk, Reason: helpful household hints » Logged
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