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

[Alef]

Another bloated Mandelbrot and almoust the same equation. From distance it looks the same exept have angle in its rear;) But non-analitical abs function does it's job quite well, it have triangles on antenna and square box minibrots on it's stalks looking like old TVs with antennas.

Code:

else if (formula == "MandelTeleBrot")
{
z= z^2+c;
z=z - 0.3*z/(abs(real(z))+abs(imag(z)));
}

Switching -0.3 with -1 and having julia seed x= 1.009, y=0.5 reveals julia set of angled spirals and almoust squares.

Works well in another powers, exept that squares are not so accurate as rings.

[Alef]

Downloaded the latest UF package and found this implemented in UF. Thanks Ikmitch, nice implementation, default cos function in formula creates very floral patterns. Was implemented just before the squares;)

Scratching mandelbrot fractal revealed these very long julias. 2nd and 3rd are as they are (r=-3.2, i=0.5), 1st (r=-1.95, i=1.45) is rotated to fit in the screen.

p.s.
Hmm, I allways tought mandelbrot an julia fractals as some sort of pair, I was disapointed realising that Julia actualy are some frenchmen Zhuly.
 

[tit_toinou]

(via bunnee mail)

The 1st pic below contains M8, M4, & M2 shapes and is obtained w. the formula f(z)->(((((z^2)+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(x-2)=sqrt(2).  In fact, practically ALL the most well-known 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 8-4-2 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 512-degree 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!!!

Hi.

First, if this is true, this is something awesome.
Second, that is not a mandelbrot-like iteration formulae. Where is the "c" variable ?
That formulae (and the other you gave in the http://www.fractalforums.com/index.php?action=gallery;sa=view;id=6995) does not work (i tested it!).

When i iterate over a polynom of z, then add c i never get smooth mandelbrot (except when i have z^n + c of course).
Your fractal seem to be awesome...
I want to explore theses Mandel-8/4/2-coexisting brots !
Thanks .

[fracmonk]

tit_toinou-  Terribly sorry to have only found out about a notation mistake I have made (& fixed) before!  It is f(z)->...((((((z^2)c)-1)^2)-1)^2)-1...  generally.  I'll fix it where you quoted it from too.  With some review, I noticed that post 38 generalizes it in another way, maybe a bit clearer to some, depending on notation style...                -And thanx for pointing it out.

[Alef]

I put together talis and tricorn. Could not find the name, but bumblebrot would be OK.
z= flip(sqr(z*z/(1+1i+z)+c))

Or easyer to see:
z= z*z/(1+1i+z)+c
z=z*z
z=flip(z)

Zoom of this reveals everything alredy seen spirals, ^4 mandelbrots and ^4 tricorns, but owerall shape is quite an interesting and exactly the same as in quaternion numbers. This don't remaind talis, but it have 2 kinds of nonsymmetric julias, smooth ones and branched.

[fracmonk]

Asdam-  I wonder if you examined your tricorn-like object far enough to determine whether it's connected.  I can already see that the Julias you showed aren't.  Pretty wild, in any case.

Later.

[Alef]

There are connected julias. At large it mostly resemble normal power 4 tricorn, and it looks all small features are connected. Here are another julia:

[fracmonk]

Very cool. Remember those are 2-d, that's generally harder to accomplish.

Later.

[Alef]

There alredy are  tonns of 2D stuff. Each Fractint formulas file contains 50 entries;)

But I think, there are gap as most of Fractints formulas are initialised by z=pixel. Thi is first iteration of mandelbrot starting z=0, but some formulas don't like beeing started with z=pixel.

[fracmonk]

Asdam-   So I was wondering-  don't you do your own formula files?  Then you can assign your own init. z, and try different ones.
Depending on the function, with some, zero won't work at all.  You need to find out at least some of the critical points.

[Alef]

I once coded some formulas in Fractal Explorer, and it allowed just z=pixel initial z. But looking at Fractins files most of stuff are initialised by z=pixel. I think, most of people are pretty laisy and just copy paste;)

*   *    *
Going back to Unit Vectors.
z= formula(z)
z=z+factor*z/function(z)

If function is cabs and the factor is some -0.3, Mandelbrot set have circles on antenna. Cabs function defined using power 8 instead ow 2 generates pillow like squares with round edges. So modulus function with power 8 are different than with power 2 ( (real(z)^8+imag(z)^8)^0.125 ) Not very different, but is different. The same thing appears in insides.

[fracmonk]

I'm making pix again, finally, w. various forms of Fractint on 2 machines.  Interesting story about that in Help and Support here, in "Still need help with DOSBox in Windows 7".

While I can foresee FEW opportunities to post pix, if I do nothing else, I will illustrate post 153 in this thread here.  I promise it will be SO cool...

Later.

[yv3]

Hello all,

as i tried to integrate Asdams formula into my tool i found another formula that produce intresting images:

z = z + ((c2*z)/z.sin()) + c;

This is an example of the Julia iteration method:



And this video that i made in 10 minutes describes how the change of c and c2 affects on the image:
<a href="http://www.youtube.com/v/3H0zjJwGnEk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3H0zjJwGnEk&rel=1&fs=1&hd=1</a>

I called this Formula "rotten apple"
Greetings

[fracmonk]

pix via bunny express

The 1st pic is of most of the index set obtained by BOTH formulae in post 153, where, in this case, d=2.  Results are identical.  It is centered on the same coords. as the 2nd pic, which is a 100,000,000 magification of the first.  If you were to zoom from the first to the 2nd, you would first go into a filament of an M4 mini on the antenna into an M2 there.  Then, just missing it in the vicinity of its "seahorse valley", you would continue to the left tip of the M4 in the 2nd pic.

The Julia sets for both formulae are shown in pix 3&4.  The 2 sym. version has a J2 at its center, and the J4 corresponding to the coords. at its non-zero critical points at 4 and -4.

The 4 sym. version has a J4 at its center, and others at 2, -2, 2i, and -2i, in similar correspondence to the index set coords.

Look closely, and you will see how the Julia sets could be "morphed" into each other's shapes, preserving the structure of each.  Both have infinite numbers of both kinds of minis in them, but the higher power minis dominate in both size and number in most views at any scale.  Zooming into their non-zero critical points reflects a journey similar to the one taken in the index set.

Stay out of trouble with your prof. if you are doing a paper on these Multipowerbrots, by kindly acknowledging their discoverer, Jeffrey Barthelmes, and this thread in Fractal Forums, in your footnotes.  Thanks.

 

[Alef]

Apple Ipod;)

Actualy this brot in the first page

looks exactly as Fractal Explorer's (R.I.P.) Talis formula with the same torch lights on stalks. A pretty popular formula, number of times reimplemented in UF and once mandelbulb 3D.
init z=0; z=z*z/(complex+z) +c;
Or formula found in Fractal Science Kit "Tails" init z=1; z=z+1/z+c; Maybe  'Talis' is coined from "Tails", probably a sheep fat tails.