Very simple formula for fractal patterns

[s31415]

By the way, maybe you can answer this: why the "Kaliset" formula is not a map of this patterns in "mandelbrot" mode?

Mmhhhh it looks like it is, no?

[Kali]

By the way, maybe you can answer this: why the "Kaliset" formula is not a map of this patterns in "mandelbrot" mode?

Mmhhhh it looks like it is, no?

I attached two images of the Kaliset Mandelbrot mode after 10 iterations... concenctric curves inside the negative part of the plane (where the julia patterns are), and bifurcations touching this lines in the border (just noticed the nice effect of exp.smoothing on this with low iterations).

More iterations makes more and more packed lines, and you will see noise because of aliasing... because zooming enough into the noise patterns reveals it's only this lines... no patterns visible, as they are in the regular burning ship, why?

[Kali]

This is a zoom at a 100 iterations version

[s31415]

Strange, I though that the pattern was simply very stretched, so appeared as lines. To check it, I took the log so that the circles appear as lines and then tried to stretch them back. But it turns out that these are really lines... So I don't know right now.

[Kali]

Strange, I though that the pattern was simply very stretched, so appeared as lines. To check it, I took the log so that the circles appear as lines and then tried to stretch them back. But it turns out that these are really lines... So I don't know right now.

I thought the same at first, but now with this check you did there are no doubts they are just lines. Thanks.

Pretty strange, I should check for Julias on some formulas I've tried before and were discarded for not being interesting in "Mandelbrot mode"
(is another way of saying this? I mean Julia & Mandelbrot modes?)

This is a Julia of z=1/z+c - so it's the same transform of my formula without abs (the 'folding' part):

[Softology]

I tried a version of the kaliset and found an Apollonian packing.  In my formula, I used the simple one-statement loop:
z=1/abs(z)+c

Nice circle packing result.  When I try z=1/abs(z)+c I do not see any areas like that.  If you do have the Julia coordinates I would be interested in them too.

To confirm, the abs(z) is just setting the real and imagnary components of z to their absolute values?  For each pixel z is set to the pixel coordinate and c is constant.

These are the type of Julias I get at the moment.  Unique results, but no apollonian like shapes.  I realised the first one uses a fractint palette file that you created.





Jason.

[ker2x]

By the way, maybe you can answer this: why the "Kaliset" formula is not a map of this patterns in "mandelbrot" mode?

Mmhhhh it looks like it is, no?

I attached two images of the Kaliset Mandelbrot mode after 10 iterations... concenctric curves inside the negative part of the plane (where the julia patterns are), and bifurcations touching this lines in the border (just noticed the nice effect of exp.smoothing on this with low iterations).

More iterations makes more and more packed lines, and you will see noise because of aliasing... because zooming enough into the noise patterns reveals it's only this lines... no patterns visible, as they are in the regular burning ship, why?

hooooooooo nice \o/ 

[Kali]

No time for pics right now, but try out z=abs(z)^-3+c and z=abs(z^-3)+c - very interesting results
And also try fractional powers, no discontinuities at all... Is nice for altering the results of a Julia by varying it in small steps.

Will post something more later...

[Kali]

z=abs(z)/abs(c)+c

with Julia values: -0.85,-0.23





Also try z=abs(z)/c+c and z=abs(z/c)+c

[lkmitch]

@lkmitch: Very nice, what julia values? (if you saved them )

The Julia parameter is -1 + i.  I think the big difference is how the image was colored.  If you use Ultra Fractal, it was my Statistics coloring, coloring by the mean of imag(z)/real(z).  I've noticed that this brings out different structure from colorings based on the magnitude of the iterate.

[Softology]

No time for pics right now, but try out z=abs(z)^-3+c and z=abs(z^-3)+c - very interesting results
And also try fractional powers, no discontinuities at all... Is nice for altering the results of a Julia by varying it in small steps.

Will post something more later...

Those are nice variations...









Jason.

[DarkBeam]

Softology, those renders are simply stunning! Great

[Kali]

@lkmitch: Nice coloring method

@Softology: As Luca said, great renders


Well, I discovered a new strange property of the original formula. In order to obtain the Mandelbrot map of patterns, we must swap real and imag values of C after starting the iterations (or cx, cy in the real number version).

This is the resulting image:



The link for bigger version: http://img405.imageshack.us/img405/7984/ksetbrotbig.jpg

And a quick render of a cut on the 3D version, M3D parameters below.



Mandelbulb3Dv16{
O.....Y/..kj1...Y/...26...U2RTYjKu/wz6xBTpeF6C1Eb4CbVvXVnzfJA29gL2owzKNEjSkNa3.E
................................bGojcFOZ7.2........yz...................Y.2.....
................/M.0/....6Uw2...12...2E2.....Ur/XjZXKhnD/..........c./...w1/OaNa
z.UaNadD1E..0..........wz0........zj........kz9..................y1...sD...../..
.w1...sDVCRV06.tCxfi1fjVZTjHzkh/E4fkjMfDOLaYwsSZBxXjuEH27tLHzUwoRdGMibnDGfOa6YDH
mw18sR7.awZAz2e90J9CXHpj......Yq5.....................sD.6kz0oAnAnAnAnyjBnAnAnAn
gz9...........................MR27.4qFY.kO5F0UoR27.srFY.sV5F0.FS27..............
................/0...Axmf4kJ.g2......I3...EB....M....E3....F....8/...k17...UJtwG
...U.ShYcNxqCHeo....d0....66.c...06U.w9UCW0tRhyDMo0FIhT67.2..........E268.EFwI0.
tSnlX3Gflzfz46r6GUPwzSOz/kQckFxD/6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................ULS00EJJd3.zzFTqJ5.MG7Z.szD3SsS.sHD
t.EzT7MSo/k6Uw/.QsbVwR6.mhKQ..bT.ObR.oV4N..lydMTA0kn7vA.MwLVuZ6.Y2m6.kqTzhLS.606
T..kzFMUA0URoR5....cU08.zz/cU08.yz1cU08.xz3cU08.................................
E....A....E.....I....2....kL4lKOkVJKX/US.............................6..........
...................wz...........................................................
................................................................................
.....................2.....3..../....wZFgZ4QMZpM..kRZ75.........................
0...............................aM8feGUhtB1.....................................
................................................................................
................................/....E/...U.....934PdBLNoBLA....................
........................................kz1.....................................
................................................................................
............................................}

[Kali]

And this is also flipping Y-Z... a really quick render, I'm out of time!

Please post a decent one 

Mandelbulb3Dv16{
O.....c...UT/...m....22...kYOKbEC2Cxzm1qpZ79im4EI5y/Q6FKjzv6ixN7qu/vz0knbccNH5yj
................................11UGZGuwH.YaNaNaNaNyz...................Y.2.....
................/MU//....6Um/...R1...2E2......2lFu0.dLoD/..........c./...w1/BnAH
z.EnAnYD1E..0..........wz0........zj........kz9..................y1...sD...../..
.w1...sDmSlr0mPmYwXVuu8G/aKBzsPiq1/zheojFd7ONh2JAx1XuuDCLzU8zOeLJ3WzuQlD90M96nqW
XwfYEhfaxBaGz4jVl0E0JQnj......on......................sD.Ekz0oAnAnAnAnyjBnAnAnAn
gz9...........................MR27.4qFY.kO5F0UoR27.srFY.sV5F0.FS27..............
................/0...Axmf4kJ.g2......I3...EB....M....E3....F....8/...k17...UJtwG
...U.ShYcNxqCHeo....d0....66.c...06U.6UkKZMxkayDZBL9s6JkUz9..........I268.EFwI0.
tSnlX3Gflzfz46r6GUPwzSOz/kQckFxD/6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................ULS00EJJd3.zzFTqJ5.MG7Z.szD3SsS.sHD
t.EzT7MSo/k6Uw/.QsbVwR6.mhKQ..bT.ObR.oV4N..lydMTA0kn7vA.MwLVuZ6.Y2m6.kqTzhLS.606
T..kzFMUA0URoR5....cU08.zz/cU08.yz1cU08.xz3cU08.................................
E.E..I....E.....I....2....kL4lKOkVJKX/US.............................6..........
...................wz...........................................................
................................................................................
.....................2.....3..../....wZFgZ4QMZpM..kRZ75.........................
0...............................aM8feGUhtB1.....................................
................................................................................
................................/....E/...E.....TN2Pd/LKOB4.....................
..........U.............................kz1.....................................
................................................................................
..........................................E.....I....2....kL4lKOkZZKX/..........
.....................6..........................................................
................................................................................
.....................................................2.....3....0....gIMgZqQZFrQ
l.............................................................zD................
................................................................................
................................................................}

[Kali]

I couldn't wait and had to make this at work  - Sorry boss...