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I just found a way to produce nice patterns, it's pretty simple and without using complex numbers.

I took a little from Samuel Monnier's Ducks, and another little bit from Tglad's ballfold, but I think is still an original "creation" ;)

Using inside coloring method (I use "exponential smoothing"), the iteration is:

x=abs(x)
y=abs(y)
m=x*x+y*y
x=x/m+cx
y=y/m+cy


In "Mandelbrot" mode (cx and cy equal to the point coordinates), this is the result:

(http://img813.imageshack.us/img813/6193/81216182.jpg)


A close up of the border:

(http://img827.imageshack.us/img827/4894/12419702.jpg)


The outside looks interesting, but insides looks like random noise, wich with lower iterations turned to be just curved lines with irregular spacing. So this noise is caused by aliasing.

The good stuff comes when picking some "julia" values (cx and cy being a constant value for all points)

This is cx=-0.5, cy=-0.5 :

(http://img716.imageshack.us/img716/5465/33064422.jpg)


cx=-0.2, cy=-0.1 :

(http://img850.imageshack.us/img850/8650/24102730.jpg)


Let's do a zoom on the last one, because I think I've seen this "entangled trees" before... :)

(http://img801.imageshack.us/img801/3183/73173618.jpg)


More examples, always just changing cx and cy only, and choosing some colors:

(http://img546.imageshack.us/img546/1921/95457548.jpg)
(http://img684.imageshack.us/img684/4855/27128103.jpg)
(http://img714.imageshack.us/img714/711/61423058.jpg)
(http://img708.imageshack.us/img708/5814/52478376.jpg)


Ok, enough examples, try it and see the variety of patterns it produces!

One thing: the number of iterations and color density must be constantly adjusted to get good results.


I'm trying more variations based on this (i.e. adding scale factors to alter the patterns), and I will also try 3D later.

To be continued...

Excellent discovery!!

Thanks :)

This is some great stuff, congratulations :)

Thanks, Brutaltoad... For 3D fans like you, I'm about to try this in Mandelbulb 3D using Luca's transformations, let's see what happen!

good work!

beautiful! well done

Nice system. Attached a test render.
The second one was made by it square cousin: z = abs(z)/(z.x*z.y) + c;

In this period I am busy, anyway Mandelbulb3d don't like all inside formulas, so have fun with 2d version ;)

Thank you for your comments, and thanks Syntopia for that renders, looks nice!

For you, Luca: I've managed to do this in M3D, using the transforms _absx, _absy, _absz, _sphereinv and addC

(http://img691.imageshack.us/img691/1427/kalibox1.jpg)

(http://img33.imageshack.us/img33/8104/kalibox2.jpg)

Julia values are : x=-0.3  y=-1   z=-1
The second one has a cut at z=0, y=0
Iterations and bailout values had to be tweaked.

I'll do more with better resolution later...

And I'll ask Jesse if he can make an optimized formula for this!

This are the parameters for M3D:

Mandelbulb3Dv16{
N.....Y/...E4...I....26..............1.......s1E................................
........................................I.2........../..................y.2.....
................/M.0/....6k3/...00....E2.....M8bjMjLw3nD/..........c./...wX/OaNa
z.UaNadD....02kAnAnAnAnoz0........zj........kz9..................y1...sD...../..
.w1...sDYsAIxzzzjznam5zggaB2zstjuFZpIEljTcFzUvMS3t9VsNWPqD6uzkplDbe3FGyDMaO9c9iB
OwXT/M/eK27tzqx3Mkax1WyD......YVC.....................sD.2kz0...................
..............................LRR6.0qpV.UOL50UnRR6.orpV.cVL50.ESR6..............
................0....wzzz1UL.22.H1...g1...kD....r1...w2....F....8/...cXz..UWGV2Q
...U.yzzzzzzzzzz./6Uz16U.0M..c../DtQ.6UkKZMxkayDPt41pmERLz1............68.kzzzD.
z.BC4POwozvV2RiGMQZtz0..........36W0zzzzz1UjJZPmF4WQzEislFs.YNxj3wLYcXY8NzH.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1....................................aBT..Do21.rslP6R1.P/YB.YFGa728.gnB
q.Esn3bBT.EEUA/.8xLM7t1.mlKO.UqTVl36.Uk/3..WzNqET.EX0a5.WyrGqI2.6Mk/.QwTpxH6.EaD
e.UvzRoBD/UYAG7....y3q/.zz/k.1A.yz1yAT2.xzpaqa9.................................
E....M..........zzzzz1......iFLNbJaQU.pPrJaQ.........................c..........
...................0./........zj................................................
................................................................................
.....................2.....3..../....wJEWB5K.656ExqRZ75.........................
........................................kz9.....................................
................................................................................
................................/....E/...E.....T3YMnZ3.........................
................................................................................
................................................................................
..........................................E.....I....2....kL/7qQO/..............
................................................................................
................................................................................
.....................................................2.....3....2....wpQkVKNmJKO
iN5.Vl4.........................................................................
..............zD................................................................
................................................................/....E/...k.....
T34NYB2.iR4QgJrQ.c3.....................................................kz1.....
...wz.........zD........kz1........wz.........................................zD
.........................................................................../}

That looks nice, i make a formula tomorrow...

 :-*

Please add scale and minscale for the ballfold, I'm playing with them in 2D and works nice! Thanks!!


---


For who want to try in the 2D formula I posted, replace:

x=x/m+cx
y=y/m+cy

with:

if m<r
   x=x/(r*r)+cx
   y=y/(r*r)+cy
else
    x=x/m*s+cx
    y=y/m*s+cy
endif

Then play with s and r values as you do with mandelbox

New renders:

(http://nocache-nocookies.digitalgott.com/gallery/7/3869_02_05_11_5_49_41.jpeg)

(http://nocache-nocookies.digitalgott.com/gallery/7/3869_02_05_11_3_04_26.jpeg)

Thanks for sharing the formula.  Always good to have something new to try out.

Here are a few quick samples I found.

(http://farm6.static.flickr.com/5148/5678472255_02b7e213bf_o.png)

(http://farm6.static.flickr.com/5308/5678472683_afa84ef83d_o.png)

(http://farm6.static.flickr.com/5028/5678473413_52c47807b6_o.png)

(http://farm6.static.flickr.com/5101/5678474437_76cc15e5d8_o.png)

Do you have a name in mind for these types?  Other than Ducks Alternative Kali Variation or something similar.

Jason.

Very nice images, Jason, great coloring.


I don't know... I'm bad at naming  :embarrass:

My idea comes from something that Samuel Monnier, creator of Ducks, said about the "entangled trees" patterns. But I don't think it's a Ducks variation, because Ducks uses complex number and complex log function, something that also makes it impossible the translation to 3D. I only borrowed the use of abs function for "mirroring".
It's more close to mandelbox, because abs function is like folding negative values to the positive side, combined with sphere inversion, wich is Tglad's ballfold, but without the conditional statements.
In conclusion, I blended two ideas into a single and simpler one. But no idea how to name this! :)

Ok, I know it's pretty vain... but I'm gonna call this "Kaliset"  :embarrass:

Or maybe plural "Kalisets", because the interesting things are the sets of Julias.

Man, you are killing me  :)


So r stands for minscale?

One good thing about the mandelbox is the limiting of sphere folding above 1, so there are no big changes of scales and rendering is smoother.
The sphereinv without limitations brings much noise into the render, i guess because the scalings change between iterations and maybe 1 iteration is skipped for next bailout condition... or something.  Maybe one can improve normals calculation by keeping the same iterationcount, like in the 2 or 4 point DE?

Nevertheless, i had already the simpler version tested, and the results differ from the 'handmade' formula with addons, because the bailou condition is only tested after all transforms.
The bailout value itself is also very crucial for the result, but i have not tested the new method yet.
Values of 2 downto 1 have a grfeat effect on the image, at least for 3d.

ok, to much talking...

ps:
the mandelbox compares m (=x*x+y*y+z*z) with sqr(minscale), should it be if m < sqr(minscale)?

I have made a little twist to four formula and ported it to the Quadray dimension..here how it looks at julia point c=(-0.8,-0.1,-0.8,-0.1)
(http://www.rfractals.net/share/DragonDimension.png)
The render is ugly, but it was fast made..
As always, just bare escapetime algorithm..

@Jesse: Many thanks you are working on this. To be honest, I'm doing the 2D ballfold as I read in one post I don't recall now, but feel free to try what you think it works best!

@Aexion: Just wonderful... Thanks! The Quadray system is really nice!

I've just realized that when I was doing a combination of what I call Mandelbrot on real numbers (wich is just x^2+cx and y^2+cy), with a ballfold (sphere/circle inversion), I was actuallly doing something like this, because instead of abs function I was turning negatives to positives with the exponentiation. The results of this are pretty distorted but nice anyway.

This one uses that method, colored with exponential smoothing:

(http://nocache-nocookies.digitalgott.com/gallery/6/3869_07_04_11_3_35_57.jpeg)

Wait a minute... look at this:

z=abs(1/z)+c

Using complex values, Julia mode, with c.real = -0.2 and c.imag = -0.35, the result is...

(http://img833.imageshack.us/img833/7508/complexkaliset.jpg)

 :o (well, not so surprised, I did it because I was expecting this to happen)

z=abs(z^-1)+c

So we are dealing with kind of an "absoluted negative power mandelbrot" <- good name!  ;D

Yet only the one i already made:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944 (http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944)

To get rid of some noise you could try the 'Normals on Zbuffer' postprocessing.

The complete box spherical folding looks to much boxlike, i think.  The benefit would be a fast analytic render method, maybe there is another way to get something more new.

I integrated the Formula into my rendering tool. Called it "Kali DuckKaliset"  :)

(http://yv3.bplaced.net/gallery/yFractalExplorer/24929.gif)

Hmm, lost a bit the track, but 3d version of this...

Mandelbulb3Dv16{
O.....Y/...33...S....26....oN3vOMZmzzuS1D/Vv0U/EkqdgPVeRJwnzan4M.7MmzmqPs76KncyD
................................nL6o6m2AuzXaNaNaNaNyz...................y.2.....
................/c.0/....6EA3...Z/....E2.....kGLB165peoD/c...............w1/BnAn
z.EnAncD1...02EnAnAnAnArzeNaNaNaN4yj........Ez9..................y1...sD...../..
.w1...sDX.HnySNEax1..........YLlWvwRv6djbh47MXeC0uHR6INU9OoMzwIsNela.NpD/SWBc6AZ
DuXzyS55P0YJzay3rzrW4BqD......ow8...U0................sD..E.0...................
..............................MRb7.4qRa.kOrN0UoRb7.srRa.sVrN0.FSb7..............
.....................w7e21EM.s3.c....E4....D....60...M4...E3....2/...M1...k/r.22
...U.eaKtwdOOZ1eD2XG21.3U.6.1c..zrhe.AWAZRBEi7yjX6HNL1YPWz1..........Ek.8..Xo/2.
4352Zs9U4y1CJrLV1AqJzU1qXIyGntwj36.3xzzzz1EFOhytTU7vzMyTadEdO/xDcXUU3BPr2zP/lcEz
uUXA.27bqQf.DuyD5fIDN9w5Fz1FK16adRghzKEA8oDU.06.uaKaZ6Aeiz1x58nQJ6TnzcSRpBK.iAwj
/6V0z5FAZ............oklrlkCx9yjW9wTJHz2Vz1..I6Vo/.DuI1.bxER7C7.G6/3.Yg5VBrU.kbU
70UOhQtVS0E4KY/.1aHWt75.a3qL.QRF7abQ.MKMT/k6G74VY/kN2W4.Pt3Rf/7.9ck0.gbO7StN.gl4
J.UqqxrXb/.Of/4...EMg75.eeWRAW2.IJJMg75.wybRAW2.................................
E....2....E.....I....6....kGVlKOnJ4Rn31.rJaQ....................................
..........UaNaNaNaNwzMWdge8/qarA................................................
................................................................................
........................}

@Jesse:
Thanks for the formula for M3D, I will try it later cause I'm not at home...
And also the params you posted... Is that the complex formula I mentioned translated to triplex numbers?

@yv3:


No problem with the name :) - It shares the Ducks concept anyway, the main difference is that mine has a direct 3D analog.

It is the "Kalisets1" formula i made, of course  :)

Ah, ok... looks nice!

I get it wrong, because you quoted my post about the "z=abs(1/z)+c" version... maybe using triplex math this could be good also ;)

Also I asked Sam for permission on naming it KaliDucks, not really important but change it if you can  :embarrass:

I don't want to even try installing M3D in this netbook, so I can't wait to get home and give it a try :)

That makes it much simpler for other people to code who are used to complex number maths.

A variation on that is z=log(abs(z^-1)+c) which also leads to interesting structures.

(http://farm6.static.flickr.com/5067/5681948386_e42660df6c_o.png)

(http://farm6.static.flickr.com/5029/5681947170_096ea440fe_o.png)

Jason.

Another variation z=log(sin(abs(z^-1))+c)

(http://farm6.static.flickr.com/5102/5682307930_252e39d9fe_o.png)

(http://farm6.static.flickr.com/5026/5682308462_9b8b390c41_o.png)

Jason.

@Softology: Very good variations... they are more close to Ducks, because of the log function, but the neg power makes the difference. Nice images.


@Jesse: tried your formula and works better than the custom I made and also faster, nice!

Still think about r parameter... I mean m<r part, and not using m<1...

Just this way:

if m<r
---
---
else
---
---
endif

Did you try somthing like this and it looks such the same as Mandelbox with all julias?
Asking this cause I get different stuff in 2D

Wow Got to try the MB3D formula out!
Great work Kali & Jesse!!!  :beer: :beer: :beer:

Well, it is the same formula:
so


There is already an 'abs' version of the power-2 Mandelbrot - the Burning Ship Fractal.
So your fractal is actually a power -1 burning ship :-)

By the way, I think the similarity to the Ducks fractal is mainly due to the folding (abs) operator. You will find similiar motifes using folding (abs), scaling, and rotation - and many of these are also found in the 3D KIFS of Platonic Solids, which also use such operations.

I logged in to post exactly that but you did it for me! :)

Thanks for additional info... I know a little about KIFS, very interesting stuff.

@Kraftwerk: Thanks, but keep in mind is quite different from others M3D formulas, you must use julia mode with negative values, bailout values between 1 and 3, and maybe do some cuts with some Julias, or zooming out because this are space-filling patterns. Also I found difficult to navigate, do it with "fixed zoom and steps" mode. Have fun! :beer:

I added some rotation for this (a quick render, sorry, I'm on a netbook!  :embarrass:)

It seems there's a cold front coming  :D (Is not a post-work!  :hmh:)

Yes, looks like rain...  O0
Ill keep your tips in mind Kali, Thanks!  :beer:

The one you mentioned looks not that much like a common mandelbox, i will try it in the next days if i find the time.

Just looked at the origin mandelbox spherefolding because you could use the analytic DE method, what gives a 4x speedup.
But i guess the 1/z thingy has no such easy analytic method, it was just a thought to forget...

Well, I think I missed something, or this is totally unknown? (At least I did a google search on this and didn't find it):

The inside of the regular "burning ship" fractal is actually a collection of this kind of patterns!

I used exponential smoothing for coloring, as most of my 2D images, look:

(http://img215.imageshack.us/img215/936/burningship.jpg)

Zoom:
(http://img814.imageshack.us/img814/9853/burningship2.jpg)

This is a Julia sample:
(http://img31.imageshack.us/img31/8845/juliaship.jpg)

Perhaps nobody used before this coloring method with the Ship? Or maybe didn't find it interesting?  :hmh:
Who knows... please let me know if this is already known, I'm only recently become a "fractalist" after all  :embarrass:

the glynn julia shapes in the last image are fantastic!
*just a side note, continue discussing*

(Thanks Chris)

This is a variation with a bigger set of patterns... z=abs(z^2)+c (instead of abs(z)^2+c):


The first are in Google Earth mode:     <-  ;D

(http://img218.imageshack.us/img218/6461/gooearth2.jpg)(http://img850.imageshack.us/img850/6462/gooearth.jpg)


Nice structures:

(http://img855.imageshack.us/img855/4175/absz2b.jpg)


Emerging patterns:

(http://img576.imageshack.us/img576/5176/absz2a.jpg)


Gold patterns:

(http://img834.imageshack.us/img834/278/absz2c.jpg)

Actually burning:      :dink:

(http://img820.imageshack.us/img820/9475/absz2d.jpg)


And a beatiful Julia:

(http://img861.imageshack.us/img861/4884/beatifuljulia.jpg)



I think I should open a new thread on this!

This is not extremely surprising. If you take a random conformal transformation together with a folding operation, you are very likely to get some region of parameter space looking like this and producing standard Ducky-Thalis patterns. What I found special with Ducks is that some of the "Julia sets" are infinitely extended. This seems to be the case as well with the Kaliset formula you first proposed.

I'm glad to see more variations on this these patterns, and disappointed that in general inverse methods don't seem to produce clean attractors.

My surprise is because this is supposed to be a well known fractal and I didn't find any references on this patterns appearing inside of it.
I suspected this just because of what you are telling (I learned it from you on the other thread ;))

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

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

and I found this image in a Julia version.

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

I was exploring the burning ship fractal a moment ago (I will not open a new thread because it's actually related to my formula),
and I think this are minibrot-like structures among the patterns ("miniships"  :D)

I raised the iterations, and they are, indeed... also I raised more the iterations and color density, and there are patterns inside the miniship that is among the patterns of the main ship. So there must be miniships inside the patterns of the miniship that is among the patterns of the main ship...
... and also there must be patterns that... Ok enough!  ;D

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?

This is a zoom at a 100 iterations version

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):

(http://fc00.deviantart.net/fs70/i/2011/124/7/5/spiratually_by_fractkali-d3flfme.jpg)

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.

(http://farm6.static.flickr.com/5310/5689136960_c24df8ed54_o.png)

(http://farm6.static.flickr.com/5105/5688565943_f94d5bcbde_o.png)

Jason.

hooooooooo nice \o/  :o

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...

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

with Julia values: -0.85,-0.23


(http://th03.deviantart.net/fs70/PRE/i/2011/125/3/2/the_birth_of_a_pattern_by_fractkali-d3fn6kd.jpg)


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

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.

Those are nice variations...

(http://farm4.static.flickr.com/3652/5691999737_882960ddda_o.png)

(http://farm4.static.flickr.com/3510/5692570310_02109c767d_o.png)

(http://farm3.static.flickr.com/2063/5691999315_faa12c391a_o.png)

(http://farm4.static.flickr.com/3541/5691999611_8f20bac71f_o.png)

Jason.

Softology, those renders are simply stunning! Great :D

@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:

(http://img808.imageshack.us/img808/4947/ksetbrotsmall.jpg)

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.

(http://img862.imageshack.us/img862/2477/kalisetbrot.jpg)

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.....................................
................................................................................
............................................}

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

Please post a decent one  :embarrass:

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................
................................................................................
................................................................}

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

(http://nocache-nocookies.digitalgott.com/gallery/7/3869_06_05_11_5_35_58.jpeg)

Interesting, so you see the julia shapes before you choose a location for the julia seed?  Have to try it out.
Cool results so far!

I included now the modified formula with folding:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944 (http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944)

Thanks Jesse!! I attached a quick render of the new KaliDucks...  O0

I mixed Kaliset with Kaliducks, I'll render another scene in full size later.
I found that high iteration values helps with noise (>1000)

Using the stalks orbit trap also works for these when in 2D.

(http://farm6.static.flickr.com/5148/5701636333_feb94d380e_o.png)

(http://farm6.static.flickr.com/5222/5702205632_f20b8cc804_o.png)

(http://farm3.static.flickr.com/2099/5702207356_f69724c6a2_o.png)

(http://farm6.static.flickr.com/5067/5702206700_46e7f0d864_o.png)

Jason.

The 3D variant of http://www.fractalforums.com/index.php?action=gallery;sa=view;id=7198

m=x*x+y*y+z*z
x=x/m+0.05
y=abs(y)/m-0.04
z=abs(z)/m-0.16

Outside and inside view.

The black crescent in the center of the last one is a sort of entangled trees (third pic in the first message of this thread).

(Here I change the julia parameters to (0.05,-0.15,-0.24))

@Trafassel: Thanks for the renders... How did you implement the bailout condition?
I have some ideas to do with Gestaltlupe later, but I will really appreciate if you can add vb.net as you mentioned before ;)

@Jason: Nice traps, I tried some with good results also. Thanks.

---

Well, in the last days I have tried lot of things with variations from my original formula, and played a little in 3D also (you may see some of the images in the gallery). But today I found something that I wanted to show here because I think the images of this new fractal are pretty good and interesting. Also they are quite different from previous ones.

The formula is based on one of the mentioned in fracmonk's thread about his research on connected sets with coexisting multiple-power shapes.

z=abs(z*c+1)+1/abs(z*c+1)

(I added the abs function to the original formula)

This are some quick monochrome renders:

(http://img641.imageshack.us/img641/9118/003xc.jpg)
(http://img84.imageshack.us/img84/3651/004fcj.jpg)
(http://img193.imageshack.us/img193/47/005cf.jpg)
(http://img852.imageshack.us/img852/858/006fht.jpg)
(http://img233.imageshack.us/img233/1504/007ks.jpg)

Just a few samples, there's a great variety of shapes...

---

Later I decided to test the original formula (without abs), using also exponential smoothing coloring instead of escapetime (Why I never did that before with fracmonk's formulas?)... I was very intrigued about the strange bifurcation maps I made yesterday for this and other formulas (fracmonk asked me), so I expected to see something different and interesting with this coloring method, and I was right... but...
 I think I will open a new thread for this later... :)

Kali- Very glad you dragged me over here.  A supremely excellent find!  But I'm afraid that because of time constraints on me elsewhere this time of year, I'm so poor, I can barely pay attention...

Just VERY cool...

Later.

Some more good variations Kali.

A couple of samples I found when exploring.

z=abs(z*c+1)+1/abs(z*c+1) images

(http://farm4.static.flickr.com/3259/5714900360_bef1f5bbd6_o.png)

(http://farm3.static.flickr.com/2022/5714901012_1efe189327_o.png)

And a few without the abs() included
z=(z*c+1)+1/(z*c+1)

(http://farm3.static.flickr.com/2682/5714338997_aba728fb36_o.png)

(http://farm3.static.flickr.com/2186/5714901516_fd6234abd4_o.png)

Jason.

Adding log into the most recent formulas works too (brings them more back into Ducks level of space filling and complexity).

z=abs(log(z*c+1))+1/abs(log(z*c+1))

(http://farm4.static.flickr.com/3310/5715233524_a71af9caa4_o.png)

(http://farm4.static.flickr.com/3487/5715233824_3f9515ec7d_o.png)

z=log(abs(z*c+1))+1/log(abs(z*c+1))

(http://farm4.static.flickr.com/3272/5715234216_d6687ecc05_o.png)

(http://farm4.static.flickr.com/3291/5715234488_b71feecc61_o.png)

The strange things with these formulas is the "Mandelbrot" versions have no detail, so you don't get a mandelbrot template to find interesting julia coordinates within.  Unless I made a coding error these would have to be the first julia set variations that have no mandelbrot detail.

Jason.

Thanks for your contributions Jason, nice images you found ;)

I'm about to start a new thread on revisiting some escapetime fractals by using abs and inner coloring, I'm counting on you (and others who were interested) for further colaborations.


I guess that happens if you start Z initial value = (0,0). Try with Z=C, you should get the Mandelbrot version.

Thanks Kali.  Setting the initial Z to C did the trick.

Jason.

Can you give me another tip on how this coloring works?

Keeping track of the average imag(z)/real(z) is easy enough.  Total imag(z)/real(z) during the iterations, and then divide total by the iteration count after all the iterations for the pixel have been calculated.

How do you then index a color palette entry for the resulting average floating point value?  For the curent color methods I use colval=trunc(avmag*colscale)mod 255 to get the floating point value into the 0-255 color palette range, but when using the imag(z)/real(z) value I am not getting those awesome appolonian-like shapes you are?  The colscale is needed (I found) otherwise if you have a very smooth palette, you don't see all the details.

Thanks for any tips.

Jason.

Got it working.  I had to use the average of abs(imag(z)/real(z)).

(http://farm6.static.flickr.com/5215/5724672752_53b549e36a_o.png)

Jason.

Softology, Reply #73 looks like a solution of a special circle packing problem.

The are some sphere packing fractal ideas (i.e. some indra pearls), but most with a complicatd generation algorithms.


Cool finding,

Trafassel

Hi Jason,

It's simply the arithmetic mean of imag(z)/real(z), over all of the iterations.  One "trick" is the clamping: Ultra Fractal clamps negative values to 0 and I (in this image, not in the coloring) clamp values larger than 1 to 1.  Values between 0 and 1 are mapped to a ramp between white and black.  But I see that you've got it figured out already.  :-)

Kerry

Thanks, it was Kerry that found it.  The julia coords just happen to have that sphere packing result.  The rest of the julia sets don't seem to have the same patterns.  Still a very cool find.

I have Indra's Pearls on my to read pile to implement one of these days.

Jason.

Really fantastic renders, Kali!  And growing fast!  Kudos!

I'm going to try a FractInt implementation, since I'm so used to doing that...(& I'm old & in the way...)... 1st chance I get...

Later!

The corresponding video to reply #64 with changing julia parameters.

http://www.youtube.com/watch?v=1q45CN-3SuA

Nice and interesting video, Per! thanks.

PS: I'll be trying the new version of Gestaltlupe, thanks for the advice ;)

I'd like to see support for this in Mandelbulber.

Jason, I can't reproduce these using the formula provided. I mostly don't get space-filling patterns with exponential smoothing and 1000 iterations, and when I do they seem to be much noisier and less structured. What were the c parameters for those images?

OK, here are the images (I realised I messed up the order of images linked to with the formulas in the other post) and formula details.

(http://farm4.static.flickr.com/3259/5714900360_2e87604227.jpg)
Formula z=abs(z*c+1)+1/abs(z*c+1)
The C values are 0.288235294117647 real 0.842261306532663 imaginary.
This one uses "Dual Real<0" symmetry.  Before doing the usual complex math calcs each iteration mirror z by using if z.x<0 then z.r:=-z.r

(http://farm3.static.flickr.com/2186/5714901516_c457a98467.jpg)
Formula z=abs(z*c+1)+1/abs(z*c+1)
The C values are -0.370588235294118 real 0.872613065326633 imaginary.
This one uses "Dual Real>0" symmetry.  Before doing the usual complex math calcs each iteration mirror z by using if z.x>0 then z.r:=-z.r

(http://farm3.static.flickr.com/2682/5714338997_2fa7bcdf18.jpg)
Formula z=(z*c+1)+1/(z*c+1)
The C values are -0.347058823529412 real -0.508391959798995 imaginary.
This one uses "Full Tetrahedral" symmetry.  Before doing the usual complex math calcs each iteration perform the following code (which was an experiment with using the KIFS 3D fold techniques in 2D)
                 if(z.x+z.y<0) then
                 begin
                      x1:=-z.y;
                      z.y:=-z.x;
                      z.x:=x1;
                 end;
                 if(z.x-z.y<0) then
                 begin
                      x1:=z.y;
                      z.y:=z.x;
                      z.x:=x1;
                 end;

(http://farm3.static.flickr.com/2022/5714901012_2493ab63ba.jpg)
Formula z=(z*c+1)+1/(z*c+1)
The C values are -0.323529411764706 real -0.0834673366834171 imaginary.
This one uses "Octahedral" symmetry.  Before doing the usual complex math calcs each iteration perform the following code (which was an experiment with using the KIFS 3D fold techniques in 2D)
                 //octahedral
                 z.x:=abs(z.x);
                 z.y:=abs(z.y);
                 if (z.x-z.y<0) then
                 begin
                      x1:=z.y;
                      z.y:=z.x;
                      z.x:=x1;
                 end;
Also I have an option to raise z to a specified power each iteration.  In the above image the power is 1.11.  The others are power 1, so no change is needed for them.

Hope that helps.

Jason.

Thanks for the information. I'll check this out.

@Paul: you are doing too much iterations, try with <50
@Jason: Nice variations and images. And thanks for the mention in your blog  :dink:

I have tried a lot of variations with good results, like adding transformations, foldings, rotations, translations, and so on...

But this is a more simple and still nice add-on to original formula, just multiplying by a constant complex number:

z=abs(1/z)*m+c

The attached example is for c=-1-2i and m=1+2i

Also try z=abs(1/z)*c+m in mandelbrot mode for some given m value. Good for a "preview" of the patterns to pick up Julia values.

another apollonian packing

c=1
m=i

Kali-  Love your last one- it's a classic subject...wish I had time to investigate it.  Very frustrating for me...hardly have time for my my own fave.  What you're doing here looks very fruitful, though.  I'm v. happy 4 U.  I'll be in touch.

(http://farm3.static.flickr.com/2186/5714901516_c457a98467.jpg)


Ah, the original post didn't mention this step, only the z=abs(z*c+1)+1/abs(z*c+1) part.

However, though I now come closer with my own code, I still can't quite reproduce the above. Here's the closest I've got:

(http://nocache-nocookies.digitalgott.com/gallery/7/511_05_07_11_12_51_59.jpeg)

That's with 3x3 AA and no serious attempt to replicate the exact color scheme. And I don't know what center coordinates you used.

There's a resemblance to your image now, and space-filling structure, but don't see the "over/under bridges" aspect that is very pleasing in your image, so I'm not apparently getting the same exact Julia set. Which means something is still not exact.

The loop is


and the various complex manipulation macros are all definitely correct (a normal Mandelbrot comes out correctly, for starters).

Any more information you might have would be useful.

Here's another one, on the Y axis this time and with a slightly closer color scheme. I still see a resemblance, but not the over/under thing. I wonder if there's still a mirroring step missing somehow.

(http://nocache-nocookies.digitalgott.com/gallery/7/511_05_07_11_1_49_41.jpeg)

Your code does look right.

This is how I do it... the references to .x and .y are for the real and imaginary components of the complex number.
cone is initialised to real=1, imag=0

if z.x>0 then z.x:=-z.x;
tmpc:=cadd(cmult(z,c),cone);
tmpc.x:=abs(tmpc.x);
tmpc.y:=abs(tmpc.y);
z:=cadd(tmpc,cdiv(cone,tmpc));
CSet(z.x,z.y,cfrectangular);

The real (x axis) coords for Z go from approx -4 to +4
The imaginary (Y axis) coords go from approx -7 to -2

Jason.

Using that framing rect I still don't get exactly the same shapes:

(http://nocache-nocookies.digitalgott.com/gallery/7/511_05_07_11_11_32_00.jpeg)


OK so far...


Whoa, what's that?

...

On closer inspection, there is a clear parallel between my image and yours if one of them is flipped vertically. It looks like the coloring is what's not matching up. Yours has four-armed spirals where blue and brown alternate going around the spiral; mine have such in corresponding spots but they have blue in the center and brown outside instead. So either my implementation of exponential smoothing is going nuts or you used something slightly different from that.

Also, how many iterations did you use? That last one of mine used 70.

I've got it! I tried, for some reason, changing the coloring code to apply the divergent form of exponential smoothing (despite it being trapped points being rendered) and got nearly identical results to yours:

(http://nocache-nocookies.digitalgott.com/gallery/7/511_05_07_11_11_55_37.jpeg)

Now to explore this beautiful fractal world some more. :) Thanks!

  Awesome images (read the whole thread)!  Didn't even notice this thread until today.... wow.

  There are a few 3d BS formulas that Jesse coded into Mandelbulber (from an old thread)... don't recall if we took 'em to negative powers however.  Gonna take one for a spin now.  :D

Could the be modified to generate quazicristaline middle eastern like pattern?

http://en.wikipedia.org/wiki/Girih_tiles#Mathematics_of_girih_tilings (http://en.wikipedia.org/wiki/Girih_tiles#Mathematics_of_girih_tilings)

 2007, Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings (presentation 1974, predecessor works starting in about 1964) predating them by five centuries.

This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th century Persian scrolls. However, we have no indication of how much more the architects may have known about the mathematics involved.

Well, it looks similar, but not realy so. Maybe the angle?

Thanks a lot for this topic i'll definitely play with these kind of fractals :) .

Love this fractal !
Changing a little bit the c, and with a different coloring method i get (zooming and zooming) :

I think, this is the best formula ever, of corse exept the original mandelbrot;)
 Without your post I wouldn't found it in this garganthuan thread.

Do it have any differences compared with z=abs(z)/c+c; do it realy needs second abs? With just one abs it alsou generates some sloped pattern in mandelbrot mode.

@Asdam: Mathematically the formulae is not the same..
However a wide range of iteration formulae with Kali's abs folding function is showing interesting fractal patterns (a lot of theses "entangled trees" by the way).
So your formulae will also generate a fractal :) .

I've found a formulae for the general case of abs, letting new fractals to be explored !
http://www.fractalforums.com/new-theories-and-research/extended-kaliset-t10372/ (http://www.fractalforums.com/new-theories-and-research/extended-kaliset-t10372/)

Well,

first I want to thank everybody who commented here and all who were interested on this thread, I can believe the huge amount of reads it has! I want to credit again Sam Monnier and Tom Lowe, it was because of their works that I found this tiny formula and it's variations.

However, the 3D version was a bit dissapointing, I expected better results. One reason is that the formula included in Mandelbulb3D is not working very well, is also slow and not easy to handle.

I've got some good results anyway, I think the best is this inside rendering:


(http://nocache-nocookies.digitalgott.com/gallery/9/3869_05_11_11_9_38_28.jpeg)

Also Traffasel's made some nice videos using his own tool, "Gestaltlupe", with infinite patience I guess, because is very slow as the renderer doesn't use a DE system.

http://www.youtube.com/watch?v=1q45CN-3SuA&feature=plcp&context=C46764c4VDvjVQa1PpcFNV5yLIuyE6RrPDtu1buavJ2mzdNK8gGVM%3D

http://www.youtube.com/watch?v=o3H0iDTf9oU&feature=plcp&context=C4bc8a84VDvjVQa1PpcFNV5yLIuyE6RnSBBliD44Id-LEhM21W4SE%3D

(in the last one the kaliset is on the first part only)
I think on this images the bailout condition is not quite right, but they still show some interesting shapes.


I've tried several times -with very poor results- to render a 3D kaliset using Syntopia´s Fragmentarium. I couldn't find the right DE to be used, the unconditional sphere inversion is a problem... and with the conditional inversion is not Kaliset anymore, but more close to the box - that's why I called it the "Kalibox" (adding also a translation after the abs fold).

But today I had an idea, to use the 2D formula including 3D rotations. I was doing something like this but with the 2D renderer, using the 3D rotations to alter the results in 2D. In this case the 2D formula is applied on the x-y axis of a rotating 3D space using the DE-renderer and the Mandelbox DE method.

It's maybe just a curiosity, but I liked the strange landscapes this method produces:


(http://img215.imageshack.us/img215/6140/3drotated2dkaliset.jpg)


(http://img214.imageshack.us/img214/2381/3drotated2dkaliset2.jpg)

(I attached the .frag files)


Perhaps other 2D formulas can be "3D rotated" giving interesting results, I'll try later...

Regards,

Kali

Including the original z² + c?

I tried but the problem is the distance estimator, still don't know how to handle it  :sad1:

Hi guys

Trying to replicate the original post image in FractInt, with the following formula:


But all it gives me is the following image:

(http://www.needanother.co.uk/uploads/fract129.gif)

Bearing in mind that FractInt works with complex numbers in its formula parser, I wasn't sure if I had to split out the x and y (real and imaginary) and perform different calculations on each, or to leave them combined.

Also, I've had to guess what the bailout condition is.  Anyone like to correct me?  I would love to get to see this image properly.  Also, what other  settings (for example inside and outside colouring) do I need?

Thanks

Simon

There's an error in your implementation, replace the lines after

y = abs(y)


I don't know the syntax for constructing a complex number from its components in FractInt, hence the comment where code should be.

It is worth having a variable for limit as it alters the appearance of the image, the higher value the closer to a quarter circle it becomes.

@ Kali

do you think that this can be done also with M3D ?

(http://img215.imageshack.us/img215/6140/3drotated2dkaliset.jpg)

I forgot ...

Kali's original formula doesn't have a bailout and all colouring is "inner" and he likes to use exponential smoothing.

I implemented Kaliset using both Mandelbrot and Julia algorithms in Saturn & Titan as "Hybrid" using C++, the Mandelbrot form is below:


The Julia form add epsilon instead of c. The code fragment is used in the usual Mandelbrot anf Julia algorithm loops.

z, c, alpha, beta, gamma, delta & epsilon are all declared as LongComplex (std::complex<long complex>), complex_i is of course 0.0 + 1.0i, the pow functions raise a long double to a LongComplex power.

The transform function, applies whatever transforms have been configured, for Kaliset the transform is set to strip both the real and imaginary components of their signs. Saturn has several transforms defined for moving the value of z in the complex plane, rotating it about a point, changing signs of the components of z, scaling, circle fold in, circle fold out etc., any number of transforms can be configured in two sets, A and B, one set at a time can be applied to an iteration of the formula, the sequence in which the sets is applied is determined by a sequence string, "A" always applies set A, "AB", first applies set A and then set B i.e. the sets are applied to alternate iterations.

Here are some examples using "Hybrid" (Mandelbrot algorithm version), the last one uses parameters of 1.0 for alpha to delta and hence is the same as Kaliset.

(http://fc09.deviantart.net/fs71/i/2011/128/6/2/shades_of_grey_and_amber_by_element90-d3fvgzz.jpg)

http://fav.me/d3fvgzz (http://fav.me/d3fvgzz)

(http://fc04.deviantart.net/fs71/i/2011/128/e/c/decay_by_element90-d3fv9g8.jpg)

http://fav.me/d3fv9g8 (http://fav.me/d3fv9g8)

(http://fc00.deviantart.net/fs71/i/2011/126/b/d/twisted_by_element90-d3fpi7q.jpg)

http://fav.me/d3fpi7q (http://fav.me/d3fpi7q)

(http://fc00.deviantart.net/fs71/i/2011/127/8/b/beyond_the_fence_by_element90-d3fs1vz.jpg)

http://fav.me/d3fs1vz (http://fav.me/d3fs1vz)

Sorry but there's no unconditional circlefold transform in M3D...

maybe if Luca or Jesse are reading...  ;D


@element90 - thanks for replying for me when I don't have the time to do it myself :)
nice images btw

Fractint formula lacks colouring revealing patterns. Without additional colour method only z=abs(z)/c+c julias will give some result.

For so popular in FF formula(s) it is suprising, that there is no good Ultra Fractal implementation of this formula. There are just some formula, who needs to load another formula  :hurt:

I'm a latecomer to this, and I'm old fashioned as well - I still have more fun with Fractint than anything else.  So, I've followed along with this and tried to implement a Fractint version of the Kaliset fractal.  The original formula (z = real(x) + imag(y)) can be set up in Fractint's formula editor to come up with images similar to what was posted by Kali, but many of the others don't come up with anything like what was posted.  At least, not the way I've written them up.  Oh well.

In any case, in the process of debugging the Kaliset formula, I noticed how important the bailout amount was.  I created a short animation with the bailout varying logarithmically from 0 to about 170 (modulus squared).  Here it is - the pattern of the lakes is interesting, especially the way they suddenly appear.  This is not an artifact of having the number of iterations insufficient to meet a higher bailout - I calculated the image with 100,000 iterations and the lakes remain exactly the same as they are at the end of this video.

Ryan

http://vimeo.com/moogaloop.swf?clip_id=49573142&amp;server=vimeo.com&amp;fullscreen=1

To stay with Kali's original intent on his Kaliset formula (no bailout, using inside colouring), I took a stab at a Fractint version without a bailout.  Fractint has limited choices available to it for inside colouring, and not too many of them were interesting.  For the majority of the Kaliset Julias, using Fractint comes up with an image that is just too busy.  I started off animating a path using relatively high iteration count, and the images were nice, but there were far too many tiny strands and the aliasing was awful.  So, here's a stab with the iterations reduced to 64.  The path starts near the origin and heads off to the outer edge of the Kaliset along the negative X axis at a constant rate, with the Y values located barely on the negative side.  It gets very busy towards the end - the rapid bifurcations come ever quicker, so perhaps an inverse exponential rate of travel would slow things down and show interesting views.

Ryan

http://vimeo.com/moogaloop.swf?clip_id=49677881&amp;server=vimeo.com&amp;fullscreen=1

You can use any inside coloring you can imagine, with this trick:

First, in the loop section of your formula, if you're about at the iteration limit (say, one less) calculate the coloring you want and assign it to z as a real number. Something like


Then use inside=real coloring.

You can, for example, implement exponential smoothing that way.

Nice video!

I was about to suggest what Paul did, I used that trick for doing exp.smoothing with Fractal Explorer

Here's a small generalization for one of Kali's formulas (http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/msg31977/#msg31977) from earlier in the discussion: z = abs(z/c₁) + c₂

Example: c₁ = (0.5, 0.5), c₂ = (-0.8, 0.2)
(http://i1122.photobucket.com/albums/l540/Fractal_Ken/trees.png)

Coloring is by an exponential smoothing variation where the magnitudes of the z's real components are subtracted, rather than the magnitudes of the z's themselves.

Edit: Zoomed versions of this fractal are here (http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/parts-of-a-tree/msg54815/#msg54815).

Wow, excelent image Ken... such a clean render, good coloring. So you used only reals for the exp.smoothing? Nice result, I'll try that.

Thanks, Kali. The image is sharp because I was able to do 100 iterations.  On most of these non-escaping fractals, that many iterations leads to a noisy looking picture, but not here. The orbits are somehow well-behaved; it's probably related to the tree structures.

I often use just the real components or just the imaginary components.

these are fantastic!

i made some webgl animations i find them quite mesmerising to watch  :crazyeyes:

• z = abs(z)/m + c (http://glsl.heroku.com/e#5654.6)
• z = abs(z /(c1 + c2))/m + c (http://glsl.heroku.com/e#5748.1)
• z = (c + mouse) / abs(z)^2 + c (http://glsl.heroku.com/e#5749.0)

(click the 'hide code' button and set the resolution drop-down to '1' if your video card can handle it)

Very nice, spongman! I should get into webgl to do some stuff... this will help me for learning, thanks for sharing!

thanks! although, be aware that you're only going to get medium precision floats with webgl, so no deep zooming. but it's good for demos.

one question, though. i don't quite understand why sometimes these animations get quite noisy (grainy), for example on http://glsl.heroku.com/e#5654.6 (http://glsl.heroku.com/e#5654.6) just after the background turns a black/white checkerboard. is that just the nature of the fractal given those parameters, or maybe something wrong with the way i'm doing the coloring?

Those are some really neat animations :)

Just thought I'd write in here and say that I (accidentally) found what turned out to be a rotation-ish of the Kali set. Well, it's actually a variation on the formula (arrived at accidentally) which, if the point is rotated by 90 before you add c then you get the Kali set, so I suppose this is a 'different' fractal:

ax = abs(p.x); ay = abs(p.y); k = x*x+y*y;

p.x = ay/k;
p.y = ax/k;
//if you rotate here by 90 degrees then you get the Kali set
p += c;

Enjoy  ^-^

EDIT: For some nice trees, try the Julia version of this formula at -1.08410,-0.21192 and its neighbourhood.

very nice.

i extended this to an arbitrary rotation:

z = abs(z . exp (i.theta)) / |z|^2 + c

here's an animation of the M-set with a continuously varying theta:
http://glsl.heroku.com/e#5742.1 (http://glsl.heroku.com/e#5742.1)


and here's a navigable julia of the same rotating/capsizing ship thing:
http://glsl.heroku.com/e#5758.0 (http://glsl.heroku.com/e#5758.0)

In wikipedia I found this fractal I never heard about, but it looks intriguing square pattern, somewhat like ducks/kalisets:
A Fibonacci word fractal by Samuel Monnier
(http://upload.wikimedia.org/wikipedia/commons/9/99/FWF_Samuel_Monnier_d%C3%A9tail.jpg)

Forumla/Link?

Hi,

The Fibonacci fractal was studied by Alexandre Monnerot-Dumaine, see this article:
http://hal.archives-ouvertes.fr/hal-00367972/fr/
Concerning my image above, I explained the idea behind the algorithm here:
http://algorithmic-worlds.net/blog/blog.php?Post=20090802
This algorithm essentially piles squares to draw it, but it is not the most natural way to draw it, see the article.

Best,

Sam

FIrstly I want to say how glad I am to have found this thread today, while started some time ago it is a great reminder of the fractal genius that lis within the forums here.....being an artist and not a mathematician, I love the work you have all done here and really find it very stimulating. One day I hopwe I can steal some stem cells from Pablo and many others here to try to understan all this and implement it in my art.

Friday April 28th, 2013 9:15 EST, my hat is off to you all

and this might be interesting as well, a collection of sams formulas in ultrafractal wiki:
http://ultrafractalwiki.fractalforums.com/Gallery_for_Samuel_Monnier%27s_Formulas

Random? -   Lucky
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Ed Algra uploaded some new version of Ducky a Lucky. Sounds as random numbers involved but it kind of interesting coz it started with ducky.