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Author Topic: Very simple formula for fractal patterns  (Read 47685 times)
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Kali
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Posts: 1138


« on: May 01, 2011, 07:33:14 PM »

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" wink

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:




A close up of the border:




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 :




cx=-0.2, cy=-0.1 :




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




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







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...


« Last Edit: May 01, 2011, 08:32:57 PM by Kali » Logged

Fractal Ken
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Proud to be 2D


« Reply #1 on: May 01, 2011, 08:10:20 PM »

Excellent discovery!!
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Fortran will rise again
Kali
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Posts: 1138


« Reply #2 on: May 01, 2011, 08:36:52 PM »

Thanks smiley
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Lee Oliver
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Posts: 314



« Reply #3 on: May 01, 2011, 08:37:17 PM »

This is some great stuff, congratulations smiley
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rte’pi twam na bhavisyanti sarve
ye’vasthitah pratyanikesu yodhah
Kali
Fractal Supremo
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Posts: 1138


« Reply #4 on: May 01, 2011, 09:46:35 PM »

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!
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yv3
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WWW
« Reply #5 on: May 01, 2011, 09:49:32 PM »

good work!
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knighty
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« Reply #6 on: May 01, 2011, 10:38:41 PM »

beautiful! well done
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Syntopia
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syntopiadk
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« Reply #7 on: May 02, 2011, 12:12:01 AM »

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


* kali.jpg (126.83 KB, 514x403 - viewed 1462 times.)

* square.jpg (131.5 KB, 640x409 - viewed 1876 times.)
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #8 on: May 02, 2011, 12:31:38 AM »

In this period I am busy, anyway Mandelbulb3d don't like all inside formulas, so have fun with 2d version wink
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No sweat, guardian of wisdom!
Kali
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Posts: 1138


« Reply #9 on: May 02, 2011, 12:33:17 AM »

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





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
.........................................................................../}
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Jesse
Download Section
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Posts: 1013


« Reply #10 on: May 02, 2011, 01:16:50 AM »

That looks nice, i make a formula tomorrow...
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Kali
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Posts: 1138


« Reply #11 on: May 02, 2011, 02:18:26 AM »

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

 kiss

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


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Kali
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Posts: 1138


« Reply #12 on: May 02, 2011, 05:51:42 AM »

New renders:





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Softology
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« Reply #13 on: May 02, 2011, 07:46:17 AM »

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

Here are a few quick samples I found.









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

Jason.
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Kali
Fractal Supremo
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Posts: 1138


« Reply #14 on: May 02, 2011, 08:18:16 AM »

Very nice images, Jason, great coloring.


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

Jason.

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! smiley
« Last Edit: May 02, 2011, 08:23:48 AM by Kali » Logged

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