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Author Topic:  Some old stuff, maybe fractal nostalgia  (Read 6060 times)

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Offline Alef

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Re: Phony Mandelbrot with satelites
« Reply #45 on: November 07, 2021, 07:12:54 PM »
This is formula by Pauldebrot. Equation looks like kind of multi powered Lambda or Chebishev. Closer examination showed it is pretty explorable. It have some more unusual quadratic shapes, power 2 minibrots, branched julias like Adam Majewski renders and julia like satelites around main set, unconnected, repeating and with the minibrots at the center. This formula should be generalised.

Quote
Newsgroups: sci.fractals
Date: Thu, 4 May 1995 06:45:52 GMT

This Julia set looks like a bunch of Mandelbrots! ; Close examination shows this to be illusion.

The formula file must be named 'pgd.frm' Note: this formula can be slow. If you have fractint 19, use floating point.

This Julia formula (PhonyMandelJ) with the parameter set to 1 produces ersatz mini Mandelbrots! They are actually illusory. The (true) Mandelbrot formula is printed below with it, and a parameter entry showing the phony Mandelbrots.

Initialisation of parameter space / mandelbrot set:
Code: [Select]
C = pixel
Z = 1
f = 15/8

It have some arbitrary parameter of f = 15/8. I don't know a mathematical background of this but I tested some other numbers with magical properties like 108/72 or 12/7, alsou 1, 1.5 and 2. Result of 15/8 still was better. Alsou it have arbitrary - 0.25. Is it 2/8?

Formula:
Code: [Select]
Z = ( Z^4 / 4 - f*Z^3/3 - Z^2/2 + f*Z - 0.25) * C
Bailout:
Code: [Select]
|z|<=127Kind of large value for fractint.
p.s.
Sitting in covid curfew.
« Last Edit: November 07, 2021, 10:17:50 PM by Alef »
by Edgar Malinovsky aka Edgars Malinovskis.

Offline 3DickUlus

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Re: Phony Mandelbrot with satelites
« Reply #46 on: November 07, 2021, 10:30:13 PM »
p.s.
Sitting in covid curfew.

 ??? :P :-[ Be strong my friend, a difficult iteration.

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia
« Reply #47 on: November 07, 2021, 11:29:09 PM »
 3DickUlus: thanks. :thumb_up:
« Last Edit: November 07, 2021, 11:40:01 PM by Alef »

Offline Alef

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Re: magnet_minimand
« Reply #48 on: November 07, 2021, 11:39:25 PM »
Magnet like by Puskas Istvan jr.  E-mail: pataki.v@******.hu 

Initial condition??? :confused:
By searching for good corespondence to a julia sets I found a value:
Z = 0.1
But probably it have multiple critical points like this:
Z= #pixel + 0.1

Iteration:
Code: [Select]
Z=( (Z^2 + C) / (2*Z + C - 1 ) )^2 - 2*Z(slightly optimised)

Magnet or combined divergent && convergent bailout conditions didn't give any added value so |Z|<@bailout.

It is just a Z=Magnet(Z) - 2*Z but -2*Z kind of eliminates most of it's magnet's character. I removed C=(C-1) as here it only moves fractal around.

It have some julia sets with very elaborate holes maybe of second order (doubled herman rings?). Maybe it is too spiked and complex to have an esthetic value. Formula Mset alsou have satelites around main sets, they are like julia sets, exactly repeating in the zooms. I had same satelite shape at the magnification levels of 878050 and 8.584E12. Some julia sets at the same areas have them too.

Quote
magnet_minimand (XAXIS) {
z=c=pixel:
z=( (z*z+(c-1)) / (2*z+(c-2)) )^2-z-z
|z|<127
}
« Last Edit: November 22, 2021, 02:11:40 AM by Alef »

Offline Alef

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6D from Fractint! This fractal appeared in number of formula files by Jos Hendriks or Lee H. Skinner, pretty cool shape, kept during zoom on antenna, but alsou zooms are OK, mostly of julia sets, just that iteresting features are rotated what often don't fit in screen and they still are like of mandelbrot set.

Parameter set, zoom on antenna and 3rd dim slice of zHeight=-0.20


Julia sets
(-0.725, 0.135, -0.03, 0)
(-0.9, 0 , -0.17, 0) slice zHeight= (-0.14, 0)
(-0.8, 0, -0.117, 0) slice zHeight= (0.15,0)
(-0.75, 0, 0, 0) this seems to be central in many aspects
(-0.8, 0, 0.117, 0)



Russell Walsmith publicated triternion group 3D multiplication rules and rendered 2D slices of it using Fractint. Like:
z_x= ...
Z_y= ...
Z_z= ...
pdf is still downloadable throught a web archive https://web.archive.org/web/20040524032840/http://www.fibonacci-arrays.com/Triternions.pdf
It is just a rotated by 45 degrees extruded mandelbrot.

Then owning to an elasticity of Fractint's formula language small change to the equation (pixel_y * i) turned each 3D variable into complex equation. Thus came out 6D algebraic group where Z*Z orbit travels throught all the dimensions. I guess, maybe this fractal could lead to some deeper insights about universe, sutch as extra dimension parallel universes or ghostly extraterrestrials of little females.



I multiplied this on paper and made it algebraicaly transparent. Long but easy (repeating):

Initialisation (for this shape Y variable of pixel is put into Yi part):
Code: [Select]
Mandelbrot set
 Z_x  = 0
 Z_xi = 0
 Z_y  = 0
 Z_yi = 0
 Z_z  = 0
 Z_zi = 0

 C_x = real(#pixel)
 C_yi = imag(#pixel)

 C_z  = real(@zHeight) ;a variable, slice in 3rd dimension
 C_zi = imag(@zHeight) ;a variable, slice in 3rd dimension

Julia set
 C_x = real(@julia)
 C_yi = imag(@julia)
 C_z = real(@zJulia)
 C_zi= imag(@zJulia)

 Z_x  = real(#pixel)
 Z_xi = 0
 Z_y  = 0
 Z_yi = imag(#pixel)
 Z_z  = real(@zHeight) ;a variable, slice in 3rd dimension
 Z_zi = imag(@zHeight) ;a variable, slice in 3rd dimension

Iteration:
Code: [Select]
  tempZ_x  =   Z_x * Z_x + 2*Z_y * Z_z   - Z_xi * Z_xi - 2*Z_yi * Z_zi
  tempZ_y  =   Z_z * Z_z + 2*Z_x * Z_y   - Z_zi * Z_zi - 2*Z_xi * Z_yi
  tempZ_z  =   Z_y * Z_y + 2*Z_z * Z_x   - Z_yi * Z_yi - 2*Z_zi * Z_xi

  tempZ_xi = 2*Z_x * Z_xi   + 2*Z_y * Z_zi + 2*Z_yi * Z_z
  tempZ_yi = 2*Z_z * Z_zi   + 2*Z_x * Z_yi + 2*Z_xi * Z_y
  tempZ_zi = 2*Z_y * Z_yi   + 2*Z_z * Z_xi + 2*Z_zi * Z_x

Z_x  = tempZ_x    + C_x
Z_xi = tempZ_xi   + C_xi ;here c=0
Z_y  = tempZ_y    + C_y  ;here c=0
Z_yi = tempZ_yi   + C_yi
Z_z  = tempZ_z    + C_z
Z_zi = tempZ_zi   + C_zi

;make complex Z (for color methods):
  Z = Z_x + flip(Z_y)

bailout:
Code: [Select]
abs(Z_x) + abs(Z_xi) + abs(Z_x) + abs(Z_y) + abs(Z_yi) + abs(Z_z) + abs(Z_zi) < @bailout




There were other versions, less smooth. Here is the original Fractint formula, each variable is automaticaly turned complex by addition of the "*(0,1)":
Quote
somename { ; Jos Hendriks <***@hexaedre-fr.com>
                  ; Fri, 24 Jan 2003 21:38:21
 ; Lee H. Skinner <***@thuntek.net>
                  ; Wed, 22 Jan 2003 10:27:11
                  ;
  c1=real(pixel),c2=imag(pixel)*(0,1),c3=p1
  z1=z2=z3=0:
  t1=z1*z1+2*z2*z3
  t2=z3*z3+2*z1*z2
  t3=z2*z2+2*z3*z1
  z1=t1+c1,z2=t2+c2,z3=t3+c3
  z=(z1^2+z2^2+z3^2)^.5
  z < 16
 }
It is somewhat turned by 45 degrees, so without rotation often don't go well in screen but the result is interesting. Alsou it turns by 90 degrees, is have some connection to split-complex aka minkowski space - square mandelbrot set. Set must be disconected in 2D  becouse it is connected in higher number of dimensions. It appears, disapears and then repeats. 

Shapes into mandelbrot set shapes:


I would like to see what it is in 3D but so far I was too lazy to implement that. Maybe later if I have an inspiration.
p.s.
Quote
"Did the Mandelbrot set exist before computers were invented?"
« Last Edit: December 08, 2021, 08:16:06 PM by Alef »

Offline Alef

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Zoom on cuted out more mandelbrot like region of mostly square basilica like 6D julia set of (-0.925, -0.05833333333) realy a (-0.925,0,-0.05833333333,0,0,0). The fractal have some turns by 90 degrees and disapearances and behaves like partialy complex and partialy split complex. Set must go in higher dimensions. On further zoom on its spike intresting features disappears.



Here escaping and non-escaping regions are coloured with different methods with insides having fluid mathematicaly generated colours by exp and sin functions aplied on orbit values and outsides painted with slightly hairy textured gradient.

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia
« Reply #51 on: December 03, 2021, 11:46:06 PM »
Few zooms on julia sets of cross-dimensional 6D formula from the «Reply #49». Escaping and non-escaping areas are coloured by very different methods to keep them distinct enought, gradient and direct colour method. Different inside colours - different orbit paths.
 
Formula is old but it's julia sets is not explored. Maybe simmilar formula could be made in 4Dimensions, with some 2D number system and parallel i dimension for each ordinary dimension and y axis being i of the i.

Images.
- Pacman Fish, zoom of julia set (-0.925, 0, 0, -0.058, 0, 0), at the non-connected area next to bulb like arm:


- Julia set of (-0.75, 0, 0, 0, 0, 0), from the Mset point where "arms", "head" and "skirt" of the fractal connects:


- Zoom to the disconected pieces on a tip of an "arm":


- Zoom to the small disconected part on a tip of the smaller "arm":




Consecutive zooms into a julia set (-0.8, 0, 0, 0, 0, 0).
- Zoom at the non-conected structure near "open mouth" bulb on the "arm", another bulb with a "mouth":


- Zoom on the right bottom bulb of a 1st zoom:


- Zoom on the top bulbs of a 2nd zoom:



10 Ultra Fractal parameters with formula included:

Offline gerson

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Re: Some old stuff, maybe fractal nostalgia
« Reply #52 on: December 04, 2021, 02:22:19 AM »
@ Alef All images are beautiful, shapes and colors. Thanks for sharing the par.

Offline hgjf2

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Re: Some old stuff, maybe fractal nostalgia
« Reply #53 on: December 04, 2021, 08:43:23 AM »
Those recent fractals look like Julia set of classic tetrabrot and of hexagonal tetrabrot (intersect of 3 Mandelbrot cylinders not 2) and of octogonal tetrabrot (intersect of 4 Mandel cylinders).
Fractal researcher

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia
« Reply #54 on: December 04, 2021, 04:20:12 PM »
gerson: thanks : )

hgjf2: It is related. The whole 6D dimensional figure must be simmilar to tetrabrot in that it have square (split complex) and mandelbrot slices (but not at XY plane like of square tetrabrot). 

Lots of images seems to be slowing down a thread. Anyway, tetrabrot with julia sets

https://commons.wikimedia.org/wiki/File:Tetrabrot_with_Julia_sets.png


. . .
There are 2, 3 interesting formulas left I found in Orgform including very interesting newton. It just take time to research them or at least make them in some readable equation ; )
« Last Edit: December 07, 2021, 01:54:07 AM by Alef »

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia / Regula Falsi Newton
« Reply #55 on: December 08, 2021, 07:59:32 PM »
Pretty cool...
By Rui S. Parracho, ~ 2001 in Fractint. "Regula Falsi" translates as "False Rule" in romanian. Slightly more complex && elaborate newton like set, imaginary root coefitients curves threads nicely && imaginary initial conditions makes fractal non- mirrored on x axis. This infinite fractal is something in between julia set(newton) && Mset (nova), so there is no real parameter space || julia set distinction.






Initialisation:
Code: [Select]
Z = #pixel
C = #pixel
ZoldER = @seed

Starting "seed" value could be anything including (0,0). As "Seed" aproaches +/- infinity fractal takes it's furthest form.

Iteration loop:
Code: [Select]
Zold = Z

Z = Z - @root*(Z - ZoldER)*(Z^@power + C) / (Z^@power - ZoldER^@power )
ZoldER = Zold

by default:
@root  = (1.65, 0)
@power = 2

Imaginary value of "root" curves fractal. Imaginary || fractional powers - pure uglyness. Larger powers makes it more dense creating scenery like Pauldebrot's deep zoom.


Bailout conditions:
Code: [Select]
|Z - Zold| > @bailoutbailout must be at least 0.000000000000001 small to work with MDLD.


 *  *  *


; generalised "Regula Falsi":
; Function ( complex^2 + C )
; Z = Z - root*(Z - ZoldER)* Function (Z) / (Function (Z) - Function (ZoldER) )




Originaly:
Quote
Regula_falsi2 { ;Rui S. Parracho
  IF(|p1|==0), r=1.65, ELSE, r=p1, ENDIF
  IF(|p2|==0), z=pixel, ELSE, z=p2, ENDIF
  x0=(1,1), x1=pixel, oz=0
  :
  fx0=x0*x0+pixel, fx1=x1*x1+pixel,
  oz=z, z=z-r*(x1-x0)*fx1/(fx1-fx0),
  x0=x1, x1=z
  cabs(z-oz) > .0000001
  }


All was colored with Mega Discrete Lagrangian Descriptors. No antiaiasing just a printscreen. Quick UF parameters with formula is in attachment.
« Last Edit: December 08, 2021, 08:09:34 PM by Alef »

Offline quadralienne

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Re: Some old stuff, maybe fractal nostalgia
« Reply #56 on: December 14, 2021, 10:31:14 PM »
If you have 'beep' on a desktop PC with Linux, here's some nostalgia: beep -f 1047 -l 100 -n -f 1109 -l 100 -n -f 1175 -l 100

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia / Luna and Lyra Newton
« Reply #57 on: December 23, 2021, 01:49:27 AM »
Here are 2 pretty cool fractal formulas from (no longer aviable) Chaos pro database. (the original file is at the bottom  of reply https://fractalforums.org/share-a-fractal/22/some-old-stuff-maybe-fractal-nostalgia/4213/msg28576#msg28576) They are different but are in the same style so I united them together. I alsou made formulas in more standart shape so that they would work with most color methods.

Cool owerall shape, nice zooms and julia sets.

Quote
; Bernd Lehnhoff 2004 f.b.*******@gmx.de
;The following fractals were derived from Heron's formula for
;calculating square roots with an added perturbation.

Initial conditions:
Code: [Select]
IF (@settype=="Mandelbrot type")
  z = #pixel
  c = #pixel
ELSE ;julia set
  z=#pixel
  c=@julia
ENDIF
The further away (from 0,0 ) julia seed is, the more julia set resembles standart Newton fractal.


Iterations loop:
Code: [Select]
Zold = Z
;formula
IF (@formula=="Luna")
;Luna
  Z = Z - Z / @Power * (1 + @A*1i - ( C / Z)^@Power ) + 0.5

ELSEIF (@formula=="Lyra")
;Lyra
    Z = (@Power * Z) / ( (Z / C )^@Power + @Power - 1 + @A*1i) + 0.5
ENDIF
Default power = 3. With power = 2 it don't have newton like features in the zooms, it is like mandelbrot set spread ower crescent - just too boring to use.
@A = 0 but imaginary value helps to curve a set.
0.5 - is needed for fractal to work. Any real number exept 0 could be placed there. It just changes size of the set and 0.5 idealy fit into the standart fractal screen.

Bailout condition:
Code: [Select]
|Z - Zold| > @bailoutBailout value < of 1E-16 is needed for them to to work with MDLD coloring.


Luna have more newton like foam structures in the zooms:


Lyra have more rings and polynomial like features:


Alsou after the formula could be placed a modifications (works better with Luna):
Code: [Select]
; modifications
IF (@function=="None")
...
ELSEIF (@function=="Abs")
z=abs(z)
ELSEIF (@function=="Ducks logabs")
z= log(abs(z))
ENDIF

Ultra Fractal parameters with formula included:
« Last Edit: December 23, 2021, 02:02:54 AM by Alef »

Offline Alef

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Re: Some old stuff, maybe fractal nostalgia
« Reply #58 on: December 27, 2021, 11:39:05 PM »
Luna with abs julia set. Newton type fractals with abs formulas seem to be not mutch explored topics. Here it have not very abs like julia sets (what is typical newton) and unsighty ducks like m sets (coresponding to nova fractal).

Slightly tweakeed parameter Luna_n_Lyra_JuliaAbs_6_Luna from file above:


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