• December 03, 2021, 08:32:42 AM

Author Topic:  Some old stuff, maybe fractal nostalgia  (Read 4438 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: November 20, 2021, 11:39:56 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.


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