 • December 03, 2021, 09:07:05 AM

### Author Topic:  Some old stuff, maybe fractal nostalgia  (Read 4439 times)

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#### Alef ##### Some old stuff, maybe fractal nostalgia
« on: May 17, 2021, 04:24:59 PM »
There were and old thread well before perturbation http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/

Multipowerbrot:
Z=(((((Z2xC) +1 )2) -1 )2) -1  or  ( Z=((((Z^2xC+D)^2)-D)^2)-D  where D is real number)
http://www.fractalforums.com/index.php?topic=4881.msg25998#msg25998
I guess maybe zooms of this could be good after all.

Z=(((((Z3)xC+i)3)+i)3)+i  or ( Z=((((z^3xC+D)^3)+D)^3)+D  where D is imaginary value)
http://www.fractalforums.com/index.php?topic=4881.msg27524#msg27524

Soap?
Z=(ZxZxC+1)+1/(ZxZxC+1) or (Z=(Z2xC+D)+1/(Z2xC+D)  where D is real)
http://www.fractalforums.com/index.php?topic=4881.msg25998#msg25998

There were a formula from Fractal Explorer, Talis:
z=z2/(z+complexvalue) + c
I guess Talis is from sheep tails it resembles. Not very good in pure form but nice shape.

Some stuff is generated by Mandelbrot and Talis combined.
z=z2 +c
z=z2/(z+0.5)

There were lots of fractals generated by mutations of this.
Realy older than this:
https://www.deviantart.com/ultra-fractal-redux/gallery/45625392/September-2013-Talis-and-Friends-Challenge
Maybe this. Not the formula itself but attached function abs and maybe alsou log was responsible for images:
http://www.algorithmic-worlds.net/blog/blog.php?Post=20110227#comment-893671968

What lead to ducks and kalis fractals.

Szegedi butterfly.
Interesting story behind this fractal. It was recreated after war in Balkans with floppy disk left. Good for generateing butterflies.
z = sqr(y) - sqrt(abs(x)) + 1i * (sqr(x) - sqrt(abs(y))) + C

(sqr - square, sqrt - square root)

Logic Turtle
http://www.fractalforums.com/mandelbrot-and-julia-set/shattered-fractal/
Not that good exept in buddhabrot render:

IF (|z| < |c|)
z= (z)power + c
ELSEIF (|z| == |c|)
z= (z)power
ELSE
z= (z)power - c
ENDIF

Fractalus Techna
http://www.fractalforums.com/new-theories-and-research/solidifying-transformation/msg78184/#msg78184
Better shattered fractal is generated by
z=z*z
z=z-z/(ceil(z) +1.0e-20)    //or any other round() function
z=z+c

Unit Vector Mandelbrot (with p-norm where p=1)
http://www.fractalforums.com/mandelbrot-and-julia-set/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/165/
z=z2+c
z =z - z/(abs(real(z))+abs(imag(z))) * 0.4
more generalised
Z= Zn + C
Z = Z + value * Z/|Z|

Pauldebrots formula:

Something like this
complex w = @seed2
z =  @seed
(non distorted mandelbrot by w = 1 ; z = 1i or w = 1i ; z = 1 )
loop:
ws = sqr(w)
w = w/z*(@wfactor)
z = sqr(z) + ws

Devaney sierpinsky carpets
z=  zn + C / (z - a)d
http://math.bu.edu/people/bob/papers.html
with critical points (formula don't works if initialised with z=0)
http://www.fractalforums.com/index.php?topic=18526.30
z=(d*c/n) (1/(n+d) )

Still maybe the best but largely unexplored maybe is General Quadratic
http://www.fractalforums.com/index.php?topic=11008.0
With abs or without
x1=x*x + y*y
y2=2.5*x*y
z=1i*y1 + x1 + c

With this there is no problem to do deep zooms in KF, it just needs value to be adjusted so it woun't be pure square.

Nova Moon
Not very usefull for fractal exploration, but generates nice lunar crescents.
z=(K+1)*z- (z^N - z)/(N*z^(N-1)-1) + 1/C
with standart crescent N = 1 K = 0;0.
bailout:
(|z-z0|>nwtbailout)

No images throught, and some threads alredy don't have image files from upload services. I wanted to sugest first one formula for zooms as it was very complex formula indeed but then remembered other formulas. I could do that in UF6 but don't want to go deeper in programming, now I want to learn other mathematical/chaotic stuff by now that would be more profitable (tradeing that is). And maybe something could have some good use.

« Last Edit: May 17, 2021, 05:36:07 PM by Alef »
by Edgar Malinovsky aka Edgars Malinovskis.

#### Alef ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #1 on: May 17, 2021, 08:52:32 PM »
More:

Peterson
Two headed mandelbrot. Of corse this can be parametiresed.
z=z^2 * c^0.1 + c;

Fractovia
Proposed flag of the state Fractovia (so long agou that this can't be found in net).
z=0.9 * z * atanh(z) + c

Star
Good only for generation of fractal stars. Sides is power - 1.
z=z/K-z^N+1/C

Collatz Conjecture
https://en.wikipedia.org/wiki/Collatz_conjecture
From wikipedia, some mathematical theory.
c=0.25*(2+7*c-(2+5*c)*cos(pi*c));
z=c;

Algorithms using zold variable:
During iteration:
oldestz=oldz;
oldz=z;
z=z*z+c

Manowar
z=z+oldz;
Phoenix
z=z+oldz*distortion;
Simurgh
z=z+oldz*distortion+oldestz*distortion2;

distortion  = 0.5
distortion2 = 0.25
Code: [Select]
; Simurgh/Phoenix/Double Mandelbrot Family;; This fractal type encompasses many variants in the; Mandelbrot family of fractals. The basic Mandelbrot; equation is:;;     z[n+1] = z[n]^a + c;; where z=0, a=2 and c varies with pixel location. The; Phoenix fractal discovered by Shigehiro Ushiki in 1988; is avariation of this, where the input of two iterations; is used:;;     z[n+1] = z[n]^a + c*z[n]^d + p*z[n-1];; The classical Phoenix fractal is the julia form, where; z varies with pixel, d=0, c=0.56667, and p=-0.5. Note; that if d=0 and p=0, the classical Mandelbrot equation; is still there. So Phoenix fractals are a superset of; the Mandelbrot fractals.;; The Simurgh fractal is a fairly straightforward extension; of the idea I wrote in November 1999; it uses three; iterations as input:;;     z[n+1] = z[n]^a + c*z[n]^d + p*z[n-1] + q*z[n-2];; If q=0, the Phoenix equation emerges, so again Simurgh; fractals are a superset of Phoenix fractals.;; A different direction of extension to the classical; Mandelbrot extension is DoubleMandel, which uses two; separately-scaled z[n] terms:;;     z[n+1] = s*z[n]^a + t*z[n]^b + c;; We can include this extension into the Simurgh equation:;;     z[n+1] = s*z[n]^a + t*z[n]^b + c*z[n]^d + p*z[n-1] + q*z[n-2];; This becomes the Simurgh equation when s=1 and t=0, so; again it is a superset. The above equation is the one used; in this formula.Damien M. Jones
Spider
initialisation:
c = pixel;
z = c;
p = c;
w=0.5;      //+-1/100 gives results.
iteration loop:
z=sqr(z)+c+p;   // (but maybe z=sqr(z)+c;)
c=w*c+z;

Glynn formula
; formula by Earl Glynn in Computers and the Imagination
initialisation:
z=pixel
loop:
z = z ^ 1.5 - 0.2
Could be considered julia set of z= z^1.5 +c . Nice trees.

;ChebyshevT4
Could be good for buddhabrot, mandelbrot like
z = c*(sqr(z)*(sqr(z)*8+8)+1)

;Chebyshev4Axolotl
Good for buddhabrot, mandelbrot like
z = (sqr(z)*(35 * sqr(z) - 30) + 3) / 8 + c

4th modulus Mbrot
Good for buddhabrot but alsou barnsley like cutted, suny julias
z=sqr(z)
z=(z^4+z^4)^0.25+c

;Rotated Mandelbrot
Kind of hairy mandelbrot
z=z^2+c
z=sqr(z)/cabs(z)

Carlson continiued fraction function
lots of disconected mandelbrot stuff
v = z^Power + c;
z = v - (1/(v + 1/(v + 1/(v + 1/(v + 1/(v + 1))))));

;z = z^z - z^5 + c;  //by Clifford A. Pickover
Not very good I think.

z = real(z)/cos(imag(z)) + 1i*( imag(z)/cos(real(z))) + c; //thorn fractal aka "Secant Sea"
Could be very good. Area filling glyph like.

;Pinwheels
z = (1, 0.5) * sin(z) + c;
Spirals. This could be parameterised, multiplication by imaginary value is rotation.

Moon like Nova fractals. Nova luna probably could generate same more paterns and anti-luna must be Nova.
Copied from no longer aviable formula file of Chaos Pro, would take some time to rewrite in more universal formal.
Code: [Select]
comment{The follwing fractals were derived from Heron's formula forcalculating square roots with an added perturbation. You should switchoff periodicity algorithms, when using them. As default value for the power (p1) I have used p1 = 2, though complex powers are possiblyeven more interesting.Bernd Lehnhoff 2004 !nospam!f.b.!nospam!lehnhoff!nospam!@gmx!nospam!.de}Luna {  parameter complex p1, p2, p3;  parameter int settype;  complex c, u, v;  real r;  void init(void)  {    if (((real(p1)==0) || (real(p1)==1)) && (imag(p1)==0))    {      p1 = 2;    }    if ((real(p2)==0) && (imag(p2)==0))    {      p2 = 0.5;    }    if (settype=="Mandel")    {      c = pixel;    }    else    {      c = p3;    }    v = pixel;  }  void loop(void)  {    u = v;    v = u - u / p1 * (1 - (c / u)^p1) + p2;    r = log10(|v - u|);  }  bool bailout(void)  {    return(r >= -12);  }  void description(void)  {    this.title = "Luna";    settype.caption = "Set type";    settype.enum = "Mandel\nJulia";    settype.default = 0;    p1.caption = "Degree of root";    p1.default = (2,0);    p1.hint = "Change this value as you like (<> 0 and <> 1).";    p2.caption = "Perturbation";    p2.default = (0.5,0);    p2.hint = "Change this value as you like.\nIt must not be equal to zero!";    p3.caption = "Radicand";    p3.default = (1,0);    p3.hint = "Change this value as you like.\nIt should not be equal to zero!";    p3.visible = (settype=="Julia");  }}NovaLuna {  parameter complex p1, p2, p3;  parameter int settype;  complex c, d, u, v, w;  real r;  void init(void)  {    if (((real(p1)==0) || (real(p1)==1)) && (imag(p1)==0))    {      p1 = 2;    }    if ((real(p2)==0) && (imag(p2)==0))    {      p2 = 0.5;    }    if (settype=="Mandel")    {      c = pixel;    }    else    {      c = p3;    }    v = pixel;    d = .5 * (p1 - 1) / p1;  }  void loop(void)  {    u = v;    w = 1 - (c / u)^p1;    v = u - u / p1 * w * (1 + d * w) + p2;    r = log10(|v - u|);  }  bool bailout(void)  {    return(r >= -12);  }  void description(void)  {    this.title = "NovaLuna";    settype.caption = "Set type";    settype.enum = "Mandel\nJulia";    settype.default = 0;    p1.caption = "Degree of root";    p1.default = (2,0);    p1.hint = "Change this value as you like (<> 0 and <> 1).";    p2.caption = "Perturbation";    p2.default = (0.5,0);    p2.hint = "Change this value as you like.\nIt must not be equal to zero!";    p3.caption = "Radicand";    p3.default = (1,0);    p3.hint = "Change this value as you like.\nIt should not be equal to zero!";    p3.visible = (settype=="Julia");  }}Lyra {  parameter complex p1, p2, p3;  parameter int settype;  complex c, d, u, v;  real r;  void init(void)  {    if (((real(p1)==0) || (real(p1)==1)) && (imag(p1)==0))    {      p1 = 2;    }    if ((real(p2)==0) && (imag(p2)==0))    {      p2 = 0.5;    }    if (settype=="Mandel")    {      c = pixel;    }    else    {      c = p3;    }    v = pixel;    d = p1 - 1;  }  void loop(void)  {    u = v;    v = (p1 * u) / ((u / c)^p1 + d) + p2;    r = log10(|v - u|);  }  bool bailout(void)  {    return(r >= -12);  }  void description(void)  {    this.title = "Lyra";    settype.caption = "Set type";    settype.enum = "Mandel\nJulia";    settype.default = 0;    p1.caption = "Degree of root";    p1.default = (2,0);    p1.hint = "Change this value as you like (<> 0 and <> 1).";    p2.caption = "Perturbation";    p2.default = (0.5,0);    p2.hint = "Change this value as you like.\nIt must not be equal to zero!";    p3.caption = "Radicand";    p3.default = (1,0);    p3.hint = "Change this value as you like.\nIt should not be equal to zero!";    p3.visible = (settype=="Julia");  }}AntiLuna {  parameter complex p1, p2, p3;  parameter int settype;  complex c, u, v;  real r;  void init(void)  {    if (((real(p1)==0) || (real(p1)==1)) && (imag(p1)==0))    {      p1 = 2;    }    if ((real(p3)==0) && (imag(p3)==0))    {      p3 = 1;    }    if (settype=="Mandel")    {      c = pixel - p3;    }    else    {      if ((real(p2)==0) && (imag(p2)==0))      {        p2 = 0.5;      }      c = p2 - p3;      if (c==0)      {        c = p3 / 2;      }    }    v = pixel;   }  void loop(void)  {    u = v;    v = u - u / p1 * (1 - (p3 / u)^p1) + c;    r = log10(|v - u|);  }  bool bailout(void)//  {    return(r >= -12);  }  void description(void)  {    this.title = "Anti-Luna";    settype.caption = "Set type";    settype.enum = "Mandel\nJulia";    settype.default = 0;    p1.caption = "Degree of root";    p1.default = (2,0);    p1.hint = "Change this value as you like (<> 0 and <> 1).";    p2.caption = "Perturbation";    p2.default = (0.5,0);    p2.hint = "Change this value as you like.\nIt must not be equal to zero!";    p2.visible = (settype=="Julia");    p3.caption = "Radicand";    p3.default = (1,0);    p3.hint = "Change this value as you like.\nIt should not be equal to zero!";  }}SinusLuna {  parameter complex p1, p2;  parameter int settype;  complex c, u, v;  real r;  void init(void)  {    if (((real(p1)==0) || (real(p1)==1)) && (imag(p1)==0))    {      p1 = 2;    }    if ((real(p2)==0) && (imag(p2)==0))    {      p2 = 0.5;    }    if (settype=="Mandel")    {      c = sin(pixel);    }    else    {      c = sin(p2);    }    v = pixel;  }  void loop(void)  {    u = v;    v = u - (sin(u) - c) / cos(u) + p2;    r = log10(|v - u|);  }  bool bailout(void)//**  {    return(r >= -12);  }  void description(void)  {    this.title = "Sinus-Luna";    settype.caption = "Set type";    settype.enum = "Mandel\nJulia";    settype.default = 0;    p1.caption = "Degree of root";    p1.default = (2,0);    p1.hint = "Change this value as you like (<> 0 and <> 1).";    p2.caption = "Perturbation";    p2.default = (0.5,0);    p2.hint = "Change this value as you like.\nIt must not be equal to zero!";    p2.visible = (settype=="Julia");  }}
« Last Edit: May 18, 2021, 09:17:29 AM by Alef »

#### Sabine62

• 3f
•      • • Posts: 1249 ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #2 on: May 18, 2021, 06:35:00 PM »
Wow, Alef, that is quite a repository!
Thank you very much for sharing   To thine own self be true

#### C0ryMcG ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #3 on: May 27, 2021, 04:16:49 AM »
Nice collection! I've been going through and trying out a few of these, some really fun results.

I'll share a few that I like so far, and their coloring formulas

Thorns with smoothed Escape Time coloring,
a Julia of Butterfly with Lagrangian Descriptor coloring,
a zoom of Fractovia Flag with cross-shaped orbit trap,
and a Julia from the Pickover Formula with Lagrangian coloring again.

I'll probably post more as I try them out.

#### gerson ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #4 on: May 28, 2021, 02:57:42 AM »
Good post.
Used Fractal Zoomer build in formula Szegedi butterfly to do an image:
https://fractalforums.org/index.php?topic=2507.msg28762#msg28762

#### Alef ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #5 on: May 28, 2021, 11:32:53 PM »
gerson:
Nice The most suprising thing about Szegedi butterfly formula is that it actualy looks like butterfly There is a second version, with x and y switched in places. It's tail is kind of less flying like:

z = sqr(x) - sqrt(abs(y)) + 1i * (sqr(y) - sqrt(abs(x))) + C

The web archive had saved the original page https://web.archive.org/web/20120308231030/http://www.szegedi.org/fractals/butterfly/index.html
Now I 'm going throught Fractal Explorer formulas. At firt I must pick the better ones.
« Last Edit: May 29, 2021, 12:40:30 PM by Alef »

#### Alef ##### Re: Fractal formulas by Dr. Joseph Trotsky
« Reply #6 on: May 29, 2021, 02:35:19 AM »
"Cracks" of the Mandelbrot set on the iteration N by Dr. Joseph Trotsky

Iterate:

If iteration_number = N Then begin
C=1/C;
end;
Z=Z2+C

I hadn't seen this before, but this is good one. I think it is more interesting with N being small number, say 1 as in example. Could work differently with julia sets. Alsou in FE mandelbrot set is initialised with Z= pixel not Z=0 so then N=2 would corespond to N=1. 1/C have some mathematical basis.
This stuff needs some good name. Maybe a "trotskyist inlaid", "trotskyist insertion" or something like that  * * *

Formula must be initialised with Z=pixel or else it will not work.
Z=Z^3 - 3*Z*C^2 + Complex_Number
Kind of a mandelbrot shape generator based on Complex_Number.

* * *

Dr. Joseph Trotsky formulas

In the software calculation is initialised with z= pixel. P1, P2 by default is complex number (0.5;0.5) Fn1, Fn2 is some function, by default none.

I made these formulas simpler and more simmetric as I don't like messy stuff. Most of trigonometric function using formulas looks like normal power sets but have a small satelites around the main set, most noticably in julia sets.

JT1 - Eyes
Z=Fn1(Z); Z=P1*(Z/23)*Fn2((Z+CTan(Z)))+Z+P2*C/1.1
Z= Z*(Z+tan(Z)) / N + Z + C   very different with N =2 or 3

JT2 - Orbits
Z=Fn1(Z); Z=Fn2(P1*Z*CTan(Z)) * sqr(Z) + P2*C/2.14
Z= tan(Z)*Z^3+C
Mandelbrot set like with satelites.

JT7 - Star
Z=Fn1(Z); Z=Fn2(Z*Z*CSin(Z+P1)) + P2*C
Z= Z*Z*Sin(Z)+C
Mandelbrot set like with satelites.

JT9 - Dragon
Z=Fn1(Z); Z=Z*Z + Fn2((Z+C)*Z*Z) - CSin(Z+P1) + P2*C
Z=  (Z+ C +1)*z^2 - Sin (Z + 0.5) +  C
Super! Some locations resemble IFS dragon. Initialise formula with Z=0.

JT13 - Dahlia
Z=Fn1(Z); Z=Z*(C+P1) + Fn2(Z*Z*Z + Z*Z*P3 + P2) + C
Z= z*C + z^3 + z^2  + C
Mandelbrot like, "dahlia" could be becouse of shapes in zooms.

JT15 - Trisome
Z=Fn1(Z); Z=Fn2(CSinh(Z*3.14)^2 + P1) + C
Z= 0.5*Sinh (Z*Pi)^2 + C
Small mandelbrot sets with a satelites.

JT16 - Rose Garden
Z=Fn1(Z); Z=Fn2(Z*Z + Z*P1) - Z*C
Z= Z^2 + Z/2 - Z*C
Looks like Lambda fractal but is different in small features. Must be initialised with Z=pixel.

JT119 - Hearts
Z=Fn1(Z); Z=(Z+P1)^Fn2(Z+P2) + C
Z= (Z + 1 ) ^Z + C
Simple https://en.wikipedia.org/wiki/Tetration of order 1 but (Z+1) gives more fractal features, including mandelbrot sets. No hearts throught Maybe there are better critical points for some of these formulas.
« Last Edit: July 07, 2021, 05:22:50 AM by 3DickUlus, Reason: removed high security risk link »

#### Alef ##### Re: More of old formulas from FE
« Reply #7 on: May 31, 2021, 01:37:19 PM »
More from the Fractal Explorer.
Formulas don't get outdated unlike rendering methods do. They needed to be slightly adjusted as in FE some were in quite a messy format. Only interesting ones included, I kept the original names however lame they are. Some of these formulas needs to be started with z= pixel (not z=0 like most M-sets). Often it is becouse of division by zero or sometimes becouse of other reasons. However some other formulas only works if initialised with z=0.

Stalkless mandelbrot sets:
Kind of looks like julia sets, but these have their own julia sets.

Four
Z= Z^3 - C
C = Z
; works with any power, 3,4,5... Power 2 is like julia set (-0.7, 0), odd powers resemble stalkless mandelbrot sets.

Mandel variation3
Z=Z^3 - C
C=C^3 - Z
; Shape is slightly distorted version of "Four". Resembles respective mandelbrot sets with odd powers only.

Mandel variation 5
Z=Z^2 + Temp
Temp = C - Z
C= C* (0.72, 0)
; Like previous but distortion of shape depends on multiplicator

Talis, was very popular formula of software, mostly as a julia sets. Maybe worth it, shape is unique. Keep bailout low, or it will need hudge numbers of iterations. Higher powers gets more interesting.

Talis
Z=Z^N / (1 + Z^(N-1) )  + C    ;  N = 2, 3, 4 ....

Talis variation 2
Z= (Z^3 + 1 ) / (z^2 + 1) + C

Authors formulas.

Karl Geigl 105 (Good with Julie)
Z = Sin ( C / Z )

Karl Geigl11
Generates foam with mandelbrot sets mutch like the Devaney formulas, initialisation must start with z= pixel
Z= ( ( C^2/Z + Z) * C)^2

E.Malinovskis
My first formula;) ha ha. Simmilar to some Trotsky stuff.
Z= Z*Sinh(Z) - C^2

Non linear formulas.

Reminder 1
This have some good julia sets, kind of like ducky / kali or sierpinsky but it don't requires inside color methods. Initialisation must be with z=pixel

Z= Abs (Z^N)^(1/C)
with N = 0.5, 1, 2, 3...   On power N depends number of symmetry axes and the size of fractal.

Curfew variation
Mandelbrot set of cardoid only, initialisation with z=0
Z=Z + C + 1
Z=Z^2/C

Sine - cosine formulas
There are just 2 good clean formulas and they are very simmilar.

polinomial 3
initialisation must be with z= pixel, kind of a world of spirals with 3 simmetry axis.
N = 5 ... 50 !
Temp = N*Z
Z= Sin(Z) - C
C = 1/(Temp)

BWP 4
initialisation z = 0, less dense spirals, 4 fold simmetry.
N = 5 ... 50 !
Z = Sin(Z) - C
C = 1/(N*z)

There are Newton type formulas left, something called supernewton and formulas from Sterlingware, alsou probably a newton based. Some of them must be good, however I had not explored newton fractals.
Alsou some of those could have some critical points generating true to type sets.
Maybe a catalogue of formulas? That would require a lot of a work, too mutch I think.
« Last Edit: June 01, 2021, 10:31:11 PM by Alef »

#### Alef ##### interesting outlines
« Reply #8 on: June 25, 2021, 03:09:10 PM »
Found in not working renderer: TS Fractal Explorer 3.0 beta

Michelitsch
Less than exciting repeating long mandebrot set (with less than beautifull biomorph thorns). Hovewer it's interesting that this formula generates typical mandelbrots without any power function, and alsou typical julias with biomorphs. Cos is enought.
Z=2*(1-cos( Z)) - C

Cool but creepy kind of little girl with chicken legs from  MATOUŠ STIEBER blog
z= (0,-1)* c*z^5 +1
z= c*z^5 +1 - is same but on side. Zooms are ordinary.

Julia set post is more interesting, still, mostly of the same as in wikipedia images: https://matousstieber.wordpress.com/2016/01/12/julia-set/ Angel
(Fond in contests, by FractalAlex) Alsou interesting outline.
Initial conditions:
z=1
Iteration:
z= 0.25*z^4-z+c

Simmilar fractal family is somewhere at element90 blog with critical points: https://element90.wordpress.com/category/mathematics-2/
alsou
https://element90.wordpress.com/tag/mathematics/
« Last Edit: June 25, 2021, 03:25:28 PM by Alef »

#### claude ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #9 on: June 25, 2021, 04:15:30 PM »
Cos is enought.
Z=2*(1-cos( Z)) - C

$2 (1 - \cos(Z)) = 4 \sin\left(\frac{Z}{2}\right)^2 \approx Z^2 + O(Z^4)$ so quadratic minibrots will appear (smallest non-linear power is 2)

#### Alef ##### Re: Some old stuff - Newton archeology
« Reply #10 on: June 28, 2021, 09:32:18 PM »
Newton type formulas from FE and Sterlingware 2, some edited a lot beyond recognition. All were initialised with z=pixel, maybe some have better critical points. Alsou I had not tested julia sets.

Supernewton
Interesting set. Am I wrong, but there seems to be a mathematical error. Correction of derivatives led to mutch too dense and uninteresting set.
z=Z-(Z^3-1)/(3*Z^2)
Z=Z-(Z^4-1)/(4*Z^2)

Adding C is a mirracle! Now it's super large mandelbrot set with same features in certain places.
maybe a Triton: (newt in my language ). initialisation z= 1 seems to work better.
z=Z-(Z^3-1)/(3*Z^2)
Z=Z-(Z^4-1)/(4*Z^2) + C

Could be simplified to:
Z=Z-(Z^4-1)/(4*Z^2) +  C
(correction of derivatives turns this into Nova fractal)

Newton in Mandelbrot maybe a Triton:
N = 0 ... 1. On 0 it is just large empty ideal mandelbrot set, on 1 it gets more interesting. 0.1 - nova inside a mandelbrot.
Z=Z-(Z^3*C +N)/(3*Z^2*C)
Z=Z^2+C
It could be simplified:
Z=Z-(Z^3 + N)/(3*Z^2)
Z=Z^2+C

Small nova, maybe a mirrage:
Z=Z-(Z^3+Z*C )/(3*Z^2+ C ) + C
or 2:
Z=Z-(Z^3+Z^2*C-Z)/(3*Z^2+2*Z*C-1)+C

Newton foam, like Devaney:
Not so good for zooms, but maybe interesting curiosity. It's a seprinski ...something in publications of Devaney.
Z=(Z-(Z^3-1)/(3*Z^2))^3*C

Somehow then they very mutch liked sine function. Lots of sine newtons. Error correction of derivatives led to worst results.

Sterlingware 22:
Interesting shapes but repeating and slow.
z  = z-(z^2*c + z^2*c + z + c - sin(z))/(4*z^2*c + 3*z*c + 1 - cos(z))

Sterlingware 43:
Bubbles with holes, OK but slow . Anyway interesting figures.
z = z-(z^5*sin(z)-z^4*sin(z)-sin(z)*z*c-z)/(5*z^4*cos(z)-z^4*cos(z)-cos(z)*z*c-1)

Sterlingware 54:
The same as above.
z = z-(z^3*sin(z)-z^2*sin(z)-c)/(4*z^2*cos(z)-z*cos(z) )

First of two must be the most explorable. Now I should render some images for some of the better formulas. I just can't make pics and good zooms for all of them.

EDITED:
Rendered things I found the most interesting, Triton fractal. Some of power 3 julia sets are just like hermann rings in wikipedia. Alsou inside of pow2 cardoid there is barelly visible Nova set.
« Last Edit: August 17, 2021, 07:38:19 PM by Alef »

#### Alef ##### Re: Some old stuff, maybe fractal nostalgia
« Reply #11 on: June 29, 2021, 01:04:34 PM »
From the post above.

Z=Z-(Z^3*C +0.05)/(3*Z^2*C)
Z=Z^2+C
Julia set and zoom on nova.

and
Z=Z-(Z^3*C +0.05)/(3*Z^2*C)
Z=Z^3+C
julia set

and simpler formula
Z=Z-(Z^3 +0.05)/(3*Z^2)
Z=Z^3+C
julia set. I can't decide wich is better.

Unfortunately ultra complexity means lots of colouring algorithms dont work nicely (with newton bailout) and automatic calculation optimisations like periodicy checking and guessing creates artefacts.
« Last Edit: June 29, 2021, 06:22:25 PM by Alef »

#### Alef ##### Triton
« Reply #12 on: July 04, 2021, 08:25:03 PM »
I decided, which version is better, and I think the simplest.
Triton (a newt in my language)

initialisation:
z= pixel
(There could be a better critical points but derivatives and roots of this would be beyod my skills. This works well. It would not work on z=0.)

iteration:
Z=Z-(Z^3 + M)/(3*Z^2)
Z=Z^N+C

Here M = 0.1 and N = 2
(newton and mandelbrot fractal formulas.)

bailout:
|z| < @bailout
(I wanted to explore just it's mandelbrot set like features.)

#### Alef ##### more newtonian Triton
« Reply #13 on: July 04, 2021, 09:41:23 PM »
Z=Z-(Z^3 + M)/(3*Z^2)
Z=Z^N+C

Here M = 0.5 and N = 2 and one zoom with M=1
and

bailout:
|z - zold| >= @bailout

Increasing of M increases newton like features very evident in certain areas and less so in other. If M=0 it is just mandelbrot set. There is one problem, no color method works resonably well (for both newton and mandelbrot type) and I just don't know good method for newton roots fractal so I used Discrete Langrangian Descriptors. Alsou then it could be slightly slower.

#### Alef ##### Triton
« Reply #14 on: July 06, 2021, 10:54:34 AM »
I made Ultra Fractal "Triton" formula good enought to be sheared. If I 'll make more formulas I 'll share whole file.

Code: [Select]
Fractal1 {::DA04ygn2lFVzuxJMQ47Ix7glvndtpLESj8h0D9QlaO17ROwAMNGbL7Bth3+OAbTqU9BLrv/m  Z8Mksdk191yChgQyBG53Pg0SxVsnmMNXUiJAHnIzlWlwZXhU2o3c0lgekyG5TOY4x7P3cuSV  pllF7i2DtzGJM4NyvZ7ebMFW89SRIa7Qa1oVKxMQThez8ijwoNnLLmtxI6HPsDeCSm70nqPz  atjej6UdZxQIxOs7amtvjbi2SLCpuJo7NTYYQMgOwbn5Za+ZITiXI++0ywsUwxmWNyflQK4l  bpEfJGuypUxvShAx1RzPf1iuwCtGBzzwVW8nYsC1+Rf4PD0uO5Pt+ew9aKQiNA5G3Vk6m+Qx  PWco9g8w7v3AM3VdWVWg+M2DHrkk1nH4uyH8QZBX0/nyhewmKLGT2eknrd28MPCTmVILQuZe  3oEdBXIZavX3U12eDd7L7AX3or0t113IqUq9v/duLN1PovR8lPdU3wd0xm8frpP8RJ/7euqm  XZ/BdtbxlB==}mNest__test.ufm:Triton {;; Simplification of idea from Fractal Explorer; https://fractalforums.org/share-a-fractal/22/some-old-stuff-maybe-fractal-nostalgia/4213/msg29319#msg29319;Calculates newton formula embeded in mandelbrot set, initialised with pixel values.; maybe there is better critical points of calculation start.::BbPqfjn2FWVUvpNMQ43jU+Pcj9CsuShWtX2mnQdDkYaraa0nQTVykcQ8mTcWsDQR9H/ObnES  gqOkQK+y5777u77uIyEm3HGMfG0fiGNmHzRGr335ZxocVhyAWL9GEGAwBgB5i9oMMweMSlmL  x9Q0RzAM9bLmGGcgVZIiN53lSBPMY6dfZ+MrHhBrUKJEryQ6ir5SN6NLVqciJ1h9gSGTOcwC  2S2yL7v8hbgLgJFKlZwV9v5NLf46BuX9wkc1Os4iPbd1Gr5z6PZFXIVlWyDMG0LD3ZUZ+8Aq  Bv/THgLdA9E8Ro+KOfmKt5RbPfC+EM+quO5zJKj8GJ2/K7FCDix18Sp1AAGhRajRv7LEWOYt  ljFCVsISYekezIrpU0kosZcKdRROXrdW5byYjG+O7zfIlvXYwCynxjG5LaJIPWkthKb8cjQl  ZxZmqgiBHuVqi+DBnB3bs2nT3lbQwXLgWF1f2umCUvnwtRAYrw/wXgBShY77rEZYMs2jzQCC  MLuhJ+eg9/apibgceBPFctIf1vFVdxtXVTxXzIzX7NkIyc8eROGJWLQNYSQA3nTlY6Nq1+z/  tk7CHdmeMmSRRUj1hugT0zxiTJlVLdGnsFjzo0ohjfJSZD0ZEykQAJ0gWJ3ixnykwAPHOKU7  0Ev1bGu3O+1IoA2IbxuMl84OvkGaNs67ENAAQHIOLTrxYLXWiPTKPy9rKxTFZ1S1jVh7To8z  BCaFm0lJlhGSU7osmAhTqlstYxGXDrYlw4IjGswuLRQDG/yHx6fCnIlESDhFpcpkiqjeaYjY  LCpqCEyLoavmq6omoqGWRBrUXSe/4phrg6Giiq7VFbVm+cZxzPK9V7yrmBpuqcfttanZnbtA  fp+2p7WrRx4vx5KE9OhJK55Ay/GwoajEM2DF8/xarQLWJbtHur6peVsnGVLybphYVRbBixdl  Ps+Xe9bhRVrbPCT7vx0iJDOvbAVpNtC15/xtnOewed1XYqj3pVpOHZNmtDqM/YvbjErav0xp  QW7ByqnZNTQ/Dwy9y4K=}

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