Fractal 058 A – The Big Juggler

Part of BEYOND series – 12 2021

alchemy . architect . artist . cosmos . creation . god . juggler
Fractal 058 A | The Big Juggler
Fractal 058 A zoom1

Fractal 058 A – zoom level 1

Fractal 058 A zoom2

Fractal 058 A – zoom level 2

Data

This digital works is created exclusively from fractals, with 0% AI generated. It can therefore be converted into a set of functions and parameters that can be the basis for the training of an AI.

The mathematical formulas and parameter combinations corresponding to each fractal are presented below. In each case, the numerical characters have been replaced by ■ to prevent unauthorized reproduction. If you are interested in using the full data set, please contact Philippe.

Fractal 058 A - The Big Juggler - Part of BEYOND series - 12.2021

Fractal 058 A | The Big Juggler

Fractal_■■■_A {
fractal:
  title="Fractal_■■■_A" width=■■■■ height=■■■■ layers=■
  credits="Philoxerax;■/■/■■■■"
layer:
  caption="Background" opacity=■■■ mergemode=addition
mapping:
  center=-■.■■■■■■■■■■■■■■/-■.■■■■■■■■■■■■■ magn=■■■.■■■■■
  angle=-■■■.■■■■
formula:
  maxiter=■■■ filename="lkm■.ufm" entry="extra-foam"
  p_pixel_type="all in" p_wq=yes p_zc=yes p_circuit_type=series
  p_bailout=■■■■ p_qinit=■/■ p_cinit=■/■ p_winit=■/■ p_zinit=■/■
  p_wpowerinw=■/■ p_zpowerinw=-■/■ p_wpowerinz=■/■ p_zpowerinz=■/■
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes rotation=-■■■ index=■■ color=■■■■■ index=■■■ color=■■
  index=-■■■ color=■■■■■■■ index=-■■ color=■■■■■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Background" opacity=■■■ mergemode=difference
  method=multipass
mapping:
  center=-■.■■■■■■■■■■■■■/■.■■■■■■■■■■■ magn=■■.■■■■■■ angle=-■.■■■■
formula:
  maxiter=■■■ percheck=off filename="lkm.ufm" entry="gap-mandelbrot"
  p_manparam=■/■ p_n=■.■ p_type="between ■ lines" p_c■=-■.■ p_c■=■.■
  p_theta=■■.■ p_center=■/■ p_radius■=■.■ p_radius■=■.■
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes rotation=■■■ index=■■■ color=■■■ index=■■■ color=■■■■■■■
  index=■■■ color=■■■■■■■■ index=■■■ color=■■■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Background" opacity=■■■
mapping:
  center=-■.■■■■■■■■■■■■■■■/-■.■■■■■■■■■■■■■■■ magn=■■■■■.■■■
  angle=-■■■.■■■■
formula:
  maxiter=■■■ filename="lkm■.ufm" entry="compound-mandelbrot"
  p_initialz_type=pixel p_initialz=■/■ p_npower=■/■ p_mpower=■/■
  p_ppower=■/■ p_bailout=■■■■
inside:
  transfer=none solid=■■■■■■■■■■
outside:
  transfer=linear
gradient:
  smooth=yes rotation=-■■■ index=■■ color=■■■■■ index=■■■ color=■■■
  index=-■■■ color=■■■■■■■ index=-■■ color=■■■■■■■■
opacity:
  smooth=no index=■ opacity=■■■
}

extra-foam { ; Kerry Mitchell ■■ June ■■■■
;
; Generalization of "Mandelbrot Foam" from fractal forums
; https://fractalforums.org/fractal-mathematics-and-new-theories/■■/mandelbrot-foam-math/■■■■/msg■■■■#msg■■■■
; original:
;   w = a■w/z
;   z = z^■ + w^■ + c
;
; powers of w and z in w and z formulas are parameters
;
; added parallel/series modes:
;   parallel: compute new w and new z independently
;   series: compute new w and use that for new z
;
; pixel_type parameter allows the pixel to represent either complex parameter (or both) or the initial values
;   of either complex variable (or both).
;
init:
  complex q=(■,■)
  complex c=(■,■)
  complex w=(■,■)
  complex wnew=(■,■)
  complex z=(■,■)
  complex znew=(■,■)
;
; Use pixel value to set parameters (like a Mandelbrot set) and/or
;   initialize variables (like a Julia set)
;
  if(@pixel_type=="q")
    q=#pixel
    c=@cinit
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="c")
    q=@qinit
    c=#pixel
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="q = c")
    q=#pixel
    c=#pixel
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="initial w")
    q=@qinit
    c=@cinit
    w=#pixel
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="initial z")
    q=@qinit
    c=@cinit
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    z=#pixel
  elseif(@pixel_type=="w = z")
    q=@qinit
    c=@cinit
    w=#pixel
    z=#pixel
  elseif(@pixel_type=="all in")
    q=#pixel
    c=#pixel
    w=#pixel
    z=#pixel
  endif
loop:
;
; Series: compute new w and use that for new z.
; Parallel: compute new w and new z independently.
;
  if(@circuit_type=="series")
    w=q■w^@wpowerinw■z^@zpowerinw
    z=z^@zpowerinz+w^@wpowerinz+c
  elseif(@circuit_type=="parallel")
    wnew=q■w^@wpowerinw■z^@zpowerinw
    znew=z^@zpowerinz+w^@wpowerinz+c
    w=wnew, z=znew
  endif
bailout:
  |z|<@bailout
default:
  title="Extra Foam"
;
; General Parameters
;
  heading
    caption="General Parameters"
  endheading
  param pixel_type
    caption="pixel type"
    default=■
    enum="q" "c" "q = c" "initial w" "initial z" "w = z" "all in"
  endparam
  bool param wq
    caption="w = q"
    default=true
    visible=((@pixel_type=="q")||(@pixel_type=="c")||(@pixel_type=="q = c")||(@pixel_type=="initial z"))
  endparam
  bool param zc
    caption="z = c"
    default=true
    visible=((@pixel_type=="q")||(@pixel_type=="c")||(@pixel_type=="q = c")||(@pixel_type=="initial w"))
  endparam
  param circuit_type
    caption="circuit type"
    default=■
    enum="series" "parallel"
  endparam
  float param bailout
    caption="bailout"
    default=■■■■
  endparam
;
; Initializations
;
  heading
    caption="Initializations"
  endheading
  complex param qinit
    caption="q"
    default=(■,■)
    visible=((@pixel_type=="c")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
  endparam
  complex param cinit
    caption="c"
    default=(■,■)
    visible=((@pixel_type=="q")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
  endparam
  complex param winit
    caption="initial w"
    default=(■,■)
    visible=((@wq==false)&&(@pixel_type!="initial w")&&(@pixel_type!="w = z"))
  endparam
  complex param zinit
    caption="initial z"
    default=(■,■)
    visible=((@zc==false)&&(@pixel_type!="initial z")&&(@pixel_type!="w = z"))
  endparam
;
; Variable Powers
;
  heading
    caption="Variable Powers"
  endheading
  complex param wpowerinw
    caption="w power in w"
    default=■
  endparam
  complex param zpowerinw
    caption="z power in w"
    default=-■
  endparam
  complex param wpowerinz
    caption="w power in z"
    default=■
  endparam
  complex param zpowerinz
    caption="z power in z"
    default=■
  endparam
}

gap-mandelbrot { ; Kerry Mitchell ■■dec■■
;
; z^n+c Mandelbrot
; bails out when orbit falls into gap
; either between ■ circles or ■ lines
;
; updates:
;   ■■dec■■ to add general starting point
;   ■■feb■■ to add complex exponent
;   ■■feb■■ to add switching to gap-julia
;   ■■mar■■ removed complex exponent for backwards compatibility
;
init:
  z=@manparam
  c=#pixel
  float a=■.■
  float b=■.■
  float gap=■.■
  float radsqr■=sqr(@radius■)
  float radsqr■=sqr(@radius■)
  float x=■.■
  float y=■.■
  float rmax=■e■■
  float tempr=■.■
  int done=■
;
; set up line/circle parameters
;
  if(@type==■)                       ; lines
    tempr=@theta/■■■■pi
    a=sin(tempr)
    b=-cos(tempr)
    gap=@c■-@c■
  else                               ; circles
    gap=radsqr■-radsqr■
  endif
loop:
  z=z^@n+c
  x=real(z)
  y=imag(z)
;
; check for falling into gap
;
  if(@type==■)                       ; lines
    tempr=a■x+b■y
    if((tempr>@c■)&&(tempr<@c■))
      done=■
      tempr=(tempr-@c■)/gap
      z=tempr■z/cabs(z)
    endif
  else                               ; circles
    tempr=|z-@center|
    if((tempr>radsqr■)&&(temprrmax))
    done=■
    z=(■.■,■.■)
  endif
bailout:
  done==■
default:
  title="Gap Mandelbrot"
  maxiter=■■■
  periodicity=■
  center=(■,■)
  method=multipass
  magn=■
  angle=■
  param manparam
    caption="Mandelbrot start"
    default=(■,■)
    hint="use (■,■) for basic Mandelbrot set"
  endparam
  param n
    caption="z exponent"
    default=■.■
    hint="Real--use Gap Mandelbrot C for complex exponents."
  endparam
  param type
    caption="gap type"
    default=■
    enum="between ■ lines" "between ■ circles"
  endparam
  param c■
    caption="line ■ c value"
    default=-■.■
    hint="must be less than line ■ c value"
  endparam
  param c■
    caption="line ■ c value"
    default=■.■
    hint="must be more than line ■ c value"
  endparam
  param theta
    caption="line angle"
    default=■■.■
    hint="angle to horizontal, degrees"
  endparam
  param center
    caption="circle center"
    default=(■,■)
  endparam
  param radius■
    caption="circle ■ radius"
    default=■.■
    hint="must be less than circle ■ radius"
  endparam
  param radius■
    caption="circle ■ radius"
    default=■.■
    hint="must be more than circle ■ radius"
  endparam
switch:
  type="gap-julia"
  n=n
  julparam=#pixel
  type=type
  c■=c■
  c■=c■
  theta=theta
  center=center
  radius■=radius■
  radius■=radius■
}

compound-mandelbrot { ; Kerry Mitchell ■■Nov■■■■
;
; f(z) = ((z^n+c)/(z^m+c))^p+c
;
init:
  complex c=#pixel
  complex z=(■,■)
  if(@initialz_type=="pixel")
    z=c
  else
    z=@initialz
  endif
loop:
  z=((z^@npower+c)/(z^@mpower+c))^@ppower+c
bailout:
  |z|<@bailout
default:
  title="Compound Mandelbrot"
  heading
    text="Uses the formula z = ((z^m + c)/(z^n + c))^p + c; ■ exponents."
  endheading
  param initialz_type
    caption="initial z"
    default=■
    enum="pixel" "constant"
  endparam
  complex param initialz
    caption="initial z value"
    default=(■,■)
    visible=(@initialz_type=="constant")
  endparam
  complex param npower
    caption="numerator power"
    default=(■,■)
  endparam
  complex param mpower
    caption="denominator power"
    default=(■,■)
  endparam
  complex param ppower
    caption="overall power"
    default=(■,■)
  endparam
  float param bailout
    caption="bailout"
    default=■■■■
  endparam
switch:
  type="compound-julia"
  julparam=#pixel
  npower=npower
  mpower=mpower
  ppower=ppower
  bailout=bailout
}
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