Fractal 026 C – Nativity

Part of BEYOND series – 09 2021

cell . microbe . nativity . night . purple . space
Fractal 026 C | Nativity
Fractal 026 C zoom1

Fractal 026 C – zoom level 1

Fractal 026 C zoom2

Fractal 026 C – 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 In each case, the numerical characters have been replaced by ■ to prevent unauthorized reproduction.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 026 C - Nativity - Part of BEYOND series - 09.2021

Fractal 026 C | Nativity

Fractal_■■■_C {
fractal:
  title="Fractal_■■■_C" width=■■■■ height=■■■■ layers=■
  credits="Philoxerax;■/■/■■■■" antialiasing=yes
layer:
  caption="Background" opacity=■■ mergemode=subtraction
mapping:
  center=-■.■■■■■■■■■■■■■■■/■.■■■■■■■■■■■■■■■ magn=■■■■.■■■■
  angle=-■■.■■■■
formula:
  maxiter=■■■ filename="Standard.ufm" entry="Mandelbrot" p_start=■/■
  p_power=■/■ p_bailout=■■■
inside:
  transfer=none offset=■■■ repeat=no solid=■■■■■■■■■■
outside:
  transfer=cube solid=■■■■■■■■■■ filename="Standard.ucl"
  entry="Smooth" p_power=■/■ p_bailout=■■■.■
gradient:
  smooth=yes rotation=■■ index=■■■ color=■■■■■■■■ index=■■
  color=■■■■■■■■ index=■■ color=■■■■■■■■ index=■■ color=■■■■■■■■
  index=■■ color=■■■■■■ index=■■ color=■■■■■■■ index=■■ color=■■■■■■■■
  index=■■ color=■■■■■■■■ index=■■ color=■■■■■■■■ index=■■
  color=■■■■■■■ index=■■■ color=■■■■■■■ index=■■■ color=■■■■■■■■
  index=■■■ color=■■■■■■■■ index=■■■ color=■■■■■ index=■■■
  color=■■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■ color=■■■■■■■
  index=■■■ color=■■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■
  color=■■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■ color=■■■■■■■■
  index=■■■ color=■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■
  color=■■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■ color=■ index=■■■
  color=■■■■■■■■ index=■■■ color=■■■■■■■ index=■■■ color=■■■■■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Background" opacity=■■■ mergemode=softlight
mapping:
  center=-■.■■■■■■■■■■■■/-■.■■■■■■■■■■■■■ magn=■■■.■■■■■ angle=■.■■■■
formula:
  maxiter=■■■■ percheck=off filename="dmj.ufm" entry="dmj-HNovaMandel"
  p_start=■/■ p_power=■/■ p_bailout=■.■■■■■ p_relax=■/■
inside:
  transfer=none solid=■■■■■■■■■■
outside:
  transfer=linear
gradient:
  smooth=yes index=■ color=■ index=■■■ color=■■■■■■■■ index=■■■
  color=■■■■■■■■ index=■■■ color=■■■■■■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Background" opacity=■■■ mergemode=hardlight
mapping:
  center=■.■■■■■■■■■■/■.■■■■■■■■■■■ magn=■■.■■■■■■ angle=-■■.■■■■
formula:
  maxiter=■■■■ percheck=off filename="dmj.ufm" entry="dmj-ManyNova"
  p_power=■/■ p_bailout=■.■■■■■ p_relax=■/■ p_scale=■.■ p_jscale=■.■
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes rotation=-■ index=■■ color=■■■■■■■■ index=■■
  color=■■■■■■■■ index=■■ color=■■■■■■■ index=■■ color=■■■■■■■
  index=■■ color=■■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■
  color=■■■■■■■■ index=-■■ color=■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Layer ■" opacity=■■ mergemode=multiply
mapping:
  center=-■.■■■■■■■■■■■■/■.■■■■■■■■■■■■ magn=■■.■■■■
formula:
  maxiter=■■■ filename="anon.ufm" entry="TZ■■■■-■■"
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes rotation=-■■■ index=■ color=■■■■■ index=■■■ color=■■■
  index=-■■■ color=■■■■■■■ index=-■■■ color=■■■ index=-■■■
  color=■■■■■■■ index=-■■■ color=■■■■■■■ index=-■■■ color=■■■■■■■■
  index=-■■■ color=■■■■■■■■ index=-■■■ color=■■■■■■■■ index=-■■■
  color=■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Layer ■" opacity=■■ mergemode=lighten
mapping:
  center=-■.■■■■■■■■■■■■■/■.■■■■■■■■■■■■■ magn=■■.■■■■■■
formula:
  maxiter=■■■ filename="anon.ufm" entry="TZ■■■■-■■"
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes rotation=-■■■ index=■ color=■■■■■ index=■■■ color=■■■
  index=-■■■ color=■■■■■■■ index=-■■■ color=■■■ index=-■■■
  color=■■■■■■■ index=-■■■ color=■■■■■■■ index=-■■■ color=■■■■■■■■
  index=-■■■ color=■■■■■■■■ index=-■■■ color=■■■■■■■■ index=-■■■
  color=■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Layer ■" opacity=■■■ mergemode=lighten
mapping:
  center=-■.■■■■■■■■■■■■■/■.■■■■■■■■■■■■■ magn=■■.■■■■■■
formula:
  maxiter=■■■ filename="anon.ufm" entry="TZ■■■■-■■"
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=■ color=■■■■■■■ index=■■ color=■■■ index=■■
  color=■■■■■■■ index=■■ color=■■■■■■■ index=■■ color=■■■■■■■■
  index=■■ color=■■■■■■■■ index=■■ color=■■■■■■■■ index=■■ color=■
  index=■■■ color=■■■■■ index=■■■ color=■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Background" opacity=■■■ mergemode=overlay
mapping:
  center=-■.■■■■■■■■■■■■/■.■■■■■■■■■■■■■ magn=■■.■■■■■■
formula:
  maxiter=■■■ filename="anon.ufm" entry="TZ■■■■-■■"
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=■ color=■■■■■■■ index=■■■ color=■■■■■■■■ index=■■■
  color=■■■■■ index=■■■ color=■■■
opacity:
  smooth=no index=■ opacity=■■■
layer:
  caption="Layer ■" opacity=■■
mapping:
  center=-■.■■■■■■■■■■■/■.■■■■■■■■■■■■ magn=■■.■■■■■■ angle=■■■.■■■■
formula:
  maxiter=■■■ filename="anon.ufm" entry="TZ■■■■-■■"
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=■ color=■■■■■■■ index=■■ color=■■■■■■■■ index=■■
  color=■■■■■■■■ index=■■ color=■■■■■■■■ index=■■■ color=■■■■■
  index=■■■ color=■■■
opacity:
  smooth=no index=■ opacity=■■■
}

Mandelbrot {
;
; Generic Mandelbrot set.
;
init:
  z = @start
loop:
  z = z^@power + #pixel
bailout:
  |z| <= @bailout

$IFDEF VER■■
perturbinit:
  #dz = ■
perturbloop:
  if @power == (■, ■)
    #dz = ■ ■ #z ■ #dz + sqr(#dz) + #dpixel
  elseif @power == (■, ■)
    complex z■ = sqr(#z)
    complex dz■ = sqr(#dz)
    #dz = ■ ■ z■ ■ #dz + ■ ■ #z ■ dz■ + #dz ■ dz■ + #dpixel
  else ; power ■
    complex z■ = sqr(#z)
    complex dz■ = sqr(#dz)
    complex zdz■ = ■■#z■#dz
    #dz = #dpixel + zdz■■z■ + ■■z■■dz■ + zdz■■dz■ + sqr(dz■)
  endif
$ENDIF

default:
  title = "Mandelbrot"
  center = (-■.■, ■)
  helpfile = "Uf■.chm"
  helptopic = "Html\formulas\standard\mandelbrot.html"
$IFDEF VER■■
  rating = recommended
$ENDIF
$IFDEF VER■■
  perturb = @power == (■, ■) || @power == (■, ■) || @power == (■, ■)
$ENDIF
  param start
    caption = "Starting point"
    default = (■,■)
    hint = "The starting point parameter can be used to distort the Mandelbrot \
            set. Use (■, ■) for the standard Mandelbrot set."
  endparam
  param power
    caption = "Power"
    default = (■,■)
    hint = "This parameter sets the exponent for the Mandelbrot formula. \
            Increasing the real part to ■, ■, and so on, will add discs to \
            the Mandelbrot figure. Non-integer real values and non-zero \
            imaginary values will create distorted Mandelbrot sets. Use (■, ■) \
            for the standard Mandelbrot set."
  endparam
  float param bailout
    caption = "Bailout value"
    default = ■.■
    min = ■.■
$IFDEF VER■■
    exponential = true
$ENDIF
    hint = "This parameter defines how soon an orbit bails out while \
            iterating. Larger values give smoother outlines; values around ■ \
            give more interesting shapes around the set. Values less than ■ \
            will distort the fractal."
  endparam
switch:
  type = "Julia"
  seed = #pixel
  power = power
  bailout = bailout
}

dmj-HNovaMandel {
;
; This is the Halley Nova fractal (Mandelbrot
; form), a modified Halley-style fractal. This
; is an adaptation of Paul Derbyshire's Halley
; Nova formulas for FractInt, which are based
; on his "Nova" formulas derived from classical
; Newton's Method fractals.
;
init:
  complex nsquaredplusn = sqr(@power) + @power
  complex nsquaredminusn = sqr(@power) - @power
  complex zton = (■,■)
  complex zold = (■,■)
  
  z = @start
  
loop:
  zold = z
  zton = z^@power
  z = z - (■■@power■z ■ (zton-■)) ■ @relax / \
      (nsquaredplusn■zton + nsquaredminusn) + #pixel

bailout:
  |z - zold| > @bailout
  
default:
  title = "HalleyNova (Mandelbrot)"
  helpfile = "dmj-pub\dmj-pub-uf-hn.htm"
  maxiter = ■■■■
  periodicity = ■
  center = (-■.■,■)
  magn = ■.■
  
  param start
    caption = "Start Value"
    default = (■,■)
    hint = "Starting value for each point.  You can use this to \
            'perturb' the fractal."
  endparam
  param power
    caption = "Exponent"
    default = (■,■)
    hint = "Overall exponent for the equation.  (■,■) gives \
            the classic HalleyNovaM type."
  endparam
  param bailout
    caption = "Bailout"
    default = ■.■■■■■
    hint = "Bailout value; smaller values will cause more \
            iterations to be done for each point."
  endparam
  param relax
    caption = "Relaxation"
    default = (■,■)
    hint = "This can be used to slow down the convergence of \
            the formula."
  endparam

switch:
  type = "dmj-HNovaJulia"
  seed = #pixel
  power = @power
  bailout = @bailout
  relax = @relax
}

dmj-ManyNova {
;
; This formula breaks the image up into a grid of
; squares, each square containing a small Nova Julia set
; using the c value from the center of the square.
;
init:
  float iscale = ■ / @scale
  c = round(#pixel ■ @scale) ■ iscale
  z = (#pixel - c) ■ @scale ■ @jscale

  complex zsquared = (■,■)
  complex zcubed = (■,■)
  complex zold = (■,■)
  
loop:
  IF (@power == (■,■)); special optimized routine for power ■
    zsquared = sqr(z)
    zcubed = zsquared ■ z
    zold = z
    z = z - @relax ■ (zcubed-■) / (■■zsquared) + c
  ELSE
    zold = z
    z = z - @relax ■ (z^@power-■) / (@power ■ z^(@power-■)) + c
  ENDIF
  
bailout:
  |z-zold| > @bailout
    
default:
  title = "ManyNova"
  helpfile = "dmj-pub\dmj-pub-uf-manynova.htm"
  maxiter = ■■■■
  periodicity = ■
  center = (■,■)
  magn = ■.■
  
  param power
    caption = "Exponent"
    default = (■,■)
    hint = "Overall exponent for the equation.  (■,■) gives \
            the classic NovaM type."
  endparam
  param bailout
    caption = "Bailout"
    default = ■.■■■■■
    hint = "Bailout value; smaller values will cause more \
            iterations to be done for each point."
  endparam
  param relax
    caption = "Relaxation"
    default = (■,■)
    hint = "This can be used to slow down the convergence of \
            the formula."
  endparam
  param scale
    caption = "Julia Density"
    default = ■.■
    hint = "Specifies the density of separate Julia sets; higher \
            numbers will produce more divisions."
  endparam
  param jscale
    caption = "Julia Zoom"
    default = ■.■
    hint = "Specifies the zoom level of Julia sets within each division."
  endparam
}

TZ■■■■-■■ {
;
;from TieraZon■ fl-■■-■■.fll
;■■.) init z = ■; z = z-(z■z■-t■zcos-z)/(■■z■-zsin-z)+c;
;
init:
  z = ■
  t = z
loop:
  z = z - (z ■ z^■ - t ■ cos(z) - z) / (■ ■ z^■ - sin(z) - z) + #pixel
bailout:
  |z| < ■
}
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