Fractal 046 G – Abyss Medusa
Part of LIFE series – 11 2021
abyss . coral . couple . jellyfish . ocean . plankton . sea . sponge . translucent . underwater
Fractal 046 G – zoom level 1
Fractal 046 G – 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 In each case, the numerical characters have been replaced by â– to prevent unauthorized reproduction. If you are interested in using the full data set to train an AI, please contact Philippe.
Fractal 046 G - Abyss Medusa - Part of LIFE series - 11.2021
Fractal_â– â– â– _G {
fractal:
title="Fractal_â– â– â– _G" width=â– â– â– â– height=â– â– â– â– layers=â–
credits="Philoxerax;â– /â– â– /â– â– â– â– "
layer:
caption="Background" opacity=â– â– mergemode=overlay method=multipass
mapping:
center=â– .â– â– â– â– â– â– â– â– â– â– â– â– â– /â– .â– â– â– â– â– â– â– â– â– â– â– â– â– magn=â– â– .â– â– â– â– â– â– angle=â– â– .â– â– â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm" entry="gap-julia"
p_n=â– .â– p_julparam=â– /â– p_type="between â– lines" p_câ– =-â– .â– p_câ– =â– .â–
p_theta=â– â– .â– p_center=â– /â– p_radiusâ– =â– .â– p_radiusâ– =â– .â–
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=-â– â– â– index=â– â– â– color=â– â– â– index=-â– â–
color=â– â– â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Background" opacity=â– â– mergemode=screen
mapping:
center=â– â– â– â– â– â– .â– â– â– â– /-â– â– â– â– â– .â– â– â– â– â– â– magn=â– .â– â– â– â– â– â– â– â– â– angle=â– â– .â– â– â– â–
stretch=â– .â– â– â– â– skew=â– .â– â– â– â–
formula:
maxiter=â– â– â– â– percheck=off filename="dmj.ufm"
entry="dmj-fBmNovaMandel" p_start=â– /â– p_power=â– /â– p_bailout=â– .â– â– â– â– â–
p_relax=â– /â– p_distortion=â– .â– p_style=radial p_distangle=â– .â–
p_distcenter=â– /â– p_centermove=no p_offset=â– /â– p_scale=â– .â–
p_angle=â– .â– p_step=â– .â– p_anglestep=â– â– .â– p_octaves=â– p_npower=â– .â–
p_noisestart=â– .â– p_noiseiter=â– â– â– â– â– .â– p_noiseskip=â– .â– p_coloronly=no
inside:
transfer=none solid=â– â– â– â– â– â– â– â– â– â–
outside:
transfer=linear
gradient:
smooth=yes rotation=â– â– â– index=â– â– â– color=â– index=â– â– â– color=â–
index=â– â– â– color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Layer â– â– " opacity=â– â– â– mergemode=overlay method=multipass
mapping:
center=-â– .â– â– â– â– â– â– â– â– â– â– â– â– eâ– â– /-â– .â– â– â– â– â– â– â– â– â– eâ– â– magn=â– .â– â– â– â– â– â– â– E-â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm"
entry="divide-and-average" p_julparam=â– .â– /â– .â– p_nexp=â– .â–
p_quotient=â– .â– /â– .â– p_bailout=â– â– â– .â– p_convtype=magnitude
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=â– â– index=â– â– color=â– â– â– â– â– â– â– index=â– â– â–
color=â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Layer â– " opacity=â– â– â– mergemode=screen method=multipass
mapping:
center=â– .â– â– â– â– â– â– â– â– â– â– â– â– /â– .â– â– â– â– â– â– â– â– â– â– â– magn=â– â– .â– â– â– â– â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm"
entry="compounding-tweaked-mandelbrot" p_manparam=â– .â– /â– .â–
p_bailout=â– â– â– â– .â– p_nexp=â– /â– p_tweaktype="c; fn(c)"
p_tweakage=â– .â– â– /â– .â– f_tweakfunction=recip
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=â– â– index=â– â– color=â– â– â– â– â– â– â– index=â– â– â–
color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â–
index=â– â– â– color=â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Layer â– " opacity=â– â– â– mergemode=softlight method=multipass
mapping:
center=â– .â– â– â– â– â– â– â– â– â– â– â– â– /â– .â– â– â– â– â– â– â– â– â– â– â– magn=â– â– .â– â– â– â– â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm"
entry="compounding-tweaked-mandelbrot" p_manparam=â– .â– /â– .â–
p_bailout=â– â– â– â– .â– p_nexp=â– /â– p_tweaktype="c; fn(c)"
p_tweakage=â– .â– â– /â– .â– f_tweakfunction=recip
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=â– â– index=â– â– color=â– â– â– â– â– â– â– index=â– â– â–
color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â–
index=â– â– â– color=â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Background" opacity=â– â– â– method=multipass
mapping:
center=â– .â– â– â– â– â– â– â– â– â– â– â– â– â– /â– .â– â– â– â– â– â– â– â– â– â– â– â– magn=â– .â– â– â– â– â– â– â– angle=â– â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm"
entry="compounding-tweaked-mandelbrot" p_manparam=â– .â– /â– .â–
p_bailout=â– â– â– â– .â– p_nexp=â– /â– p_tweaktype="c; fn(c)"
p_tweakage=â– .â– â– /â– .â– f_tweakfunction=recip
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– index=â– â– â–
color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â– index=â– â– â– color=â– â– â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
}
gap-julia { ; Kerry Mitchell â– â– decâ– â–
;
; z^n+c Julia
; bails out when orbit falls into gap
; either between â– circles or â– lines
;
; updates:
; â– â– febâ– â– to add complex exponent
; â– â– febâ– â– to add switching to gap-julia
; â– â– marâ– â– removed complex exponent for backwards compatibility
;
init:
z=#pixel
c=@julparam
float a=â– .â–
float b=â– .â–
float gap=â– .â–
float radsqrâ– =sqr(@radiusâ– )
float radsqrâ– =sqr(@radiusâ– )
float x=â– .â–
float y=â– .â–
float rmax=â– eâ– â–
float tempr=â– .â–
int done=â–
;
; set up line/circle parameters
;
if(@type==â– ) ; lines
tempr=@theta/â– â– â– â– pi
a=sin(tempr)
b=-cos(tempr)
gap=@câ– -@câ–
else ; circles
gap=radsqrâ– -radsqrâ–
endif
loop:
z=z^@n+c
x=real(z)
y=imag(z)
;
; check for falling into gap
;
if(@type==â– ) ; lines
tempr=aâ– x+bâ– y
if((tempr>@câ– )&&(tempr<@câ– ))
done=â–
tempr=(tempr-@câ– )/gap
z=temprâ– z/cabs(z)
endif
else ; circles
tempr=|z-@center|
if((tempr>radsqrâ– )&&(temprrmax))
done=â–
z=(â– .â– ,â– .â– )
endif
bailout:
done==â–
default:
title="Gap Julia"
maxiter=â– â– â–
periodicity=â–
center=(â– ,â– )
method=multipass
magn=â–
angle=â–
param n
caption="z exponent"
default=â– .â–
hint="Real--use Gap Julia C for complex exponents."
endparam
param julparam
caption="Julia parameter"
default=(â– ,â– )
endparam
param type
caption="gap type"
default=â–
enum="between â– lines" "between â– circles"
endparam
param câ–
caption="line â– c value"
default=-â– .â–
hint="must be less than line â– c value"
endparam
param câ–
caption="line â– c value"
default=â– .â–
hint="must be more than line â– c value"
endparam
param theta
caption="line angle"
default=â– â– .â–
hint="angle to horizontal, degrees"
endparam
param center
caption="circle center"
default=(â– ,â– )
endparam
param radiusâ–
caption="circle â– radius"
default=â– .â–
hint="must be less than circle â– radius"
endparam
param radiusâ–
caption="circle â– radius"
default=â– .â–
hint="must be more than circle â– radius"
endparam
switch:
type="gap-mandelbrot"
n=n
type=type
câ– =câ–
câ– =câ–
theta=theta
center=center
radiusâ– =radiusâ–
radiusâ– =radiusâ–
}
dmj-fBmNovaMandel {
;
; This is the basic Nova (Mandelbrot) type, but with a
; bit of fBm noise added at each iteration. This
; tends to distort the fractal beyond all recognition
; after just a few iterations, which may or may not
; be what you're looking for.
;
; You can also use the "Coloring Only" option to
; restrict the fBm distortion to the value passed
; to the coloring algorithm; the distortion will be
; removed before the next iteration is calculated.
;
init:
complex zsquared = (â– ,â– )
complex zcubed = (â– ,â– )
complex zold = (â– ,â– )
z = @start
complex oz = z
complex c = @distcenter
IF (@centermove)
c = #center
ENDIF
complex r = (â– ,â– ) ^ (@angle / â– â– .â– )
complex râ– = (â– ,â– ) ^ (@anglestep / â– â– .â– )
complex râ– = (â– ,â– ) ^ (@distangle / â– â– .â– )
float fiter = @noisestart
BOOL noise = false
loop:
IF (@noiseskip != â– ); we are skipping some iterations
fiter = fiter - â– ; one less to go before we add noise
WHILE (fiter < â– .â– ); iterations all used up
IF (noise); we are currently adding noise
noise = false; so stop
fiter = fiter + @noiseskip; skip this many iterations
ELSE; we aren't currently adding noise
noise = true; so start
fiter = fiter + @noiseiter; do this many iterations
ENDIF
ENDWHILE
ENDIF
IF (@coloronly); only using fBm on coloring
z = oz; restore z from un-fBm'ed copy
ENDIF
IF (@power == (â– ,â– )); special optimized routine for power â–
zsquared = sqr(z)
zcubed = zsquared â– z
zold = z
z = z - @relax â– (zcubed-â– ) / (â– â– zsquared) + #pixel
ELSE
zold = z
z = z - @relax â– (z^@power-â– ) / (@power â– z^(@power-â– )) + #pixel
ENDIF
IF (@coloronly); only using fBm on coloring
oz = z
ENDIF
IF (@noiseskip == â– .â– || noise); adding noise this iteration
complex p = z â– @scale â– r ;+ @offset
float sum = â– .â–
float freq = â– .â–
complex v = (â– ,â– )
int i = @octaves
WHILE (i > â– )
; determine integer coordinate for corners of square
; surrounding p
float bxâ– = floor(real(p)) % â– â– â–
float byâ– = floor(imag(p)) % â– â– â–
IF (bxâ– < â– )
bxâ– = bxâ– + â– â– â–
ENDIF
IF (byâ– < â– )
byâ– = byâ– + â– â– â–
ENDIF
float bxâ– = (bxâ– + â– ) % â– â– â–
float byâ– = (byâ– + â– ) % â– â– â–
float rxâ– = real(p) - floor(real(p))
float ryâ– = imag(p) - floor(imag(p))
float rxâ– = rxâ– - â–
float ryâ– = ryâ– - â–
; create a "random" index for each corner
; (this is where Intel's version differs from Perlin's;
; I used Intel's version because it doesn't require a
; pre-computed random table, which is difficult to manage
; in UF.)
float bâ– â– = (bxâ– ^@npower % â– â– â– â– â– + byâ– )^@npower % â– â– â– â– â–
float bâ– â– = (bxâ– ^@npower % â– â– â– â– â– + byâ– )^@npower % â– â– â– â– â–
float bâ– â– = (bxâ– ^@npower % â– â– â– â– â– + byâ– )^@npower % â– â– â– â– â–
float bâ– â– = (bxâ– ^@npower % â– â– â– â– â– + byâ– )^@npower % â– â– â– â– â–
; produce a "random" vector for each corner
float g_bâ– â– _â– = (bâ– â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– +â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– +â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– +â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
float g_bâ– â– _â– = (bâ– â– +â– )^@npowerâ– â– .â– â– % â– â– â– - â– â– â–
; normalize each vector
float d = â– .â– ;
d = â– / sqrt(sqr(g_bâ– â– _â– ) + sqr(g_bâ– â– _â– ))
g_bâ– â– _â– = g_bâ– â– _â– â– d
g_bâ– â– _â– = g_bâ– â– _â– â– d
d = â– / sqrt(sqr(g_bâ– â– _â– ) + sqr(g_bâ– â– _â– ))
g_bâ– â– _â– = g_bâ– â– _â– â– d
g_bâ– â– _â– = g_bâ– â– _â– â– d
d = â– / sqrt(sqr(g_bâ– â– _â– ) + sqr(g_bâ– â– _â– ))
g_bâ– â– _â– = g_bâ– â– _â– â– d
g_bâ– â– _â– = g_bâ– â– _â– â– d
d = â– / sqrt(sqr(g_bâ– â– _â– ) + sqr(g_bâ– â– _â– ))
g_bâ– â– _â– = g_bâ– â– _â– â– d
g_bâ– â– _â– = g_bâ– â– _â– â– d
; produce colors for each corner
float uâ– = rxâ– â– g_bâ– â– _â– + ryâ– â– g_bâ– â– _â–
float vâ– = rxâ– â– g_bâ– â– _â– + ryâ– â– g_bâ– â– _â–
float uâ– = rxâ– â– g_bâ– â– _â– + ryâ– â– g_bâ– â– _â–
float vâ– = rxâ– â– g_bâ– â– _â– + ryâ– â– g_bâ– â– _â–
; interpolate between corners using
; bilinear filtering
float sx = sqr(rxâ– ) â– (â– - rxâ– â– â– )
float sy = sqr(ryâ– ) â– (â– - ryâ– â– â– )
float a = uâ– + sxâ– (vâ– -uâ– )
float b = uâ– + sxâ– (vâ– -uâ– )
sum = sum + (a + syâ– (b-a))â– freq
freq = freq â– @step
p = p â– râ– / @step
i = i - â–
ENDWHILE
IF (@style == â– ); radial distortion
v = (z-c)/cabs(z-c) â– râ– ; use vector based on angle to distortion center
ELSEIF (@style == â– ); linear distortion
v = râ– ; just use rotation vector
ENDIF
z = z + v â– sumâ– â– .â– â– @distortion
ENDIF
IF (@coloronly == false); not just using fBm on coloring
oz = z; value for bailout is fBm'ed z
ENDIF
bailout:
|oz - zold| > @bailout
default:
title = "Nova (Mandelbrot) + fBm"
helpfile = "dmj-pub\dmj-pub-uf-nf.htm"
maxiter = â– â– â– â–
periodicity = â–
center = (-â– .â– ,â– )
magn = â– .â–
param start
caption = "Start Value"
default = (â– ,â– )
hint = "Starting value for each point. You can use this to \
'perturb' the fractal."
endparam
param power
caption = "Exponent"
default = (â– ,â– )
hint = "Overall exponent for the equation. (â– ,â– ) gives \
the classic NovaM type."
endparam
param bailout
caption = "Bailout"
default = â– .â– â– â– â– â–
hint = "Bailout value; smaller values will cause more \
iterations to be done for each point."
endparam
param relax
caption = "Relaxation"
default = (â– ,â– )
hint = "This can be used to slow down the convergence of \
the formula."
endparam
param distortion
caption = "Distortion Strength"
default = â– .â–
hint = "This is the amount the noise distorts the image."
endparam
param style
caption = "Distortion Style"
default = â–
enum = "radial" "linear"
hint = "This selects whether the distortion will be focused \
around a single point, or directed along a line."
endparam
param distangle
caption = "Distortion Angle"
default = â– .â–
hint = "This is the angle to rotate the distortion."
endparam
param distcenter
caption = "Distortion Center"
default = (â– ,â– )
hint = "Sets the center of distortion. If Use Screen \
Center is set, this item is ignored."
endparam
param centermove
caption = "Use Screen Center"
default = FALSE
hint = "If set, distortion will be around the center of \
the window, regardless of the Distortion Center \
setting."
endparam
param offset
caption = "Noise Offset"
default = (â– ,â– )
hint = "This is the offset of the pattern. You can use this to shift \
the pattern around on the complex plane."
endparam
param scale
caption = "Noise Scale"
default = â– .â–
hint = "This is the overall scale of the noise."
endparam
param angle
caption = "Noise Rotation"
default = â– .â–
hint = "This is the angle, in degrees, of the noise."
endparam
param step
caption = "Noise Scale Step"
default = â– .â–
hint = "This is the step in scale between noise iterations."
endparam
param anglestep
caption = "Noise Rotation Step"
default = â– â– .â–
hint = "This is the angle, in degrees, to rotate between noise \
iterations."
endparam
param octaves
caption = "Noise Octaves"
default = â–
min = â–
hint = "This is the number of iterations of the noise formula."
endparam
param npower
caption = "Noise Exponent"
default = â– .â–
hint = "This is the exponent used to scramble numbers."
endparam
param noisestart
caption = "Start Iteration"
default = â– .â–
hint = "This is the iteration at which to start adding noise."
endparam
param noiseiter
caption = "Noise Iterations"
default = â– â– â– â– â– .â–
hint = "This is the number of iterations to add noise to."
endparam
param noiseskip
caption = "Skip Iterations"
default = â– .â–
hint = "This is the number of iterations to skip adding noise \
before starting again."
endparam
param coloronly
caption = "Coloring Only"
default = false
hint = "If set, noise will only apply to pixel values passed \
to the coloring algorithm; it will not be included in \
the fractal calculation between iterations."
endparam
switch:
type = "dmj-fBmNovaJulia"
seed = #pixel
power = @power
bailout = @bailout
relax = @relax
distortion = @distortion
style = @style
distangle = @distangle
distcenter = @distcenter
centermove = @centermove
offset = @offset
scale = @scale
angle = @angle
step = @step
anglestep = @anglestep
octaves = @octaves
npower = @npower
noisestart = @noisestart
noiseiter = @noiseiter
noiseskip = @noiseskip
coloronly = @coloronly
}
divide-and-average { ; Kerry Mitchell â– â– junâ– â–
;
; Based on Arne Richter's variation of the Newton
; method for finding a square root
;
init:
znew=#pixel
c=@julparam
q=â– /@quotient
float nmâ– =â– .â– -@nexp
float r=â– .â–
loop:
zold=znew
if(@nexp==â– .â– )
znew=qâ– (zold+c/zold)
else
znew=qâ– (-nmâ– â– zold+câ– (zold^nmâ– ))
endif
#z=â– /(znew-zold)
if(@convtype==â– )
r=abs(real(#z))
elseif(@convtype==â– )
r=abs(imag(#z))
else
r=cabs(#z)
endif
bailout:
r<@bailout
default:
title="Divide and Average"
maxiter=â– â– â–
periodicity=â–
center=(â– .â– ,â– .â– )
magn=â– .â–
angle=â–
method=multipass
param julparam
caption="seed"
default=(â– .â– ,â– .â– )
endparam
param nexp
caption="exponent"
default=â– .â–
endparam
param quotient
caption="quotient"
default=(â– .â– ,â– .â– )
hint="make = exponent for standard Newton's method"
endparam
param bailout
caption="bailout"
default=â– â– â– .â–
hint="â– /(difference between subsequent iterates)--make big"
endparam
param convtype
caption="convergence type"
enum="magnitude" "real" "imaginary"
default=â–
endparam
}
compounding-tweaked-mandelbrot { ; Kerry Mitchell â– â– octâ– â–
;
; Tweaks either c or z each iteration, so that the tweaking compounds
;
init:
z=@manparam
c=#pixel
loop:
z=z^@nexp+c
if(@tweaktype==â– )
c=c+@tweakageâ– @tweakfunction(z)
elseif(@tweaktype==â– )
c=c+@tweakageâ– @tweakfunction(câ– z)
elseif(@tweaktype==â– )
c=c+@tweakageâ– @tweakfunction(z/c)
elseif(@tweaktype==â– )
z=z+@tweakageâ– @tweakfunction(c)
elseif(@tweaktype==â– )
z=z+@tweakageâ– @tweakfunction(z)
elseif(@tweaktype==â– )
z=z+@tweakageâ– @tweakfunction(câ– z)
elseif(@tweaktype==â– )
z=z+@tweakageâ– @tweakfunction(z/c)
else
c=c+@tweakageâ– @tweakfunction(c)
endif
bailout:
|z|<@bailout
default:
title="Compounding Tweaked Mandelbrot"
helpfile="lkm-help\lkm-compounding.html"
maxiter=â– â– â–
periodicity=â–
center=(-â– .â– ,â– .â– )
magn=â– .â–
angle=â–
method=multipass
param manparam
caption="initial z"
default=(â– .â– ,â– .â– )
hint="Use (â– ,â– ) for something resembling the \
standard Mandelbrot set."
endparam
param bailout
caption="bailout value"
default=â– â– â– â– .â–
endparam
param nexp
caption="exponent"
default=(â– ,â– )
hint="Use â– .â– for something resembling the \
standard Mandelbrot set."
endparam
param tweaktype
caption="tweaking type"
default=â–
enum="c; fn(c)" "c; fn(z)" "c; fn(câ– z)" "c; fn(z/c)" \
"z; fn(c)" "z; fn(z)" "z; fn(câ– z)" "z; fn(z/c)"
hint="Sets what gets tweaked, and how."
endparam
param tweakage
caption="tweaking amount"
default=(â– .â– â– ,â– .â– )
hint="Make small for something resembling the \
standard Mandelbrot set."
endparam
func tweakfunction
caption="tweaking function"
default=recip()
hint="Function of the tweaking variable."
endfunc
switch:
type="compounding-tweaked-julia"
julparam=#pixel
nexp=nexp
bailout=bailout
tweaktype=tweaktype
tweakage=tweakage
}
