Fractal 058 A – The Big Juggler
Part of BEYOND series – 12 2021
alchemy . architect . artist . cosmos . creation . god . juggler
Fractal 058 A – zoom level 1
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 In each case, the numerical characters have been replaced by â– to prevent unauthorized reproduction.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_â– â– â– _A {
fractal:
title="Fractal_â– â– â– _A" width=â– â– â– â– height=â– â– â– â– layers=â–
credits="Philoxerax;â– /â– /â– â– â– â– "
layer:
caption="Background" opacity=â– â– â– mergemode=addition
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– â– magn=â– â– â– .â– â– â– â– â–
angle=-â– â– â– .â– â– â– â–
formula:
maxiter=â– â– â– filename="lkmâ– .ufm" entry="extra-foam"
p_pixel_type="all in" p_wq=yes p_zc=yes p_circuit_type=series
p_bailout=â– â– â– â– p_qinit=â– /â– p_cinit=â– /â– p_winit=â– /â– p_zinit=â– /â–
p_wpowerinw=â– /â– p_zpowerinw=-â– /â– p_wpowerinz=â– /â– p_zpowerinz=â– /â–
inside:
transfer=none
outside:
transfer=linear
gradient:
smooth=yes rotation=-â– â– â– index=â– â– color=â– â– â– â– â– index=â– â– â– color=â– â–
index=-â– â– â– color=â– â– â– â– â– â– â– index=-â– â– color=â– â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Background" opacity=â– â– â– mergemode=difference
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– magn=â– â– .â– â– â– â– â– â– angle=-â– .â– â– â– â–
formula:
maxiter=â– â– â– percheck=off filename="lkm.ufm" entry="gap-mandelbrot"
p_manparam=â– /â– p_n=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– 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–
opacity:
smooth=no index=â– opacity=â– â– â–
layer:
caption="Background" opacity=â– â– â–
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– â– â– â– â– magn=â– â– â– â– â– .â– â– â–
angle=-â– â– â– .â– â– â– â–
formula:
maxiter=â– â– â– filename="lkmâ– .ufm" entry="compound-mandelbrot"
p_initialz_type=pixel p_initialz=â– /â– p_npower=â– /â– p_mpower=â– /â–
p_ppower=â– /â– p_bailout=â– â– â– â–
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– color=â– â– â– â– â– index=â– â– â– color=â– â– â–
index=-â– â– â– color=â– â– â– â– â– â– â– index=-â– â– color=â– â– â– â– â– â– â– â–
opacity:
smooth=no index=â– opacity=â– â– â–
}
extra-foam { ; Kerry Mitchell â– â– June â– â– â– â–
;
; Generalization of "Mandelbrot Foam" from fractal forums
; https://fractalforums.org/fractal-mathematics-and-new-theories/â– â– /mandelbrot-foam-math/â– â– â– â– /msgâ– â– â– â– #msgâ– â– â– â–
; original:
; w = aâ– w/z
; z = z^â– + w^â– + 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=(â– ,â– )
complex c=(â– ,â– )
complex w=(â– ,â– )
complex wnew=(â– ,â– )
complex z=(â– ,â– )
complex znew=(â– ,â– )
;
; 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=qâ– w^@wpowerinwâ– z^@zpowerinw
z=z^@zpowerinz+w^@wpowerinz+c
elseif(@circuit_type=="parallel")
wnew=qâ– w^@wpowerinwâ– 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=â–
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=â–
enum="series" "parallel"
endparam
float param bailout
caption="bailout"
default=â– â– â– â–
endparam
;
; Initializations
;
heading
caption="Initializations"
endheading
complex param qinit
caption="q"
default=(â– ,â– )
visible=((@pixel_type=="c")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
endparam
complex param cinit
caption="c"
default=(â– ,â– )
visible=((@pixel_type=="q")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
endparam
complex param winit
caption="initial w"
default=(â– ,â– )
visible=((@wq==false)&&(@pixel_type!="initial w")&&(@pixel_type!="w = z"))
endparam
complex param zinit
caption="initial z"
default=(â– ,â– )
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=â–
endparam
complex param zpowerinw
caption="z power in w"
default=-â–
endparam
complex param wpowerinz
caption="w power in z"
default=â–
endparam
complex param zpowerinz
caption="z power in z"
default=â–
endparam
}
gap-mandelbrot { ; Kerry Mitchell â– â– decâ– â–
;
; z^n+c Mandelbrot
; bails out when orbit falls into gap
; either between â– circles or â– lines
;
; updates:
; â– â– decâ– â– to add general starting point
; 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=@manparam
c=#pixel
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 Mandelbrot"
maxiter=â– â– â–
periodicity=â–
center=(â– ,â– )
method=multipass
magn=â–
angle=â–
param manparam
caption="Mandelbrot start"
default=(â– ,â– )
hint="use (â– ,â– ) for basic Mandelbrot set"
endparam
param n
caption="z exponent"
default=â– .â–
hint="Real--use Gap Mandelbrot C for complex exponents."
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-julia"
n=n
julparam=#pixel
type=type
câ– =câ–
câ– =câ–
theta=theta
center=center
radiusâ– =radiusâ–
radiusâ– =radiusâ–
}
compound-mandelbrot { ; Kerry Mitchell â– â– Novâ– â– â– â–
;
; f(z) = ((z^n+c)/(z^m+c))^p+c
;
init:
complex c=#pixel
complex z=(â– ,â– )
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; â– exponents."
endheading
param initialz_type
caption="initial z"
default=â–
enum="pixel" "constant"
endparam
complex param initialz
caption="initial z value"
default=(â– ,â– )
visible=(@initialz_type=="constant")
endparam
complex param npower
caption="numerator power"
default=(â– ,â– )
endparam
complex param mpower
caption="denominator power"
default=(â– ,â– )
endparam
complex param ppower
caption="overall power"
default=(â– ,â– )
endparam
float param bailout
caption="bailout"
default=â– â– â– â–
endparam
switch:
type="compound-julia"
julparam=#pixel
npower=npower
mpower=mpower
ppower=ppower
bailout=bailout
}
