Crystal Globe

From Ultrafractal Wiki

Crystal Globe (Julia) and Crystal Globe (Mandelbrot) are Fractal Formulas in sam.ufm.

From Samuel Monnier's Help File

Contents 1 What is it? 2 The Parameters 2.1 Starting point 2.2 Power 2.3 Bailout value 2.4 Inside mode 2.5 Outside mode 2.6 Inside parameter 2.7 On-the-trap parameter 2.8 Outside parameter 2.9 Inside spin 2.10 Outside spin 2.11 Shape 2.12 Flip shape? 2.13 Mirror shape? 2.14 Shape offset 2.15 Center of the Shape 2.16 Shape periodicity 2.17 Smoothy periodic? 2.18 Concentric Periodicity 2.19 Size of the shape 2.20 Ratio width/height 2.21 Shape parameter 2.22 Graph function 2.23 Extra graph function 3 Examples

What is it?

These formulas were inspired by the Orbit Traps colourings and the Barnsley 1 formula. They are based on the equation Z = (z + p)^power + #pixel where p is a parameter depending on where the current z is. It works like an orbit trap.

The Parameters

Starting point

As in the standard Mandelbrot set, it's the first value of z.

Power

The power of (z + p)

Bailout value

When the fomula decide that a point doesn't belong to the set. If you see strange black regions, decreasing the bailout can help.

Inside mode

The transfer function from the location of z to the parameter when z is inside the trap (or f(x,y) < 0 for open shapes). In the first mode, Raw, when the point falls inside the trap, p takes the "Inside Parameter" value. In the second mode, Smooth, p vary linearily between the "On-the-Trap parameter" and the "Inside parameter". In the third mode, the first derivative of the transfer function is 0 at the extremities. This give a smoother transition.

Outside mode

Idem "Inside mode" but this time when the point is outside the trap. You have to modify the outside parameter so that it's different from the On-the-trap parameter. Else it won't vary outside.

Inside parameter

The value p takes when it's inside the trap.

On-the-trap parameter

The value p takes when it's on the trap. (Useless when the inside mode and the outside mode are "Raw".)

Outside parameter

The value p takes when it's outside the trap.

Inside spin

This enables you to multiply p by a factor e^Insidespin. In the complex plane, p will move along a spiral instead of along a line. Useless if you use the "Raw mode".

Outside spin

The same as Inside spin, but when p is outside the trap.

Shape

The shape of the trap. The closed shapes are : the circle, the rectangle, the astroïd, Bernoulli's lemniscate and the folium. The other shapes are open and sometimes work differently.

Flip shape?

Enables you to flip the shape. (Horizontal lines are mapped into vertical ones and vice versa.)

Mirror shape?

Allow you to mirror the shape.

Shape offset

Only for open shapes. Such shapes are usually described by a function like that :

f(x) + g(y) = 0. 

This parameter modifies the function so :

f(x) + g(y) = "offset". 

The result is that the shape is moved and deformed.

Center of the Shape

Just an advice, if you see a black screen, try changing this parameter.

Shape periodicity

The shape is repeated periodically.

Smoothy periodic?

Use a smooth repeat function (sin) instead of the usual function.

Concentric Periodicity

Only for open shapes. Instead of placing the shapes on a gird, they repeat concentrically. For the circle, this gives a kind of ripple. I haven't manage to get this option to work with the rectangles. So it is (temporarily, I hope) disabled with them.

Size of the shape

No comment...

Ratio width/height

...

Shape parameter

A parameter for the shape. Sometimes it won't change anything. I set the default to 1 so that it works well with the astroïd, but with the pinch, set it to 3 or more to get the standard shape.

Graph function

Only with the Graph shape.

Extra graph function

Same as Graph function.

N.B.: To really appreciate the smooth and very smooth modes, use Smoothed Iterations (Mandelbrot) in dmj.ucl or a similar colouring. If the image isn't smooth, try modifying the "exponent" value of the colouring. make also the bailout of the formula and of the colouring match. Idem with the triangle inequality average and cilia coloring.

Examples

Copyright 1999 Samuel Monnier

lowo {
; Copyrights 1999 S. Monnier
; http://www.crosswinds.net/~s31415/index/index.htm
fractal:
  title="lowo" width=399 height=600 author="Samuel"
  created="September 22, 1999" numlayers=3
layer:
  method=multipass caption="New Layer 5" opacity=100 visible=yes alpha=no
  mergemode=hardlight
mapping:
  center=0.190865772584357216/0.388465799802877696 magn=91.3519320640363264
  angle=200.616456320492474
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=100 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=10000000000 p_mode1="Raw" p_mode2="Raw"
  p_p1=-0.035/0.65 p_p2=-1/0 p_p3=-1/0 p_ispin=0 p_ospin=0 p_mode="Circle"
  p_flip=no p_mirror=no p_offset=0 p_center=0/0 p_rot=0 p_period=2
  p_smper=yes p_concper=no p_size=0.2 p_r=1 p_n=1 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="mt.ucl" entry="mt-trifunctional" transfer=linear repeat=yes
  p_mode="Closest" p_flavor="1" p_coloring="Distance" p_r=0.1 f_fn1=sin
  f_fn2=log f_fn3=sqrt
gradient:
  smooth=no numnodes=4 index=54 color=0 index=152 color=0 index=250
  color=4210752 index=399 color=10263708
layer:
  method=multipass caption="New Layer 1" opacity=71 visible=yes alpha=no
  mergemode=overlay
mapping:
  center=0.190865772584357216/0.388465799802877696 magn=91.3519320640363264
  angle=200.616456320492474
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=100 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=10000000000 p_mode1="Raw" p_mode2="Raw"
  p_p1=-0.035/0.65 p_p2=-1/0 p_p3=-1/0 p_ispin=0 p_ospin=0 p_mode="Circle"
  p_flip=no p_mirror=no p_offset=0 p_center=0/0 p_rot=0 p_period=2
  p_smper=yes p_concper=no p_size=0.2 p_r=1 p_n=1 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Lyapunov" density=0.100000000000000006
  transfer=linear repeat=yes p_trackvariable="real part of z"
  p_negative="absolute value"
gradient:
  smooth=no numnodes=2 index=1 color=0 index=64 color=16777215
layer:
  method=multipass caption="New Layer 3" opacity=90 visible=yes alpha=no
  mergemode=difference
mapping:
  center=0.190865772584357216/0.388465799802877696 magn=91.3519320640363264
  angle=200.616456320492474
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=100 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=10000000000 p_mode1="Raw" p_mode2="Raw"
  p_p1=-0.035/0.65 p_p2=-1/0 p_p3=-1/0 p_ispin=0 p_ospin=0 p_mode="Circle"
  p_flip=no p_mirror=no p_offset=0 p_center=0/0 p_rot=0 p_period=2
  p_smper=yes p_concper=no p_size=0.2 p_r=1 p_n=1 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Triangle" density=2 transfer=linear
  repeat=yes p_apower=1 p_aflavor="normal" p_power=2 p_bailout=10000000000
gradient:
  smooth=no numnodes=2 index=56 color=3868 index=249 color=16777215
}

Copyright 1999 Samuel Monnier

(uses Smoothed Iterations)

globe2 {
; Copyrights 1999 S. Monnier
; http://www.crosswinds.net/~s31415/index/index.htm
fractal:
  title="globe2" width=600 height=600 author="Samuel"
  created="September 19, 1999" numlayers=4
layer:
  caption="New Layer 2" opacity=100 visible=yes alpha=no mergemode=overlay
mapping:
  center=-1.7692707111695472/0.0569176768789583808 magn=14179040788077.6144
  angle=63.4987455814236288
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=10000 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=1E20 p_mode1="Very Smooth (C1)"
  p_mode2="Very Smooth (C1)" p_p1=1/0 p_p2=0/0 p_p3=0/0 p_ispin=0 p_ospin=0
  p_mode="Agnesi's Curve" p_flip=no p_mirror=no p_offset=0
  p_center=-1.815/-0.375 p_rot=0 p_period=0 p_smper=yes p_concper=no
  p_size=0.2 p_r=0.2 p_n=0 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Triangle" density=4 transfer=linear
  repeat=yes p_apower=1 p_aflavor="normal" p_power=2 p_bailout=1E20
gradient:
  smooth=no numnodes=6 index=0 color=16777215 index=39 color=0 index=122
  color=16777215 index=355 color=16777215 index=368 color=12105912
  index=379 color=0
layer:
  caption="New Layer 3" opacity=100 visible=yes alpha=yes
mapping:
  center=-1.7692707111695472/0.0569176768789583808 magn=14179040788077.6144
  angle=63.4987455814236288
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=10000 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=1E20 p_mode1="Very Smooth (C1)"
  p_mode2="Very Smooth (C1)" p_p1=1/0 p_p2=0/0 p_p3=0/0 p_ispin=0 p_ospin=0
  p_mode="Agnesi's Curve" p_flip=no p_mirror=no p_offset=0
  p_center=-1.815/-0.375 p_rot=0 p_period=0 p_smper=yes p_concper=no
  p_size=0.2 p_r=0.2 p_n=0 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Smooth" density=0.1 transfer=arctan
  repeat=yes p_power=2/0 p_bailout=128
gradient:
  smooth=no numnodes=7 index=2 color=7252 index=3 color=394836 index=37
  color=11645361 index=60 color=460551 index=187 color=16777215 index=376
  color=16777215 index=388 color=8882055
alpha:
  smooth=yes numnodes=2 index=0 alpha=144 index=5 alpha=248
layer:
  caption="New Layer 4" opacity=100 visible=yes alpha=no mergemode=overlay
mapping:
  center=-1.7692707111695472/0.0569176768789583808 magn=14179040788077.6144
  angle=63.4987455814236288
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=10000 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=1E20 p_mode1="Very Smooth (C1)"
  p_mode2="Very Smooth (C1)" p_p1=1/0 p_p2=0/0 p_p3=0/0 p_ispin=0 p_ospin=0
  p_mode="Agnesi's Curve" p_flip=no p_mirror=no p_offset=0
  p_center=-1.815/-0.375 p_rot=0 p_period=0 p_smper=yes p_concper=no
  p_size=0.2 p_r=0.2 p_n=0 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Triangle" density=4 transfer=linear
  repeat=yes p_apower=1 p_aflavor="normal" p_power=2 p_bailout=1E20
gradient:
  smooth=no numnodes=6 index=7 color=13103615 index=26 color=1080 index=111
  color=16777215 index=343 color=16777215 index=356 color=12105912
  index=367 color=0
layer:
  caption="New Layer 1" opacity=100 visible=yes alpha=no
  mergemode=softlight
mapping:
  center=-1.7692707111695472/0.0569176768789583808 magn=14179040788077.6144
  angle=63.4987455814236288
formula:
  filename="sam.ufm" entry="cglobesmm" maxiter=10000 percheck=normal
  p_start=0/0 p_power=2/0 p_bailout=1E20 p_mode1="Very Smooth (C1)"
  p_mode2="Very Smooth (C1)" p_p1=1/0 p_p2=0/0 p_p3=0/0 p_ispin=0 p_ospin=0
  p_mode="Agnesi's Curve" p_flip=no p_mirror=no p_offset=0
  p_center=-1.815/-0.375 p_rot=0 p_period=0 p_smper=yes p_concper=no
  p_size=0.2 p_r=0.2 p_n=0 f_f1=sin f_f2=ident
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Triangle" density=4 transfer=linear
  repeat=yes p_apower=1 p_aflavor="normal" p_power=2 p_bailout=1E20
gradient:
  smooth=no numnodes=6 index=28 color=1080 index=113 color=16777215
  index=345 color=16777215 index=358 color=12105912 index=369 color=0
  index=399 color=13103615
}