# User:Soul windsurfer

 This user has uploaded images to Wikimedia Commons.

|}

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 pl Polski jest językiem ojczystym tego użytkownika.
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Wikipedia - my page in wiki

Mathematics Stack Exchange MathOverflow Wikibooks -pl - Adam majewski

 The Photographer's Barnstar foto
 This file was selected as the media of the day for 06 March 2012. It was captioned as follows: English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot set .mw-parser-output .tpl-hidden{clear:both}.mw-parser-output .tpl-hidden>.mw-collapsible-toggle{float:none}.mw-parser-output .tpl-hidden::before{display:none}html.client-js .mw-parser-output .tpl-hidden-headertext{padding-left:18px}.mw-parser-output .tpl-hidden.mw-made-collapsible>.mw-collapsible-toggle>.tpl-hidden-headertext{background-image:url("//upload.wikimedia.org/wikipedia/commons/1/10/MediaWiki_Vector_skin_action_arrow.png");background-position:left 50%;background-repeat:no-repeat}.mw-parser-output .tpl-hidden.mw-made-collapsible.mw-collapsed>.mw-collapsible-toggle>.tpl-hidden-headertext{background-image:url("//upload.wikimedia.org/wikipedia/commons/4/41/MediaWiki_Vector_skin_right_arrow.png")} Other languages English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot setМакедонски: Квадратно Жулиино множество со множества на внатрешно ниво за вредностите на c, заедно со внатрешен зрак 0 на главната кардоида од Манделбротово множество.中文（简体）：朱利亚集合

## syntaxhighlight

{{Galeria
|Nazwa = Trzy krzywe w różnych skalach
|wielkość = 400
|pozycja = right
|Plik:LinLinScale.svg|Skala liniowo-liniowa
|Plik:LinLogScale.svg|Skala liniowo-logarytmiczna
|Plik:LogLinScale.svg|Skala logarytmiczno-liniowa
|Plik:LogLogScale.svg|Skala logarytmiczno-logarytmiczna
}}

== c source code==
<syntaxhighlight lang="c">
</syntaxhighlight>

== bash source code==
<syntaxhighlight lang="bash">
</syntaxhighlight>

==make==
<syntaxhighlight lang=makefile>
all:
chmod +x d.sh
./d.sh
</syntaxhighlight>

Tu run the program simply

make

==text output==
<pre>

==references==
<references/>



# function

Mathematical Function Plot
Description Function displaying a cusp at (0,1)
Equation ${\displaystyle y={\sqrt {|x|}}+1}$
Co-ordinate System Cartesian
X Range -4 .. 4
Y Range -0 .. 3
Derivative ${\displaystyle {\frac {dy}{dx}}={\frac {1}{2{\sqrt {x}}}}}$

Points of Interest in this Range
Minima ${\displaystyle \left(0,1\right)\,}$
Cusps ${\displaystyle \left(0,1\right)\,}$
Derivatives at Cusp ${\displaystyle \lim _{x\to 0^{+}}f'(x)=+\infty }$,

${\displaystyle \lim _{x\to 0^{-}}f'(x)=-\infty }$

The program can calculate many of the objects found in Singularity theory:

• Algebraic curves defined by a single polynomial equation in two variables. e.g. a circle x^2 + y^2 - r^2;
• Algebraic surfaces defined by a single polynomial equation in three variables. e.g. a cone x^2 + y^2 - z^2;
• Paramertised curves defined by a 3D vector expression in a single variable. e.g. a helix [cos(pi t), sin(pi t), t];
• Parameterised surfaces defined by a 3D vector expression in two variables. e.g. a cross-cap [x,x y,y^2];
• Intersection of surfaces with sets defined by an equation. Can be used to calculate non-polynomial curves.
• Mapping from R^3 to R^3 defined by 3D vector equation in three variables. e.g. a rotation [cos(pi th) x - sin(pi th) y,sin(pi th) x + cos(pi th) y,z];
• Intersections where the equation depends on the definition of another curve or surface. e.g. The profile of a surface N . [A,B,C]; N = diff(S,x) ^^ diff(S,y);
• Mappings where the equation depends on another surface. For example projection of a curve onto a surface.
• Intersections where the equations depends on a pair of curves. For example the pre-symmetry set of a curve.
• Mapping where the equation depends on a pair of curves. For example the Symmetry set.

# video

### formats

YT:

• .MOV
• .MPEG-1 ( also commons)
• .MPEG-2 ( also commons)
• .MPEG4
• .MP4
• .MPG
• .AVI ( also commons)
• .WMV
• .MPEGPS
• .FLV
• 3GPP
• WebM ( also commons)
• DNxHR
• ProRes
• CineForm
• HEVC (h265)

commmons:

Kodeki

Text

# Image guidelines

FIles

## Color calibration

dwa rodzaje kalibracji:

• programową (poprzez soft dostarczany przez producenta panelu)
• sprzętową (z pomocą urządzenia pomiarowego, na przykład kalibratora do monitorów).

wynik deltaE po kalibracji

• nie powinien przekraczać 1,5 ( dobrze)
• < 1.0 ( wybornie)

W praktyce lepiej polegać na kalibracji sprzętowej, która zachowuje wszystkie tony, odcienie barw oraz płynną gradację.

Monitory

• z fabryczną kalibracją kolorów ( krzywa gamma jest regulowana w fabryce dla każdej matrycy )
• Eizo
• BenQ
• czujnik automatycznej kalibracji, który eliminuje konieczność korzystania z zewnętrznych urządzeń
• rozdzielczości ( 4k )
• gęstość pikseli ( duża to 163 punktów na cal)
• technologia eliminacji migotania obrazu
• tryb ciemni, szczególnie przydatny w postprocessingu
• tryb animacji, dbający o spójność wyświetlanych odcieni
• tryb CAD/CAM uwydatniający najdrobniejsze szczegóły znajdujące się na ekranie monitora
• gamut kolorów : 100% gamutu sRGB

zapisz ustawienia monitora w formie profilu, a najlepiej (jeśli to możliwe) kilku różnych, dopasowanych do konkretnych zastosowań.

Before reviewing the work of authors it is recommended that you calibrate your monitor. If you don't do this, keep in mind that you might not see details in very bright or very dark areas. Also, some monitors may be tinted too much towards a certain color, and not have "neutral" color.

See the image below full screen on a completely black background. You should be able to see at least three of the four circles on this image. If you see four then your brightness setting is on the high side, if you see three, it is fine, but if fewer than three are discernible then it is set too low.

Monitor gamma checkerboard

On a gamma-adjusted display, the four circles in the color image blend into the background when seen from a few feet away. If they do not, you could adjust the gamma setting (found in the computer's settings, not on the display), until they do. This may be very difficult to attain, and a slight error is not detrimental. Uncorrected PC displays usually show the circles darker than the background.

Note that on most consumer LCD displays (laptop or flat screen) viewing angle strongly affects these images - correct adjustment on one part of the screen might be incorrect on another part for a stationary head position. Click on the images for more technical information.

If possible, calibration with a hardware monitor calibrator is recommended.

It is also important to ensure that your browser is displaying images at the correct resolution. To verify this, open the full 4000x2000 version of the grid you see at the left. The grid is 8 squares wide by 4 squares high, with the squares measuring 500 pixels on each side. After zooming in to 100%, you should check if it looks reasonable given your monitor resolution. For example, if you have a 1920x1080 monitor, the horizontal width should be just short of covering 4 squares, which would be 2000 pixels. Due to scroll bars, menus, etc. the actual amount of space available will be slightly less than your screen resolution.

Firefox is known for having issues on a Windows computer where the Display setting in Control Panel have been set to anything other than 100% scaling; it is common to have a default of 125% (Medium) for large monitors. This results in Firefox upsampling everything by 25%, causing images to appear less sharp than they really are! To fix this issue, you can either change the scaling factor in Display to 100% or go to about:config (in the URL bar) and change layout.css.devPixelsPerPx to 1.0.

# commons


{{ValiCat|+|type=stop}}

[[Category:CommonsRoot]]


Help:

## Category

Image

• Static
• non-photographic
• computer graphic
• art
• AI
• human

Criteria for fractals classifications

• by method
• by fractal type
• by country ???
• by year ( ? creation ?)
• by File type ( or file format)
• static image
• raster
• vector
• animation
• video
• by quality
• by technical citeria
• by features
• Fractal images - media of the day

Help

# spike

It can be used for:

• highlight the boundary ( 1D)
• Specular reflection in Phong reflection model ( 3D )

# slope

// "Psychofract" by Carlos Ureña - 2015

mat3 tran1,
tran2 ; //transform matrix for each branch

const float pi = 3.1415926535 ;

const float rsy = 0.30 ; // length of each tree root trunk in NDC

// -----------------------------------------------------------------------------

{
s = sin( rads ) ;

return mat3(  c,  s, 0.0,
-s,   c, 0.0,
0.0, 0.0, 1.0  );
}
// -----------------------------------------------------------------------------

mat3 TranslateMat( vec2 d )
{
return mat3(  1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
d.x, d.y, 1.0 );
}
// -----------------------------------------------------------------------------

mat3 ScaleMat( vec2 s )
{
return mat3( s.x, 0.0, 0.0,
0.0, s.y, 0.0,
0.0, 0.0, 1.0 );
}
// -----------------------------------------------------------------------------

mat3 ChangeFrameToMat( vec2 org, float angg, float scale )
{
float angr = (angg*pi)/180.0 ;
return
ScaleMat( vec2( 1.0/scale, 1.0/scale ) )
* RotateMat( -angr )
* TranslateMat( -org ) ;
}
// -----------------------------------------------------------------------------

float RectangleDistSq( vec3 p )
{
if ( 0.0 <= p.y && p.y <= rsy )
return p.x * p.x;

if (p.y > rsy)
return p.x*p.x + (p.y-rsy)*(p.y-rsy) ;

return p.x*p.x + p.y*p.y ;
}
// -----------------------------------------------------------------------------

float BlendDistSq( float d1, float d2, float d3 )
{
float dmin = min( d1, min(d2,d3)) ;

return 0.5*dmin ;
}
// -----------------------------------------------------------------------------

vec4 ColorF( float distSq, float angDeg )
{
float b = min(1.0, 0.1/(sqrt(distSq)+0.1)),
v = 0.5*(1.0+cos( 200.0*angDeg/360.0 + b*15.0*pi ));

return vec4( b*b*b,b*b,0.0,distSq) ; // returns squared distance in alpha component
}
// -----------------------------------------------------------------------------

float Trunk4DistSq( vec3 p )
{
float d1 = RectangleDistSq( p );

return d1 ;
}

// -----------------------------------------------------------------------------

float Trunk3DistSq( vec3 p )
{
float d1 = RectangleDistSq( p ),
d2 = Trunk4DistSq( tran1*p ),
d3 = Trunk4DistSq( tran2*p );

return BlendDistSq( d1, d2, d3 ) ;
}

// -----------------------------------------------------------------------------

float Trunk2DistSq( vec3 p )
{
float d1 = RectangleDistSq( p ),
d2 = Trunk3DistSq( tran1*p ),
d3 = Trunk3DistSq( tran2*p );

return BlendDistSq( d1, d2, d3 ) ;
}

// -----------------------------------------------------------------------------

float Trunk1DistSq( vec3 p )
{
float d1 = RectangleDistSq( p ) ,
d2 = Trunk2DistSq( tran1*p ),
d3 = Trunk2DistSq( tran2*p );

return BlendDistSq( d1, d2, d3 ) ;
}

// -----------------------------------------------------------------------------

float Trunk0DistSq( vec3 p )
{
float d1 = RectangleDistSq( p ) ,
d2 = Trunk1DistSq( tran1*p ),
d3 = Trunk1DistSq( tran2*p );

return BlendDistSq( d1, d2, d3 ) ;
}
// -----------------------------------------------------------------------------
// compute the color and distance to tree, for a point in NDC coords

vec4 ComputeColorNDC( vec3 p, float angDeg )
{
vec2 org = vec2(0.5,0.5) ;
vec4 col = vec4( 0.0, 0.0, 0.0, 1.0 );
float dmin ;

for( int i = 0 ; i < 4 ; i++ )
{
mat3 m = ChangeFrameToMat( org, angDeg + float(i)*90.0, 0.7 );
vec3 p_transf = m*p ;
float dminc = Trunk0DistSq( p_transf ) ;

if ( i == 0 )
dmin = dminc ;
else if ( dminc < dmin )
dmin = dminc ;
}
return ColorF( dmin, angDeg ); // returns squared dist in alpha component
}
// -----------------------------------------------------------------------------

vec3 ComputeNormal( vec3 p, float dd, float ang, vec4 c00 )
{
vec4   //c00  = ComputeColorNDC( p, ang )  ,
c10  = ComputeColorNDC( p + vec3(dd,0.0,0.0), ang )  ,
c01  = ComputeColorNDC( p + vec3(0.0,dd,0.0) , ang ) ;
float h00  = sqrt(c00.a),
h10  = sqrt(c10.a),
h01  = sqrt(c01.a);
vec3  tanx = vec3( dd, 0.0, h10-h00 ),
tany = vec3( 0.0, dd, h01-h00 );
vec3  n    = normalize( cross( tanx,tany ) );

if ( n.z < 0.0 ) n *= -1.0 ;
return n ;
}

// -----------------------------------------------------------------------------

void mainImage( out vec4 fragColor, in vec2 fragCoord )
{

const float width = 0.1 ;

vec2  res  = iResolution.xy ;
float mind = min(res.x,res.y);
vec2  pos  = fragCoord.xy ;
float x0   = (res.x - mind)/2.0 ,
y0   = (res.y - mind)/2.0 ,
px   = pos.x - x0 ,
py   = pos.y - y0 ;

// compute 'tran1' and 'tran2':

vec2  org1      = vec2( 0.0, rsy ) ;
float ang1_deg  = +20.0 + 30.00*cos( 2.0*pi*iTime/4.05 ),
scale1    = +0.85 +  0.40*cos( 2.0*pi*iTime/2.10 )  ;

vec2  org2      = vec2( 0.0, rsy ) ;
float ang2_deg  = -30.0 + 40.00*sin( 2.0*pi*iTime/2.52 ),
scale2    = +0.75 +  0.32*sin( 2.0*pi*iTime/4.10 )  ;

tran1 = ChangeFrameToMat( org1, ang1_deg, scale1 ) ;
tran2 = ChangeFrameToMat( org2, ang2_deg, scale2 ) ;

// compute pixel color (pixCol)

float mainAng = 360.0*iTime/15.0 ,    // main angle, proportional to time
dd      = 1.0/float(mind) ;           // pixel width or height in ndc
vec3  pixCen  = vec3( px*dd, py*dd, 1.0 ) ; // pixel center
vec4  pixCol  = ComputeColorNDC( pixCen, mainAng ),
resCol  ;

// compute output color as a function 'use_normal'

const bool use_gradient = true ;

{
vec3 nor     = ComputeNormal( pixCen, dd, mainAng, pixCol );
vec4 gradCol = vec4( max(nor.x,0.0), max(nor.y,0.0), max(nor.z,0.0), 1.0 ) ;

}
else
resCol = pixCol ;

fragColor = vec4( resCol.rgb, 1.0 ) ;
}



# rays

// https://geometricolor.wordpress.com/2013/02/28/fractal-surprise-from-complex-function-iteration-the-code/
//Fractal surprise from complex function iteration: The code Posted on February 28, 2013 by Peter Stampfli
//  here is the program for the last post:  Fractal surprise from complex function iteration
//   simply use it in processing 1.5 (I don’t know if it works in the new version 2.)

float range, c,step;
int n, iter, count;
float rsqmax;

void setup() {
size(600, 600);
c=0.4;
step=0.002;
iter=40;
rsqmax=100;
range=1.2;
count=0;
}

void draw() {
int i, j, k;
float d=2*range/width;
float x, y, h;
float phi, phiKor;                             //  phiKor is the trivial phase change to subtract
colorMode(HSB, 400, 100, 100);
for (j=0;j<height;j++) {
for (i=0;i
y=-range+d*j;
x=-range+d*i;
phiKor=0;
for (k=0;k<iter;k++) {
if (x*x+y*y>rsqmax) {
break;
}
phiKor+=atan2(y, x);
h=x*x-y*y+c;
y=2*x*y;
x=h;
}
phi=atan2(y, x)-phiKor;    //  correction
phi=0.5*phi/PI;
phi=(phi-floor(phi))*2*PI;  //  calculate phase phi mod 2PI
pixels[i+j*width]=color(phi/PI*200, 100, 100);
}
}
updatePixels();
//  saveFrame(“a-####.jpg”);
println(count+” “+c);
c-=step;
if (c<0) {
noLoop();
}
count++;
}


${\displaystyle 0\leq \operatorname {mod} \left(\ \operatorname {floor} \left({\frac {\operatorname {floor} \left(x\right)}{2^{\operatorname {floor} \left(y\right)}}}\right)\ ,\ 2\ \right)}$


pins=[]
pins_num=150

setup=()=> {
createCanvas(windowWidth, windowHeight)
y=0
for (i=0; i<pins_num; i++ ) {
pins[i]=createVector(random(width), random(height))
}
}

draw=()=> {
if (y<height) {
for (x=0; x<width; x++ ) {
angle=0
pins.forEach(p=>angle+=atan2(p.y-y, p.x-x))
colour=map(sin(angle), -1, 1, 0, 256)
stroke(colour)
line(x, y, x, y+colour)
}
y++
}
}

mousePressed=()=>setup()



## The field lines

In a Fatou domain (that is not neutral) there is a system of lines orthogonal to the system of equipotential lines, and a line of this system is called a field line.

If we colour the Fatou domain according to the iteration number (and not the real iteration number), the bands of iteration show the course of the equipotential lines, and so also the course of the field lines.

If the iteration is towards ∞, we can easily show the course of the field lines, namely by altering the colour according to whether the last point in the sequence is above or below the x-axis, but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently (because we use the argument of the product of ${\displaystyle f'(z_{i})}$ for the points of the cycle).

For an attracting cycle C, the field lines issue from the points of the cycle and from the (infinite number of) points that iterate into a point of the cycle. And the field lines end on the Julia set in points that are non-chaotic (that is, generating a finite cycle).

Let

• r be the order of the cycle C
• z* be a point in C.
• ${\displaystyle f(f(...f(z*)))=z*}$ (the r-fold composition)
• the complex number ${\displaystyle \alpha }$ by ${\displaystyle \alpha =(d(f(f(...f(z))))/dz)_{z=z*}}$

If the points of C are ${\displaystyle z_{i}(i=1,2,...,r,z_{1}=z*)}$, ${\displaystyle \alpha }$ is the product of the r numbers ${\displaystyle f'(z_{i})}$. The real number 1/${\displaystyle |\alpha |}$ is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1/${\displaystyle |\alpha |}$ < ∞. The point z* is a fixed point for ${\displaystyle f(f(...f(z)))}$, and near this point the map ${\displaystyle f(f(...f(z)))}$ has (in connection with field lines) character of a rotation with the argument ${\displaystyle \beta }$ of ${\displaystyle \alpha }$ (so that ${\displaystyle \alpha =|\alpha |e^{\beta i}}$).

### colour the Fatou domain

In order to colour the Fatou domain, we have chosen a small number ${\displaystyle \epsilon }$ and set the sequences of iteration ${\displaystyle z_{k}(k=0,1,2,...,z_{0}=z)}$ to stop when ${\displaystyle |z_{k}-z*|<\epsilon }$, and we colour the point z according to

• the number k which gives Level Set Method (LSM)
• the real iteration number, if we prefer a smooth colouring

### FL

def

• the field lines, which are the lines orthogonal to the equipotential lines
• A field line is orthogonal to the equipotential surfaces, which are the loci for the points of constant iteration number

The colouring is determined by

• the distance to the centre line of the field line (density/across)
• and by the potential function (density/along),

This colouring can be mixed with the colouring of the background. As two colour scales are required, these must be imported.

The field lines are determined by their number,

• their (relative) thickness (≤ 1)
• a number transition determining the mixing of the colours of the field lines with the background - e.g
• . 0.1 for a soft transition and therefore indistinct and thinner looking field lines,
• 4 for more well defined field lines.

If the field lines do not run precisely coherently, the bailout number must be diminished and the maximum iteration number increased. Here is the function 1 1 0 0 1, and the background is made of one colour by setting the density to 0:

Within a field line there are two local distances:

• the distance from the centre line
• the distance from the equipotential line.

In addition there are two whole numbers:

• the number of the field line
• the iteration number.

The two local distances establish local coordinate systems within the field lines, and this means that it is possible to colour on the basis of mathematical procedures or pictures that are input.

And the two whole numbers mean that such procedures or pictures can be made to depend on the field line number and the iteration number.

When the Fatou domain is associated to a super-attracting cycle, the field lines cannot be drawn coherently.

However, it is possible to draw a system of bands that follow their courses, but the number of which increases for each increase in the iteration number. If the function is a polynomial (that is, if the denominator is a constant), the factor of increase is the degree of the polynomial. For a general rational function the factor is the difference between the degree of the numerator and the denominator, but the constellation of the partitions is not regular, because fiels lines also originate from the points that are zeros of the denominator, and from the (infinitely many) points that are iterated into one of these zeros.

For the function z4/(1 + z), the field lines are separated in three, but field lines also come from the point -1 and the points iterated into -1:

### colouring of the field lines

If we choose a direction from z* given by an angle ${\displaystyle \theta }$, the field line issuing from z* in this direction consists of the points z such that the argument ${\displaystyle \psi }$ of the number ${\displaystyle z_{k}-z*}$ satisfies the condition

${\displaystyle \psi -k\beta =\theta (mod2\pi )}$.

For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ${\displaystyle \psi }$ is increased by ${\displaystyle \beta }$, therefore the number ${\displaystyle \psi -k\beta (mod2\pi )}$ is constant along the field line.

A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we use the name field line for such coloured "interval of field lines", As a field line in our terminology is an interval of field lines

we choose a number of regularly situated directions issuing from z*, and in each of these directions we choose two directions around this direction. As it can happen, that the two bounding field lines do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the centre line of the field line, and we can mix this colouring with the usual colouring.

Let n be the number of field lines and let t be their relative thickness (a number in the interval [0, 1]). For the point z, we have calculated the number ${\displaystyle \psi -k\beta (mod2\pi )}$, and z belongs to a field line if the number ${\displaystyle v=(\psi -k\beta (mod2\pi ))/(2\pi )}$ (in the interval [0, 1]) satisfies |v - i/n| < t/(2n) for one of the integers i = 0, 1, ..., n, and we can use the number |v - i/n|/(t/(2n)) (in the interval [0, 1] - the relative distance to the centre of the field line) to the colouring.

Where

• an attracting cycle C
• r be the order of the cycle C
• the points of C are ${\displaystyle z_{i}(i=1,2,...,r,z_{1}=z*)}$
• z* be a point in C;
• it is a fixed point of ${\displaystyle f^{r}}$
• ${\displaystyle f^{r}(z*)=z*}$
• the argument ${\displaystyle \psi }$ of the number ${\displaystyle z=z_{k}-z*}$
 ${\displaystyle \psi =\operatorname {atan2} \left(\operatorname {Im} (z),\operatorname {Re} (z)\right)}$


### Pictures

• In the first picture, the function is of the form ${\displaystyle z/2+1/(z-z^{3}/6)+c}$ and we have only coloured a single Fatou domain
• The second picture shows that field lines can be made very decorative (the function is of the form ${\displaystyle z/(1+z^{3})+c}$).
• third picture: A (coloured) field line is divided up by the iteration bands, and such a part can be put into a ono-to-one correspondence with the unit square: the one coordinate is the relative distance to one of the bounding field lines, this number is (v - i/n)/(t/(2n)) + 1/2, the other is the relative distance to the inner iteration band, this number is the non-integral part of the real iteration number. Therefore we can put pictures into the field lines. As many as we desire, if we index them according to the iteration number and the number of the field line. However, it seems to be difficult to find fractal motives suitable for placing of pictures - if the intention is a picture of some artistic value. But we can restrict the drawing to the field lines (and possibly introduce transparency in the inlaid pictures), and let the domain outside the field lines be another fractal motif

# relief

Cartographic

Sketchy

with lines

define the colors used for different terrain heights:

# Digital Image Processing

• data analysis
• image processing
• image enhacement = process an image so that result is more suitable than original image for specific application
• Histogram equalization

# computer graphic

www.pling.org.uk/cs/cgv.html
www.tutorialspoint.com/computer_graphics/computer_graphics_curves.htm
github.com/jagregory/abrash-black-book
pages.mtu.edu/~shene/COURSES/cs3621/NOTES/model/b-rep.html
pages.mtu.edu/~shene/COURSES/cs3621/NOTES/notes.html


## libraries

### icc

  convert image_rgb.tiff -profile "RGB.icc" -profile "CMYK.icc" image_cmyk.tiff


## algorithm

• Render your image using correct radiometric calculations. You trace individual wavelengths of light or buckets of wavelengths. Whatever. In the end, you have an image that has a representation of the spectrum received at every point.
• At each pixel, you take the spectrum you rendered, and convert it to the CIE XYZ color space. This works out to be integrating the product of the spectrum with the standard observer functions (see CIE XYZ definition).
• This produces three scalar values, which are the CIE XYZ colors.
• Use a matrix transform to convert this to linear RGB, and then from there use a linear/power transform to convert linear RGB to sRGB.
• Convert from floating point to uint8 and save, clamping values out of range (your monitor can't represent them).
• Send the uint8 pixels to the framebuffer.
• The display takes the sRGB colors, does the inverse transform to produce three primaries of particular intensities. Each scales the output of whatever picture element it is responsible for. The picture elements light up, producing a spectrum. This spectrum will be (hopefully) a metamer for the original spectrum you rendered.
• You perceive the spectrum as you would have perceived the rendered spectrum.

## hdr

• Set your ISO to 200, set your camera to Aperture Priority
• take three photos with exposure settings as EV 0, EV-2, and EV+2. The more differently exposed photos you have, the better.
• Merge to HDR
• Select 32-bit/channel and tick Remove Ghosts
• . Click Image > Mode > 16-bit/channel
• tone mapping. Adjust the settings depending on how you want your HDR photo to look.

## image enhancement techniques

• gray-level transformation functions
• linear (negative and identity transformations)
• logarithmic (log and inverse-log transformations)
• power-law (nth power and nth root transformations).T

## bit depth and bitrate

• 8-bit = 2^8 = 256 values. 256 to create an 8-bit grayscale image
• 16-bit grayscale. This means that they capture over 65,000 shades of gray. [1]
• 24 bit = with three colour channels, that’s 256 (red) x 256 (green) x 256 (blue) for a total of around 2^8^3 = 16.7 million individual colours (RGB = true color) 16 777 216 = 256^3 = 2^8^3

## aliasing

• "A simple way to prevent aliasing of cosine functions (the color palette in this case) by removing frequencies as oscillations become smaller than a pixel. You can think of it as an LOD system. Move the mouse to compare naive versus band-limited cos(x)" Inigo Quilez

# color

  "color operations should be done ...to either model human perception or the physical behavior of light"Björn Ottosson : How software gets color wrong


## Method for domain coloring

HL plot of z, as per the simple color function example described in the text (left), and the graph of the complex function z3 − 1 (right) using the same color function, showing the three zeros as well as the negative real numbers as pink rays starting at the zeros.

Representing a four dimensional complex mapping with only two variables is undesirable, as methods like projections can result in a loss of information. However, it is possible to add variables that keep the four-dimensional process without requiring a visualization of four dimensions. In this case, the two added variables are visual inputs such as color and brightness because they are naturally two variables easily processed and distinguished by the human eye. This assignment is called a "color function". There are many different color functions used. A common practice is to represent the complex argument (also known as "phase" or "angle") with a hue following the color wheel, and the magnitude by other means, such as brightness or saturation.

### Simple color function

The following example colors the origin in black, 1 in green, −1 in magenta, and a point at infinity in white:

${\displaystyle {\begin{cases}H&=\arg z+2\pi /3,\\S&=100\%,\\L&=\ell (|z|).\end{cases}}}$

There are a number of choices for the function ${\displaystyle \ell :[0,\infty )\to [0,1)}$. ${\displaystyle \ell }$ should be strictly monotonic and continuous. Another desirable property is ${\displaystyle \ell (1/r)=1-\ell (r)}$ such that the inverse of a function is exactly as light as the original function is dark (and the other way around). Possible choices include

• ${\displaystyle \ell _{1}(r)={\frac {2}{\pi }}\arctan(r)}$ and
• ${\displaystyle \ell _{2}(r)={\frac {r^{a}}{r^{a}+1}}}$ (with some parameter ${\displaystyle a>0}$). With ${\displaystyle a=2}$, this corresponds to the stereographic projection onto the Riemann sphere.

A widespread choice which does not have this property is the function ${\displaystyle \ell _{3}(r)=1-a^{|r|}}$ (with some parameter ${\displaystyle 0) which for ${\displaystyle a=1/2}$ and ${\displaystyle 0\leq r\leq 1}$ is very close to ${\displaystyle \ell _{1}}$.

This approach uses the HSL (hue, saturation, lightness) color model. Saturation is always set at the maximum of 100%. Vivid colors of the rainbow are rotating in a continuous way on the complex unit circle, so the sixth roots of unity (starting with 1) are: green, cyan, blue, magenta, red, and yellow.

Since the HSL color space is not perceptually uniform, one can see streaks of perceived brightness at yellow, cyan, and magenta (even though their absolute values are the same as red, green, and blue) and a halo around L = 12. More modern color spaces, e.g, the Lab color space or CIECAM02, correct this, making the images more accurate and less saturated.

### Discontinuous color changing

Many color graphs have discontinuities, where instead of evenly changing brightness and color, it suddenly changes, even when the function itself is still smooth. This is done for a variety of reasons such as showing more detail or highlighting certain aspects of a function. This is also sometimes done unintentionally, for example if determining the direction of a complex number first depends on calculating the angle, say in the range [-π, π) and then assigning a color as some function of this angle, then in this case the transition across -π = π could be discontinuous. This kind of artifact rarely contrributes to the usefulness of a graph.

#### Magnitude growth

A discontinuous color function. In the graph, each discontinuity occurs when ${\displaystyle |z|=2^{n}}$ for integers n.

Unlike the argument, which has finite range, the magnitude of a complex number can range from 0 to . Therefore, in functions that have large ranges of magnitude, changes in magnitude can sometimes be hard to differentiate when a very large change is also pictured in the graph. This can be remedied with a discontinuous color function which shows a repeating brightness pattern for the magnitude based on a given equation. This allows smaller changes to be easily seen as well as larger changes that "discontinuously jump" to a higher magnitude. In the graph on the right, these discontinuities occur in circles around the center, and show a dimming of the graph that can then start becoming brighter again. A similar color function has been used for the graph on top of the article.

Equations that determine the discontinuities may be linear, such as for every integer magnitude, exponential equations such as every magnitude n where ${\displaystyle 2^{n}}$ is an integer, or any other equation.

#### Highlighting properties

Discontinuities may be placed where outputs have a certain property to highlight which parts of the graph have that property. For instance, a graph may instead of showing the color cyan jump from green to blue. This causes a discontinuity that is easy to spot, and can highlight lines such as where the argument is zero.[2] Discontinuities may also affect large portions of a graph, such as a graph where the color wheel divides the graph into quadrants. In this way, it is easy to show where each quadrant ends up with relations to others.[3]

## Color depth

• 1-bit color = binary image
• 8-bit color = 256 colors, usually from a fully-programmable palette ( VGA)
• 24-bit = True color
• 64-bit color = stores 16-bit R, 16-bit G, 16-bit B and 16-bit A. All integer, no floats

Number type

### store

Image formats to store RGB images with floating point pixels:[4]

• you cannot use PNG ( png64 stores 16-bit R, 16-bit G, 16-bit B and 16-bit A. All integer, no floats), JPEG, TGA or GIF formats
• use either

### processing

For processing such images:

• cannot use these with PIL/Pillow because it doesn't support 32-bit float RGB files.
• use :

## Dynamic range

Dynamic rang types:

• image
• display
• print

HDR and:

## color variations

Variations created by color mixing[5]

• Shades: created by adding black to a base color, increasing its darkness. Shades appear more dramatic and richer
• tints: created by adding white to a base color, increasing its lightness. Tints are likely to look pastel and less intense
• Tones: created by adding gray to a base color, increasing its lightness. Tones looks more sophisticated and complex than base colors

Other:

• Hues: refers to the basic family of a color from red to violet. Hues are variations of a base color on the color wheel
• Temperatures: Color are often divided in cool and warm according to how we perceive them. Greens and blues are cool, whilst reds and yellows are warm

4 categories related to color brightness and saturation:[6]

• vivid (light) = high saturation and brightness
• vivid dark = low brightness
• pastel = low saturation and high brightness
• pale (pastel) dark = low saturation and brightness

Shades of Color Notice that since (0,0,0) is black and (1,1,1) is white, shades of any particular color are created by moving closer to black or to white. You can use the parametric equation for a linear relationship between two values to make shades of a color darker or lighter.

A parametric equation to calculate a linear change between values A and B:

  C = A + (B-A)*t;  // where t varies between 0 and 1


To change a color (r,g,b) to make it lighter, move it closer to (1,1,1).

  newR = r + (1-r)*t;  // where t varies between 0 and 1
newG = g + (1-g)*t;  // where t varies between 0 and 1
newB = b + (1-b)*t;  // where t varies between 0 and 1


To change a color (r,g,b) to make it darker, move it closer to (0,0,0).

  newR = r + (0-r)*t;  // where t varies between 0 and 1
newG = g + (0-g)*t;  // where t varies between 0 and 1
newB = b + (0-b)*t;  // where t varies between 0 and 1
// or
newR = r*t;  // where t varies between 1 and 0
newG = g*t;  // where t varies between 1 and 0
newB = b*t;  // where t varies between 1 and 0


### tints or pastel

Names:

• tints (a mixture of a base color with white )
• pale colors = increased lightness
• pastel colors
• soft or muted type of color
• light color

Effect

• soothing to the eye
• looks less intense, pastel, pale, faded look.
• subtle, modern or sophisticated design

Algorithm

• mix base color with white = heavily tinted with white = desaturated with white
• first generate a random color. Then we sature it a little and mix this color with white color[7]
• in HSV. Take a hue. Desaturate the color a bit( = 80% saturation). Use 100% for value.
• in the HSV color space, have high value and low saturation

# https://mdigi.tools/random-pastel-color/
random_color = { r: Random(0, 255), g: Random(0, 255), b: Random(0, 255) }
pastel_color = random_color.saturate( 10% ).mix( white )


/*
* https://sighack.com/post/procedural-color-algorithms-color-variations
* Mix randomly-generated RGB colors with a specified
* base color. The mixing is performed taking into
* account a user-specified weight parameter 'w', which
* specifies what percentage of the final color should
* come from the base color.
*
* A value of 0 for the weight specifies that the 100%
* of the final RGB components should come from the
* randomly generated color while a value of 0.5
* specifies an equal proportion from both the base color
* and the randomly-generated one.
*/
color rgbMixRandom(color base, float w) {
float r, g, b;

/* Check bounds for weight parameter */
w = w > 1 ? 1 : w;
w = w < 0 ? 0 : w;

/* Generate components for a random RGB color */
r = random(256);
g = random(256);
b = random(256);

/* Mix user-specified color using given weight */
r = (1-w) * r + w * red(base);
g = (1-w) * g + w * green(base);
b = (1-w) * b + w * blue(base);

return color(r, g, b);
}
‎


## Intermediate Colour Values

Smooth colouring requires the ability to obtain a colour in between two colours picked from a discrete palette:

function:   getIntermediateColourValue
parameters: c1(R,G,B), c2(R',G',B'), m: 0 <= m <= 1
return:     [R + m * (R' - R),
G + m * (G' - G),
B + m * (B' - B)]


# Gimp

• flatpak run org.gimp.GIMP

# nested squares


int VECT = 1;

void setup() {
size(640, 640);
rectMode(CENTER);
noStroke();
}

void draw() {
background(0);
translate(width/2, height/2);
int i = frameCount*2 % width;
if (i==0) VECT = -VECT;
SCALE = dist(i, 0, 0, width-i)/width;
for (int j=0; j<16; j++) {
scale(SCALE);
fill(map(j, 0, 15, 64, 255));
rect(0, 0, width, height);
}
}


# log-polar mapping

examples

## mapping

• Twisted Mandelbrot Set „Let p is pixel location relative to image's center. Normally, c = c0 + p. Twisted, c = c0 + p^2.” and you change c0 along some circle to get the rotation ?

Exponential map [8]

### notation

${\displaystyle f:A\to B}$

is meant to say ${\displaystyle f}$ is a map whose domain is $A$ and whose codomain is $B$. $A$ and $B$ are both sets. E.g.

${\displaystyle {\text{sq}}:\mathbb {R} \to \mathbb {R} }$

means $\text{sq}$ is a real valued function defined over the real numbers.

If you have $f:A\to B$, then we also have the notation $$f:a\mapsto b$$ where $a$ is an element of $A$ and $b$ is an element of $B$. This can be used to fix the notation for the evaluation of the map: E.g. $\text{sq}:x\mapsto \text{sq}(x)$. You can also prescript the actual map in that moment, by defining what $\text{sq}(x)$, e.g. $\text{sq}:x\mapsto \text{sq}(x) := x^{2}$

Other use for this notation is to simply say to what element of $B$ a particular $a\in A$ is mapped to. E.g. $\text{sq}:8\mapsto 64$.

# Mandelbrot

## dense

• Example: c0 = -0.39055 + 0.58680 i @ 1e5 magnification, 64k iterations
• the set of all Misiurewicz points is dense on the boundary of the Mandelbrot set ( Local connectivity of the Mandelbrot set at certain infinitely renormalizable points by Yunping Jiang ) https://arxiv.org/abs/math/9508212

## period

• 0. Assume a well-behaved formula like quadratic Mandelbrot set.
• 1. A hyperbolic component of period P is surrounded by an atom domain of period P.
• 2. The size of the atom domain is around 4x larger than a disk-like component, often larger for cardioid-like components.
• 3. Conjecture: Newton's method in 1 complex variable (f_c^P(z)-z=0) can find the limit cycle Z_1 .. Z_P when starting from the Pth iterate of 0, when c is sufficiently within the atom domain
• 4. The limit cycle has multiplier (product 2 Z_k) 0 at the center and 1 in magnitude at the boundary of the component.
• 5. The atom domain coordinate is 0 at the center and 1 in magnitude at the boundary of the atom domain.
• 6. Perturbed Newton's method can be used for deep zooms.
• 7. The limit cycle can also be used for interior distance estimation.

Start from period 1 increasing, you get a sequence of atom domains. At each, if the atom domain coordinate is small, use Newton's method to find the limit cycle. If the limit cycle has small multiplier, stop, period is detected. If the iterate escapes, stop, pixel is exterior.

## 22-legged ant in the m-set

$m-describe double 1000 1000 -0.72398340 0.28671980 1 the input point was -0.723983400000000055 + +0.286719800000000025 i the point didn't escape after 1000 iterations nearby hyperbolic components to the input point: - a period 1 cardioid with nucleus at +0.000000000000000000 + +0.000000000000000000 i the component has size 1 and is pointing west the atom domain has size 0 the nucleus domain coordinate is 0 at turn 0.000000000000000000 the atom domain coordinate is -nan at turn -nan the nucleus is 0.77869 to the east-south-east of the input point the input point is exterior to this component at radius 1.0354 and angle 0.45522 (in turns) a point in the attractor is -0.497319450905372551 + +0.143745216108897733 i external angles of this component are: .(0) .(1) - a period 2 circle with nucleus at -1.000000000000000000 + +0.000000000000000000 i the component has size 0.5 and is pointing west the atom domain has size 1 the nucleus domain coordinate is 1.2361 at turn 0.500000000000000000 the atom domain coordinate is 1 at turn 0.000000000000000000 the nucleus is 0.39799 to the south-west of the input point the input point is exterior to this component at radius 1.5919 and angle 0.12803 (in turns) a point in the attractor is -0.861857110872783827 + +0.396178203198002954 i external angles of this component are: .(01) .(10) - a period 105 cardioid with nucleus at -0.723948838355664148 + +0.286852337885451558 i the component has size 3.7103e-07 and is pointing east-south-east the atom domain has size 0.00010235 the nucleus domain coordinate is 1.5924 at turn 0.128053542315208269 the atom domain coordinate is 0.39811 at turn 0.128053542315208269 the nucleus is 0.00013697 to the north-north-east of the input point external angles of this component are: .(010101010011010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001) .(010101010011010100101010101001010101010010101010100101010101001010101010010101010100101010101001010101010) - a period 116 cardioid with nucleus at -0.723983481008419805 + +0.286719746468555081 i the component has size 1.7465e-07 and is pointing east-south-east the atom domain has size 7.0286e-05 the nucleus domain coordinate is 36.617 at turn 0.370795651751681277 the atom domain coordinate is 0.73603 at turn 0.878154585895997375 the nucleus is 9.7098e-08 to the west-south-west of the input point the input point is interior to this component at radius 0.83522 and angle 0.62253 (in turns) a point in the attractor is +0.000065706660165846 + -0.000227325037589120 i the interior distance estimate is 4.491e-08 external angles of this component are: .(01010101001101001010101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001) .(01010101001101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001010101010) nearby Misiurewicz points to the input point: - 2p1 the strength is 0.38133 the center is at -2.000000000000000000 + 0.000000000000000000 i the center is 1.3078 to the west-south-west of the input point the multiplier has radius 4 and angle 0.00000 (in turns) - 3p1 the strength is 0.35938 the center is at -1.543689012692076368 + -0.000000000000000000 i the center is 0.8684 to the west-south-west of the input point the multiplier has radius 1.6786 and angle -0.50000 (in turns) - 12p1 the strength is 0.00037767 the center is at -0.724112682973573563 + 0.286456567676711404 i the center is 0.00029327 to the south-south-west of the input point the multiplier has radius 1.0354 and angle 0.45526 (in turns) - 12p11 the strength is 2.0286e-05 the center is at -0.724112682973573563 + 0.286456567676711460 i the center is 0.00029327 to the south-south-west of the input point the multiplier has radius 1.4656 and angle 0.00789 (in turns) - 2p116 the strength is 2.3938e-14 the center is at -0.723874171053648929 + 0.286847598521453362 i the center is 0.00016812 to the north-east of the input point the multiplier has radius 41405 and angle -0.45166 (in turns) - 3p116 the strength is 9.1963e-15 the center is at -0.723892769678345482 + 0.286784362650608693 i the center is 0.00011128 to the north-east of the input point the multiplier has radius 8827.5 and angle 0.13635 (in turns)$ cabal repl # while in folder with mandelbrot prelude project
...
ghci> fmap Txt.plain . Sym.angledAddress . Sym.rational =<< Txt.parse ".(01010101001101001010101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001)"
Just "1_5/11->11_1/2->22_1/2->33_1/2->44_1/2->55_1/2->66_1/2->77_1/2->88_1/2->99_1/2->110_1/2->116"


${\displaystyle 1_{\frac {5}{11}}\to 11_{\frac {1}{2}}\to 22_{\frac {1}{2}}\to 33_{\frac {1}{2}}\to 44_{\frac {1}{2}}\to 55_{\frac {1}{2}}\to 66_{\frac {1}{2}}\to 77_{\frac {1}{2}}\to 88_{\frac {1}{2}}\to 99_{\frac {1}{2}}\to 110_{\frac {1}{2}}\to 116}$

${\displaystyle 1\xrightarrow {5/11} 11\xrightarrow {1/2} 22\xrightarrow {1/2} 33\xrightarrow {1/2} 44\xrightarrow {1/2} 55\xrightarrow {1/2} 66\xrightarrow {1/2} 77\xrightarrow {1/2} 88\xrightarrow {1/2} 99\xrightarrow {1/2} 110\xrightarrow {1/2} 116}$

## alg by Claude

coloured with atom domain quadrant, exterior binary decomposition, and interior and exterior distance estimates) Double precision, no perturbation. Atom domain coordinate is smallest z divided by previous smallest z. Check for interior only if the atom domain coordinate is small (I think the smallest atom domains are for circle-like components, and are about 4x the size of the component, cardioid-like components tend to have much larger atom domains). Atom domain quadrant colouring is based on the quadrant of the atom domain coordinate, something like floor(4 * arg(a)/2pi).

How many iterations do you perform for distance estimation? For exterior distance estimation, you need a large escape radius, eg 100100. For interior distance estimation, you need the period, then a number (maybe 10 or so should usually be enough) Newton steps to find the limit cycle. Iteration count limit is arbitrary, with a finite limit some pixels will always be classified as "unknown".

Looks like your code is finding interior unexpectedly (points that should be exterior are falsely determined to be interior). But without seeing the source it's hard to tell.

extern bool m_d_compute_step(m_d_compute *px, int steps) {
76   if (! px) {
77     return false;
78   }
79   if (px->tag != m_unknown) {
80     return true;
81   }
82   double er2 = px->er2;
83   double _Complex c = px->c;
84   double _Complex z = px->z;
85   double _Complex dc = px->dc;
86   double _Complex zp = px->zp;
87   double _Complex zq = px->zq;
88   double mz2 = px->mz2;
89   double mzq2 = px->mzq2;
90   int p = px->p;
91   int q = px->q;
92   for (int i = 1; i <= steps; ++i) {
93     dc = 2 * z * dc + 1;
94     z = z * z + c;
95     double z2 = cabs2(z);
96     if (z2 < mzq2 && px->filter && px->filter->accept && px->filter->accept(px->filter, px->n + i))
97     {
98       mzq2 = z2;
99       q = px->n + i;
100       zq = z;
101     }
102     if (z2 < mz2) {
103       double atom_domain_radius_squared = z2 / mz2;
104       mz2 = z2;
105       p = px->n + i;
106       zp = z;
107       if (atom_domain_radius_squared <= 0.25) {
108         if (px->bias == m_interior) {
109           double _Complex dz = 0;
110           double de = -1;
111           if (m_d_interior_de(&de, &dz, z, c, p, 64)) {
112             px->tag = m_interior;
113             px->p = p;
114             px->z = z;
115             px->dz = dz;
116             px->zp = zp;
117             px->de = de;
118             return true;
119           }
120         } else {
121           if (px->partials && px->np < px->npartials) {
122             px->partials[px->np].z = z;
123             px->partials[px->np].p = p;
124             px->np = px->np + 1;
125           }
126         }
127       }
128     }
129     if (! (z2 < er2)) {
130       px->tag = m_exterior;
131       px->n = px->n + i;
132       px->p = p;
133       px->q = q;
134       px->z = z;
135       px->zp = zp;
136       px->zq = zq;
137       px->dc = dc;
138       px->de = 2 * cabs(z) * log(cabs(z)) / cabs(dc);
139       return true;
140     }
141   }
142   if (px->bias != m_interior && px->partials) {
143     for (int i = 0; i < px->np; ++i) {
144       z = px->partials[i].z;
145       zp = z;
146       int p = px->partials[i].p;
147       double _Complex dz = 0;
148       double de = -1;
149       if (m_d_interior_de(&de, &dz, z, c, p, 64)) {
150         px->tag = m_interior;
151         px->p = p;
152         px->z = z;
153         px->dz = dz;
154         px->zp = zp;
155         px->de = de;
156         return true;
157       }
158     }
159   }
160   px->tag = m_unknown;
161   px->n = px->n + steps;
162   px->p = p;
163   px->q = q;
164   px->mz2 = mz2;
165   px->mzq2 = mzq2;
166   px->z = z;
167   px->dc = dc;
168   px->zp = zp;
169   px->zq = zq;
170   return false;
171 }

mandelbrot-numerics] / c / lib / m_d_interior_de.c
1 // mandelbrot-numerics -- numerical algorithms related to the Mandelbrot set
2 // Copyright (C) 2015-2018 Claude Heiland-Allen
4
5 #include <mandelbrot-numerics.h>
6 #include "m_d_util.h"
7
8 extern bool m_d_interior_de(double *de_out, double _Complex *dz_out, double _Complex z, double _Complex c, int p, int steps) {
9   double _Complex z00 = 0;
10   if (m_failed != m_d_attractor(&z00, z, c, p, steps)) {
11     double _Complex z0 = z00;
12     double _Complex dz0 = 1;
13     for (int j = 0; j < p; ++j) {
14       dz0 = 2 * z0 * dz0;
15       z0 = z0 * z0 + c;
16     }
17     if (cabs2(dz0) <= 1) {
18       double _Complex z1 = z00;
19       double _Complex dz1 = 1;
20       double _Complex dzdz1 = 0;
21       double _Complex dc1 = 0;
22       double _Complex dcdz1 = 0;
23       for (int j = 0; j < p; ++j) {
24         dcdz1 = 2 * (z1 * dcdz1 + dz1 * dc1);
25         dc1 = 2 * z1 * dc1 + 1;
26         dzdz1 = 2 * (dz1 * dz1 + z1 * dzdz1);
27         dz1 = 2 * z1 * dz1;
28         z1 = z1 * z1 + c;
29       }
30       *de_out = (1 - cabs2(dz1)) / cabs(dcdz1 + dzdz1 * dc1 / (1 - dz1));
31       *dz_out = dz1;
32       return true;
33     }
34   }
35   return false;
36 }

‎


## inversion

• z^2 + c to z^2 + 1/c

First, there's the transformation from the cardioid of the body of the set to a circle. This is done in c-space (a+ib) as follows:

rho=sqrt(a*a+b*b)-1/4
phi=arctan(b/a)
a-new=rho*(2*cos(phi)-cos(2*phi))/3
b-new=rho*(2*sin(phi)-sin(2*phi))/3


Then, there's the 4 different ways on how to get from c to 1/c, that group in 3 families:

• addition: c -> c-c*t+t/c and t=0..1
• multiplication: c -> c/(t*(c*c-1)+1) and t=0..1
• exponentiation: c -> c^t and t=1..-1

The first 2 are pretty elementary to work out. The 3rd makes use of the fact that: c = a+ib = r*exp(i*phi) where r=sqrt(a*a+b*b) and phi=arctan(b/a) then c^t = r^t*exp(t*i*phi) = r^t*[cos(t*phi)+i*sin(t*phi)]

The top left is method 1, top right method 2, and the bottom 2 are variants of method 3. Hope this helps somewhat.

z_(n+1) = (z_n)^2 + (tan(t) + i*c)/(i + c*tan(t)) with t from 0.2pi to 0.35pi Another way to invert the Mandelbrot set | Closer look at t from 0.2*pi to 0.35*pi by Fraktoler Curious

# Julia

Computing the Julia set of f(z) = z^2 + i where (J(f) = K(f)). [9]

• approximating the filled Julia set from above: the first 15 preimages of a large disk D = B(0, R) ⊃ K(f)
• approximating the Julia set from below: ∪0≤k≤12f−k(β) where β is a repellingfixed point in J(f) = IIM
• a good-quality picture of J(f).

# other Julia

### 610

https://fractalforums.org/fractal-mathematics-and-new-theories/28/julia-and-parameter-space-images-of-polynomials/2786/msg16210#msg16210 marcm200 A very basic cubic Julia set p(z)=z^3+(0.099609375-0.794921875i), but with very interesting entry points into the attracting periodic cycle of length 610.

The image shows the Julia set (yellow), its attracting cycle (cyan), and some white pixels which have not (yet) entered the cycle at the current maxit of 15000 (there were much more at maxit 10000, so they will enter the cycle as expected).

I was interested to see at what point(s) the attracted numbers "enter" the cycle. Entering was defined as: If a complex number is bounded, I checked its last iterate whether it is in the attracting cycle. If so, I went backwards in the orbit until I encountered the first non-cycle point (epsilon of 10^-7). Its image was set to be the entry point.

The result was quite unambiguous. The 4k image had ~222,000 interior points, of which:

the top 3 entry points were: 0.09960906311,-0.794921357 => used 220,130 times 0.09976627035,-0.7951977228 => 251 0.09942639427,-0.7946757595 => 208

So there is a preferred point to enter the cycle - it almost looks like the not-often used entry points might be numerical errors and everything enters the cycle at the same number (or the cycle itself is a numerical error - darn, I hope my question is still valid).

Would this distribution still be accurate in the limit when one could actually test all interior points of the set and not just a finite number of rational coordinate complex numbers?

Since a point never actually goes exactly into the cycle unless it is a preimage or an image of a cycle point, but will come arbitrarily close to (some?) periodic points: Does this imply that periodic points have an event horizon - once in, never out again?

Or can an orbit point be close to a periodic point, jump out into the vicinity of another and coming closer there - and so on, so actually never getting stuck near one specific periodic point?

## z^d + c

Internal angle = 0 ( one main component)

• d = 2 c = 1/4 = 0.25
• d = 3 c = 0.384900179459751 +0.000000000000000 i period = 10000
• d = 4 c = -0.236235196855289 +0.409171363489396 i period = 10000 ??? it should be c = 0.472464424146544;
• d = 5 c = 0.534992243981138 +0.000000000000000 i period = 10000
• d = 6 c = -0.471135846013573 +0.342300228596646 i period = 10000 ?? it should be c = 0.582559084495983 +0.000000000000000 i period = 0

Internal angle 1/3 from main component

• d = 2 c = -0.125000000000000 +0.649519052838329 i period = 10000 ( Douady rabbit )
• d = 3 c = 0.481125224324688 +0.500000000000000 i period = 10000
• d = 4 c = -0.619317130969330 +0.370556691297005 i period = 10000
• d = 5 c = 0.694975311172961 +0.267496121990569 i period = 10000
• d = 6 c = -0.719547645525888 +0.256065008698348 i period = 10000
• d = 7 c = 0.758540222557608 +0.180894796695791 i period = 10000
• d = 8 c = -0.768629397583800 +0.197192545338461 i period = 10000

## cubic

• A 4k cubic Julia by ‎Chris Thomasson‎ . Here is the formula: z = pow(z, 3) - (pow(-z, 2.00001) - 1.0008875); link
• cubic
• "Let c = (.387848...) + i(.6853...). The left picture shows the filled Julia set Kc of the cubic map z3 + c, covered by level 0 of the puzzle. The center of symmetry is at 0, the point where the rays converge is α and the other fixed points are marked by dotted arrows. In this example the rotation number around α is ρα = 25 and the ray angles are 5 121 7→ 15 121 7→ 45 121 7→ 14 121 7→ 42 121 7→ 5 121 . The right picture illustrates level 1 of the puzzle for the same map. "A New Partition Identity Coming from Complex Dynamics
• Owen Maresh owen maresh(-0.4999999999999998 + 0.8660254037844387*I) - (0.2926009682749477 + 0.252068970984772*I)* z - (0.4916379276414715 + 0.2509264824918978*I)* z^2 + (0.2511839558093919 - 1.0778044459985288*I)* z^3
• f f(z)=z5+(0.8+0.8i)z4+z which has the following fixed points
• p1^=0 with multiplier |λ1|=|f′(0)|=1 (parabolic),
• p2^=−0.8−0.8i with multiplier |λ2|=|f′(−0.8−0.8i)|≈|−13.7|>1 (repelling),
• p3^=∞ with multiplier |λ3|=|limz→∞f(z)|−1=0 (super-attracting)
• f(z)=(−0.090706+0.27145i)+(2.41154−0.133695 i)z^2 − z^3 has one critical orbit attracted to an orbit of period two and one critical orbit attracted to an orbit of period three. Basic complex dynamicsA computational approach by Mark McClure

## rational

Julia sets of rational maps ( not polynomials )

### Christopher Williams

Family:

${\displaystyle z_{k+1}=z_{k}^{P}+c-\lambda z_{k}^{-Q}}$

Examples:

• Julia set of ${\displaystyle z^{2}-0.0625z^{-2}}$ The most obvious feature is that it's full of holes! The fractal is homemorphic to (topologically the same as) the Sierpinski carpet
• Julia set of z2 - 1 - 0.005z-2
• f = z^2 - 0.01z^-2
• f = z2 - 0.01z-2

Phoenix formula

• z5 - 0.06iz-2

### Robert L. Devaney

• Singular perturbations of complex polynomials

${\displaystyle G_{\lambda ,c}(z)=z^{n}+c+{\frac {\lambda }{z^{n}}}}$

# Julia sets for fc(z) = z*z + c

const double Cx=-0.74543; const double Cy=0.11301;

-0.808 +0.174i;

-0.1 +0.651i; ( beween 1 and 3 period component of Mandelbrot set )

-0.294 +0.634i

## tuned rabbit

• c = 0
• 5/13
• c = -0.407104083085098 +0.584852842868102 i period = 13
• 1/2
• c = -0.410177342420846 +0.590406710726110 i period = 10000

the 5/13 Rabbit tuned with the Basilica approximates the golden Siegel disk ( LOCAL CONNECTIVITY OF THE MANDELBROT SET AT SOME SATELLITE PARAMETERS OF BOUNDED TYPE by DZMITRY DUDKO AND MIKHAIL LYUBICH )

## spirals

• on the parameter plane
• part of M-set near Misiurewicz points
• on the dynamic plane
• Julia set near cut points
• critical orbits
• external rays landing on the parabolic or repelling perriodic points

https://imagej.net/Directionality a flat directionality histogram is a good metric for many-armed spirals (at least with distance estimation colouring)

 ${\displaystyle (r,t)\to \lambda \to c}$

 ${\displaystyle t={\frac {p}{q}}+\epsilon }$

 ${\displaystyle \lambda =re^{2\pi ti}}$


In period 1 component of Mandelbrot set :

 ${\displaystyle c:{\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}}$


In period 2 component of Mandelbrot set :

Period 1

• c = -0.106956704870346 +0.648733714002516 i , inside period 1 parent component , near period 100 child component , critical orbit is a spiral that starts , near internal ray 33/100

### table

Caption text
Period c r 1-r t p/q p/q - t image author address
1 0.37496784+i*0.21687214 0.99993612384259 0.000063879203489 0.1667830755386747 1/6 0.00011640887201 Cr6spiral.png 1
1 -0.749413589136570+0.015312826507689*i. 0.9995895293978963 0.00041047060211000001 0.4975611481254812 1/2 -0.00243885187451881 png 1
2 -0.757 + 0.027i 0.977981594918841 0.02201840508115904 0.01761164373863864 0/1 -0.01761164373863864 pauldelbrot 1 -(1/2)-> 2
2 -0.752 + 0.01i 0.9928061240745848 0.007193875925415205 0.006414063302849116 0/1 -0.006414063302849116 pauldelbrot 1 -(1/2)-> 2
10 -1.2029905319213867188 + 0.14635562896728515625 i 0.979333 0.02490599999999998 0.985187275828761422 0/1 -0.01481272417123857 marcm200 1 -(1/2)-> 2 -(2/5)-> 10
14 -1.2255649566650390625 0.1083774566650390625 0.951928 0.048072 0.992666114460366900 1/1 0.0073338855396331 marcm200 1 -(1/2)-> 2 -(3/7)-> 14
14 -1.2256811857223510742 +0.10814088582992553711 i 0.955071 0.044929 0.984062994677356362 1/1 0.01593700532264363 marcm200 1 -(1/2)-> 2 -(3/7)-> 14
14 -0.8422698974609375 -0.19476318359375 i 0.952171 0.04782900000000001 0.935491618649184731 1/1 0.06450838135081527 marcm200 1 -(1/2)-> 2 -(6/7)-> 14

### pauldelbrot

"c=0.027*%i-0.757"

period =  1
z= 0.01345178808414596*%i-0.5035840525848648
r = |m(z)| = 1.007527366616821
1-r = -0.007527366616821407
t = turn(m(z))  =  0.4957496478171055
p/q = 1/2
p/q-t = -0.004250352182894435

z= 1.503584052584865-0.01345178808414596*%i
r = |m(z)| = 3.007288448945452
1-r = -2.007288448945452
t = turn(m(z))  =  0.9985761611087214
p/q = 1/2
p/q-t = 0.4985761611087214

period =  2
z= (-0.1022072682395012*%i)-0.3679154600985363
r = |m(z)| = 0.977981594918841
1-r = 0.02201840508115904
t = turn(m(z))  =  0.01761164373863864
p/q = 1/2
p/q-t = -0.4823883562613613

z= 0.1022072682395012*%i-0.6320845399014637
r = |m(z)| = 0.9779815949188409
1-r = 0.02201840508115915
t = turn(m(z))  =  0.01761164373863864
p/q = 1/2
p/q-t = -0.4823883562613613



c=0.01*%i-0.752"

period =  1
z= 0.0049949453016411*%i-0.501011962705025
r = |m(z)| = 1.002073722342039
1-r = -0.00207372234203973
t = turn(m(z))  =  0.498413323518715
p/q = 0
p/q-t = 0.498413323518715

z= 1.501011962705025-0.0049949453016411*%i
r = |m(z)| = 3.002040547136002
1-r = -2.002040547136002
t = turn(m(z))  =  0.9994703791038458
p/q = 0
p/q-t = 0.9994703791038458

period =  2
z= (-0.06402358560400053*%i)-0.4219037804142045
r = |m(z)| = 0.9928061240745848
1-r = 0.007193875925415205
t = turn(m(z))  =  0.006414063302849116
p/q = 0
p/q-t = 0.006414063302849116

z= 0.06402358560400053*%i-0.5780962195857955
r = |m(z)| = 0.9928061240745848
1-r = 0.007193875925415205
t = turn(m(z))  =  0.006414063302849116
p/q = 0
p/q-t = 0.006414063302849116

(%o296) "/home/a/Dokumenty/periodic/MaximaCAS/p1/p.mac"
(%i297)


### marcm200

the input point was -1.2029905319213867e+00 + +1.4635562896728516e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.21186e+00 to the east of the input point
the input point is exterior to this component at
radius 1.41904e+00 and angle 0.486382891412633800 (in turns)
the multiplier is -1.41385e+00 + +1.21263e-01 i
a point in the attractor is -7.0694e-01 + +6.06308e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.20299 + +0.14636 i
the atom domain coordinates in polar form are 0.25025 to the north-west
the nucleus is 2.50250e-01 to the south-east of the input point
the input point is exterior to this component at
radius 1.00100e+00 and angle 0.400579159596292533 (in turns)
the multiplier is -8.11962e-01 + +5.85423e-01 i
a point in the attractor is +1.81557e-01 + -1.07368e-01 i

- a period 4 circle
with nucleus at -1.310703e+00 + +3.761582e-37 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.507 + +0.53837 i
the atom domain coordinates in polar form are 0.73952 to the north-east
the nucleus is 1.81719e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00200e+00 and angle 0.801158319192584956 (in turns)
the multiplier is +3.16563e-01 + -9.50682e-01 i
a point in the attractor is +1.815562e-01 + -1.073687e-01 i
external angles of this component are:
.(0110)
.(1001)

- a period 10 circle
with nucleus at -1.2103996e+00 + +1.5287483e-01 i
the component has size 2.02739e-02 and is pointing north-west
the atom domain has size 4.09884e-02
the atom domain coordinates of the input point are +0.16767 + -0.13713 i
the atom domain coordinates in polar form are 0.2166 to the south-east
the nucleus is 9.86884e-03 to the north-west of the input point
the input point is interior to this component at
radius 9.79333e-01 and angle 0.985187275828761422 (in turns)
the multiplier is +9.75094e-01 + -9.10160e-02 i
a point in the attractor is +7.0332348e-02 + -8.243835e-02 i
external angles of this component are:
.(0110010110)
.(0110011001)


the input point was -1.2255649566650391e+00 + +1.0837745666503906e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.23035e+00 to the east of the input point
the input point is exterior to this component at
radius 1.43387e+00 and angle 0.490097175551864883 (in turns)
the multiplier is -1.43109e+00 + +8.91595e-02 i
a point in the attractor is -7.15546e-01 + +4.45805e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.22556 + +0.10838 i
the atom domain coordinates in polar form are 0.25025 to the west-north-west
the nucleus is 2.50250e-01 to the east-south-east of the input point
the input point is exterior to this component at
radius 1.00100e+00 and angle 0.428714049282459153 (in turns)
the multiplier is -9.02260e-01 + +4.33510e-01 i
a point in the attractor is +1.94019e-01 + -7.80792e-02 i

- a period 4 circle
with nucleus at -1.310703e+00 + +0e+00 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.38418 + +0.4137 i
the atom domain coordinates in polar form are 0.56457 to the north-east
the nucleus is 1.37819e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00200e+00 and angle 0.857428098564918195 (in turns)
the multiplier is +6.26142e-01 + -7.82277e-01 i
a point in the attractor is +1.940184e-01 + -7.80797e-02 i
external angles of this component are:
.(0110)
.(1001)

- a period 14 circle
with nucleus at -1.2299714e+00 + +1.1067143e-01 i
the component has size 1.06543e-02 and is pointing west-north-west
the atom domain has size 1.95731e-02
the atom domain coordinates of the input point are +0.16376 + -0.15414 i
the atom domain coordinates in polar form are 0.22489 to the south-east
the nucleus is 4.96796e-03 to the west-north-west of the input point
the input point is interior to this component at
radius 9.51928e-01 and angle 0.992666114460366900 (in turns)
the multiplier is +9.50917e-01 + -4.38495e-02 i
a point in the attractor is +7.5747371e-02 + -5.129233e-02 i
external angles of this component are:
.(01100110010110)
.(01100110011001)

the input point was -1.2256811857223511e+00 + +1.0814088582992554e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.23044e+00 to the east of the input point
the input point is exterior to this component at
radius 1.43394e+00 and angle 0.490119703062896594 (in turns)
the multiplier is -1.43118e+00 + +8.89616e-02 i
a point in the attractor is -7.15576e-01 + +4.44813e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.22568 + +0.10814 i
the atom domain coordinates in polar form are 0.25025 to the west-north-west
the nucleus is 2.50253e-01 to the east-south-east of the input point
the input point is exterior to this component at
radius 1.00101e+00 and angle 0.428881674271762436 (in turns)
the multiplier is -9.02725e-01 + +4.32564e-01 i
a point in the attractor is +1.9408e-01 + -7.79018e-02 i

- a period 4 circle
with nucleus at -1.310703e+00 + +0e+00 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.38355 + +0.41286 i
the atom domain coordinates in polar form are 0.56353 to the north-east
the nucleus is 1.37561e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00202e+00 and angle 0.857763348543524873 (in turns)
the multiplier is +6.27801e-01 + -7.80972e-01 i
a point in the attractor is +1.940823e-01 + -7.790208e-02 i
external angles of this component are:
.(0110)
.(1001)

- a period 14 circle
with nucleus at -1.2299714e+00 + +1.1067143e-01 i
the component has size 1.06543e-02 and is pointing west-north-west
the atom domain has size 1.95731e-02
the atom domain coordinates of the input point are +0.15716 + -0.16242 i
the atom domain coordinates in polar form are 0.22601 to the south-east
the nucleus is 4.98108e-03 to the west-north-west of the input point
the input point is interior to this component at
radius 9.55071e-01 and angle 0.984062994677356362 (in turns)
the multiplier is +9.50287e-01 + -9.54765e-02 i
a point in the attractor is +6.2976569e-02 + -5.7543144e-02 i
external angles of this component are:
.(01100110010110)
.(01100110011001)

the input point was -8.422698974609375e-01 + -1.9476318359375e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 8.64495e-01 to the east-north-east of the input point
the input point is exterior to this component at
radius 1.11403e+00 and angle 0.526643305764455283 (in turns)
the multiplier is -1.09846e+00 + -1.85625e-01 i
a point in the attractor is -5.49225e-01 + -9.28116e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are +0.15773 + -0.19476 i
the atom domain coordinates in polar form are 0.25062 to the south-east
the nucleus is 2.50622e-01 to the north-west of the input point
the input point is exterior to this component at
radius 1.00249e+00 and angle 0.858340235783291439 (in turns)
the multiplier is +6.30920e-01 + -7.79053e-01 i
a point in the attractor is -1.0771e-01 + +2.48238e-01 i

- a period 14 circle
with nucleus at -8.4076071e-01 + -1.9927227e-01 i
the component has size 1.01164e-02 and is pointing south-east
the atom domain has size 1.66560e-02
the atom domain coordinates of the input point are +0.13324 + -0.2159 i
the atom domain coordinates in polar form are 0.25371 to the south-south-east
the nucleus is 4.75495e-03 to the south-south-east of the input point
the input point is interior to this component at
radius 9.52171e-01 and angle 0.935491618649184731 (in turns)
the multiplier is +8.75023e-01 + -3.75452e-01 i
a point in the attractor is +4.338561e-02 + +7.777828e-02 i
external angles of this component are:
.(10101010100110)
.(10101010101001)


## basilica

• San Marco Fractal = Basilica : c = - 3/4
• parabolic perturbation
• 5/11 :

### 5/11

• The angle 681/2047 or p01010101001 has preperiod = 0 and period = 11
• The angle 682/2047 or p01010101010 has preperiod = 0 and period = 11.

## disconnected

near c = -0.750357820200574 +0.047756163825227 i

## Siegel Disk

Julia set = Jordan curve Irrational recurrent cycles

• 0.59...+i0.43...
• 0.33...+i0.07...
• C= -0.408792866 -0.577405 i
• c=-0.390541-0.586788i

## Other

Feigenbaum: C= -1.4011552 +0.0 i

Tower: C= -1 + 0.0 i

Cauliflower: C= 0.25 +0.0 i

## Dendrite

Critical point eventually periodic 0 > -2 > 2 (fixed).

C= i

c^3 + 2c^2 +2c +2 =0

### 3

The core entropy for polynomials of higher degree Yan Hong Gao, Giulio Tiozzo The Julia set of fc(z) = z → z3 + 0.22036 + 1.18612i

To show the non-uniqueness, let us consider the following example, which comes from [Ga]. We consider the postcritically finite polynomial fc(z) = z3 + c with c ≈ 0.22036 + 1.18612i. The critical value c receives two rays with arguments 11/72 and 17/72. Then, Θ := { Θ1(0) := {11/216, 83/216} , Θ2(0) := {89/216, 161/21

## Other

Circle: C= 0.0 +0.0 i

Segment: C= -2 +0.0 i

## centers

• c = -0.748490484405062 +0.048290910555737 i period = 65 ( 1 -? - > 65 )
• by ‎Chris Thomasson‎
• (0.355534, -0.337292) it is a center of period 85 componnet. Adress 1->5 -> 85,
• 29 cycle power of two Julia set at point: (-0.742466, -0.107902) address 1 -> 29
• 38-cycle power of 3 Julia at point (0.388823, -0.000381453)
• c = -0.051707765779845 +0.683880135777732 i period = 273, 1 -(1/3)-> 3 -(1/7)-> 21 -(1/13)-> 273 https://www.math.stonybrook.edu/~jack/tune-b.pdf
• https://arxiv.org/pdf/math/9411238.pdf see Figure.2
• 1- (1/3) → 3 -(2/3) → 7
• 1 - (1/2) -> 2 - (1/2) → 4 -(1/3) → 7
• 1-(1/3) → 3- (1/2) → 4
• 1-(1/3) → 3 - (1/2) → 5-(1/2) → 6
• 1-(1/3) → 3-(1/2) → 5-(1/2) → 7
• 1-(1/3) → 3-(1/3) → 7
• 1-(3/4)-> 4 -(?/5)-> 20 : c = 0.300078079301992 -0.489531524188048 i period = 20
• http://wrap.warwick.ac.uk/35776/1/WRAP_THESIS_Sharland_2010.pdf

### minibrot

• c = 0.284912968784722 +0.484816779093857 i period = 84

### Superattracting per 3 (up to complex conjugate)

C= -1.75488 (airplane)

C= -0.122561 + 0.744862 i (rabbit) Douady's Rabbit Rabbit: C= -0.122561 +0.744862 i = ( -1/8+3/4 i ??? ) whose critical point 0 is on a periodic orbit of length 3

### Superattracting per 4 (up to complex conjugate)

C= -1.9408

C= -1.3107

C= -1.62541

C= -0.15652 +1.03225 i

C= 0.282271 +0.530061 i

#### Kokopelli

• p = γM (3/15)
• p(z) = z^2 0.156 + 1.302*i
• The angle 3/15 or p0011 has preperiod = 0 and period = 4.
• The conjugate angle is 4/15 or p0100 .
• The kneading sequence is AAB* and the internal address is 1-3-4 .
• The corresponding parameter rays are landing at the root of a primitive component of period 4.
• c = -0.156520166833755 +1.032247108922832 i period = 4

### Superattracting per 5 (up to complex conjugate)

C= -1.98542

C= -1.86078

C= -1.62541

C= -1.25637 +0.380321 i

C= -0.50434 +0.562766 i

C= -0.198042 +1.10027 i

C= -0.0442124 +0.986581 i

C= 0.359259 +0.642514 i

C= 0.379514 +0.334932 i

### A superattracting per 15

C= -0.0384261 +0.985494 i

## dense

### Vivid-Thick-Spiral

VividThickSpiral { fractal:

 title="Vivid Thick Spiral" width=800 height=600 layers=1
credits="Ingvar Kullberg;1/7/2014;Frederik Slijkerman;7/23/2002"


layer:

 caption="Background" opacity=100 method=multipass


mapping:

 center=-0.745655283381030620841/0.07611785523064665900025
magn=3.7328517E10


formula:

 maxiter=1000000 percheck=off filename="Standard.ufm"
entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=100


inside:

 transfer=none


outside:

 density=0.1 transfer=arctan filename="Standard.ucl" entry="Basic"
p_type=Iteration


 smooth=yes rotation=93 index=151 color=16580604 index=217
color=1909029 index=227 color=2951679 index=248 color=6682867
index=262 color=223 index=291 color=255 index=-29 color=55539


opacity:

 smooth=no index=0 opacity=255


}

### Pink-Labyrinth

PinkLabyrinth { fractal:

 title="Pink Labyrinth" width=800 height=600 layers=1
credits="Ingvar Kullberg;1/7/2014;Frederik Slijkerman;7/23/2002"


layer:

 caption="Background" opacity=100 method=multipass


mapping:

 center=-0.745655283616919391525/0.0761178553836136017335
magn=6.6880259E9


formula:

 maxiter=1000000 filename="Standard.ufm" entry="Mandelbrot"
p_start=0/0 p_power=2/0 p_bailout=100


inside:

 transfer=none


outside:

 density=0.25 transfer=arctan filename="Standard.ucl" entry="Basic"
p_type=Iteration


 smooth=yes rotation=74 index=132 color=16580604 index=198
color=1909029 index=208 color=2951679 index=229 color=6682867
index=243 color=223 index=272 color=255 index=-48 color=55539


opacity:

 smooth=no index=0 opacity=255


}

IMAGE DETAILS

### Density near the cardoid 3

• Magnification:
• 2^35.769
• 5.8552024369543422761426995117521 E10
• Coordinates:
• Re = 0.360999615968828800
• Im = -0.121129382033034400

### Density near the cardoid 3 by DinkydauSet

https://fractalforums.org/image-threads/25/mandelbrot-set-various-structures/716/;topicseen


Mandelbrot set

Again a location that's not deep but super dense.

Magnification: 2^35.769 5.8552024369543422761426995117521 E10

Coordinates: Re = 0.360999615968828800 Im = -0.121129382033034400

XaoS coordinates

(maxiter 50000) (view -0.775225602760841 -0.136878655029377 7.14008235131944E-11 7.14008235674045E-11)

https://arxiv.org/pdf/1703.01206.pdf "Limbs 8/21, 21/55, 55/144, 144/377, . . . scale geometrically fast on the right side of the (anti-)golden Siegel parameter, while limbs 5/13, 13/34, 34/89, 89/233, . . . scale geometrically fast on the left side. The bottom picture is a zoom of the top picture."

## a

https://plus.google.com/u/0/photos/115452635610736407329/albums/6124078542095960129/6124078748270945650 frond tail Misiurewicz point of the period-27 bulb of the quintic Mandelbrot set (I don't have a number ATM, but you can find that)﻿

# SVG

<?xml version="1.0" encoding="utf-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">


# ps

"Obrazy w indywidualnym wkładzie, mam z samodzielnie napisanymi programami PASCAL (FREEPASCAL) i / lub XFIG (program do rysowania pod LINUX) początkowo tworzone jako pliki EPS. Niestety bezpośrednia integracja plików EPS z Wikipedią nie jest możliwa. Przydatna jest konwersja plików EPS na pliki XFIG za pomocą PSTOEDIT. Konwersja plików EPS na pliki SVG jest możliwa dzięki INKSCAPE, a także eksportowi plików xfig do plików SVG. Jednak nie znalazłem zamiennika dla etykiety LATEX, włączając ją do pliku LATEX w środowisku rysunkowym z plikiem psfrag. Także dla licznych możliwości manipulowania krzywymi za pomocą xfig wiem w inkscape (wciąż) brak odpowiednika. Z plików PS tworzę pliki SVG z ps2pdf i pdf2svg i używam programu inkscape do dostosowania stron (marginesów) do rysunków." de:Benutzer:Ag2gaeh

# Help

|Other fields=SOUL WINDSURFER development

InfoField

This chart was created with R.

Source code

InfoField

## R code

dx=800; dy=600            # define grid size
C = complex( real=rep(seq(-2.2, 1.0, length.out=dx), each=dy ),
imag=rep(seq(-1.2, 1.2, length.out=dy),      dx ) )
C = matrix(C,dy,dx)       # convert from vector to matrix
Z = 0                     # initialize Z to zero
X = array(0, c(dy,dx,20)) # allocate memory for all the frames
for (k in 1:20) {         # perform 20 iterations
Z = Z^2+C               # the main equation
X[,,k] = exp(-abs(Z))   # store magnitude of the complex number
}
library(caTools)          # load library with write.gif function
jetColors = colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000"))
write.gif(X, "Mandelbrot.gif", col=jetColors, delay=100, transparent=0)


differences between :

<gallery> </gallery>
{{SUL Box|en|wikt}}



[[1]]

## compare with


==Compare with==

<gallery caption="Sample gallery" widths="100px" heights="100px" perrow="6">

</gallery>

</nowiki>


Change in your preferences : Show hidden categories

## syntaxhighlight

{{Galeria
|Nazwa = Trzy krzywe w różnych skalach
|wielkość = 400
|pozycja = right
|Plik:LinLinScale.svg|Skala liniowo-liniowa
|Plik:LinLogScale.svg|Skala liniowo-logarytmiczna
|Plik:LogLinScale.svg|Skala logarytmiczno-liniowa
|Plik:LogLogScale.svg|Skala logarytmiczno-logarytmiczna
}}

== c source code==
<syntaxhighlight lang="c">
</syntaxhighlight>

== bash source code==
<syntaxhighlight lang="bash">
</syntaxhighlight>

==make==
<syntaxhighlight lang=makefile>
all:
chmod +x d.sh
./d.sh
</syntaxhighlight>

Tu run the program simply

make

==text output==
<pre>

==references==
<references/>



## references

1. colortrac glossary: greyscale
2. May 2004. http://users.mai.liu.se/hanlu09/complex/domain_coloring.html Retrieved 13 December 2018.
3. (September 2012). "Domain Coloring of Complex Functions: An Implementation-Oriented Introduction". IEEE Computer Graphics and Applications 32 (5): 90–97. DOI:10.1109/MCG.2012.100. PMID 24806991.
4. stackoverflow.com/questions/71856674/how-to-save-images-after-normalizing-the-pixels/71863957?noredirect=1
5. coolors by Fabrizio Bianchi
6. https://colors.artyclick.com/color-names-dictionary/
7. random-pastel-color by Micro Digital Tools
8. theinnerframe : exponential-function/, Playing with Infinity
9. Computability of Brolin-Lyubich Measure by Ilia Binder, Mark Braverman, Cristobal Rojas, Michael Yampolsky
10. fractal_ken An "escape time" fractal generated by homemade software using recurrence relation z(n) = z(n - 1)^2 + 0.3305 + 0.06i

### GeSHi

description:

<syntaxhighlight lang="c" enclose="none">
g;
a=2;
</syntaxhighlight>


### source templetate

deprecated:

<syntaxhighlight lang="gnuplot">ssssssssssssss</syntaxhighlight>


### prettytable

Mathematical Function Plot
Description Function displaying a cusp at (0,1)
Equation ${\displaystyle y={\sqrt {|x|}}+1}$
Co-ordinate System Cartesian
X Range -4 .. 4
Y Range -0 .. 3
Derivative ${\displaystyle {\frac {dy}{dx}}={\frac {1}{2{\sqrt {x}}}}}$

Points of Interest in this Range
Minima ${\displaystyle \left(0,1\right)\,}$
Cusps ${\displaystyle \left(0,1\right)\,}$
Derivatives at Cusp ${\displaystyle \lim _{x\to 0^{+}}f'(x)=+\infty }$,

${\displaystyle \lim _{x\to 0^{-}}f'(x)=-\infty }$