User:Soul windsurfer
This user has uploaded images to Wikimedia Commons. 
}
Babel user information  

 
Users by language 
Babel user information  

 
Users by language 
Babel user information  

 
Users by language 
Wikipedia  my page in wiki
home page.  dead but web archive can help
https://tools.wmflabs.org/glamtools/glamorous.php
Wikibooks en  Adam majewski
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
Other languages
English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot set Македонски: 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[edit]
{{Galeria Nazwa = Trzy krzywe w różnych skalach wielkość = 400 pozycja = right Plik:LinLinScale.svgSkala liniowoliniowa Plik:LinLogScale.svgSkala liniowologarytmiczna Plik:LogLinScale.svgSkala logarytmicznoliniowa Plik:LogLogScale.svgSkala logarytmicznologarytmiczna }} == 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[edit]
Mathematical Function Plot  

Description  Function displaying a cusp at (0,1) 
Equation  
Coordinate System  Cartesian 
X Range  4 .. 4 
Y Range  0 .. 3 
Derivative  
Points of Interest in this Range  
Minima  
Cusps  
Derivatives at Cusp  ,

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 crosscap [x,x y,y^2];
 Intersection of surfaces with sets defined by an equation. Can be used to calculate nonpolynomial 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 presymmetry set of a curve.
 Mapping where the equation depends on a pair of curves. For example the Symmetry set.
video[edit]
formats[edit]
YT:
 .MOV
 .MPEG1 ( also commons)
 .MPEG2 ( also commons)
 .MPEG4
 .MP4
 .MPG
 .AVI ( also commons)
 .WMV
 .MPEGPS
 .FLV
 3GPP
 WebM ( also commons)
 DNxHR
 ProRes
 CineForm
 HEVC (h265)
commmons:
tools[edit]
Kodeki
Text
Image guidelines[edit]
 Commons:Image_guidelines
 :Category:Images_by_technical_criteria
 :Category:Color in computer graphics
 :Commons:Featured pictures
FIles
LCD tests[edit]
Color calibration[edit]
 color calibration in wikipedia
 askubuntu question: howtocalibratethemonitoronanubuntusystem
 help.ubuntu : colorcalibratescreen
 askubuntu question: monitorcalibrationon2004
 howtocalibratethemonitoronanubuntusystem
 xkomkalibratorydomonitorow
 xkom: jakimonitordlagrafikakomputerowego
 netflix test pattern
 astrosurf: calibrate
 cosmotography : calibrat
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.
Calibrate your monitorour monitor[edit]
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.
On a gammaadjusted 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[edit]
{{ValiCat+type=stop}} [[Category:CommonsRoot]]
Help:
 Commons:Categories#Category_names
 Category:CommonsRoot
 Category:Categories
 https://meta.wikimedia.org/wiki/Namespace
 Commons:Categories#How_to_categorize:_guidance_by_topic
 Commons:Categories#Sorting_categories
 essays
Category[edit]
Image
 Static
 nonphotographic
 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
 static image
 by quality
 by technical citeria
 by features
 Fractal images  media of the day
Help
spike[edit]
 "spike function" x > 1  exp(a*abs(mod(x,1)0.5)^b)
 periodic version of function f can be created by reducing mod 1: fp(x)=f(x−⌊x⌋)
 Exponential_decay
 https://zone.ni.com/reference/enXX/help/371361R01/gmath/spike_function/#details
It can be used for:
 highlight the boundary ( 1D)
 Specular reflection in Phong reflection model ( 3D )
peak[edit]
 https://stackoverflow.com/questions/3684484/peakdetectionina2darray
 enriched signal regions = peak
tree[edit]
 https://github.com/someuser321/TreeGenerator
 https://www.shadertoy.com/view/wtf3DB
 https://www.shadertoy.com/view/wslGz7
slope[edit]
// "Psychofract" by Carlos Ureña  2015 // License Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License. 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 //  mat3 RotateMat( float rads ) { float c = cos(rads), 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.yrsy)*(p.yrsy) ; 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, h10h00 ), tany = vec3( 0.0, dd, h01h00 ); 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 ; if ( use_gradient ) { 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 ) ; resCol = 0.8*pixCol+ 0.2*gradCol ; } else resCol = pixCol ; fragColor = vec4( resCol.rgb, 1.0 ) ; }
rays[edit]
 Quadratic Julia set with Internal binary decomposition for internal ray 0.ogv
 mobtree256.png from Arithmetic Exponential Generating Functions by Linas Vepstas
 analytic functions of interest in number theory in C by Linas
 Tree roots by Naoki Tsutae in openprocessing
 A LEVEL 3, WEIGHT 32 TRACEFORM by David LowryDuda
 ModularForm
 btree by dahart
 BinaryTreeVar
 desmos  binary decomposition
 Fractal surprise from complex function iteration Posted on February 28, 2013 by Peter Stampfli  Processing code
// https://geometricolor.wordpress.com/2013/02/28/fractalsurprisefromcomplexfunctioniterationthecode/ //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.) // you can download it from processing.org 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); loadPixels(); 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*xy*y+c; y=2*x*y; x=h; } phi=atan2(y, x)phiKor; // correction phi=0.5*phi/PI; phi=(phifloor(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++; }
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.yy, p.xx)) colour=map(sin(angle), 1, 1, 0, 256) stroke(colour) line(x, y, x, y+colour) } y++ } } mousePressed=()=>setup()
in the circle[edit]
strip[edit]
The field lines[edit]
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 xaxis, but in this case (more precisely: when the Fatou domain is superattracting) we cannot draw the field lines coherently (because we use the argument of the product of 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 nonchaotic (that is, generating a finite cycle).
Let
 r be the order of the cycle C
 z* be a point in C.
 (the rfold composition)
 the complex number by
If the points of C are , is the product of the r numbers . The real number 1/ is the attraction of the cycle, and our assumption that the cycle is neither neutral nor superattracting, means that 1 < 1/ < ∞. The point z* is a fixed point for , and near this point the map has (in connection with field lines) character of a rotation with the argument of (so that ).
colour the Fatou domain[edit]
In order to colour the Fatou domain, we have chosen a small number and set the sequences of iteration to stop when , 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[edit]
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 superattracting 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[edit]
If we choose a direction from z* given by an angle , the field line issuing from z* in this direction consists of the points z such that the argument of the number satisfies the condition
 .
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 is increased by , therefore the number 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 , and z belongs to a field line if the number (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
 z* be a point in C;
 it is a fixed point of
 the argument of the number
Pictures[edit]
 In the first picture, the function is of the form 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 ).
 third picture: A (coloured) field line is divided up by the iteration bands, and such a part can be put into a onotoone 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 nonintegral 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
smoke[edit]
relief[edit]
Cartographic
Hachures, are a form of shading using lines
Sketchy
with lines
define the colors used for different terrain heights:
map projections[edit]
Digital Image Processing[edit]
 data analysis
 image processing
 image enhacement = process an image so that result is more suitable than original image for specific application
 Histogram equalization
 image enhacement = process an image so that result is more suitable than original image for specific application
computer graphic[edit]
 https://matlabsimulation.com/imagepreprocessingprojects/
 http://learn.graphics/
 https://www.cs.cmu.edu/~kmcrane/
 https://perso.liris.cnrs.fr/david.coeurjolly/teaching/ENS2018/cgdcontour.html
 https://www.tankonyvtar.hu/en/tartalom/tamop425/0046_algorithms_of_informatics_volume2/ch10.html
 http://math.hws.edu/graphicsbook/index.html
www.pling.org.uk/cs/cgv.html www.tutorialspoint.com/computer_graphics/computer_graphics_curves.htm github.com/jagregory/abrashblackbook pages.mtu.edu/~shene/COURSES/cs3621/NOTES/model/brep.html pages.mtu.edu/~shene/COURSES/cs3621/NOTES/notes.html
libraries[edit]
icc[edit]
 http://www.imagemagick.org/Usage/formats/#profiles
 https://stackoverflow.com/questions/18272797/convertingrgbtocmykusingiccprofile
convert image_rgb.tiff profile "RGB.icc" profile "CMYK.icc" image_cmyk.tiff
algorithm[edit]
perceptually accurate rendering 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[edit]
 Set your ISO to 200, set your camera to Aperture Priority
 take three photos with exposure settings as EV 0, EV2, and EV+2. The more differently exposed photos you have, the better.
 Merge to HDR
 Select 32bit/channel and tick Remove Ghosts
 . Click Image > Mode > 16bit/channel
 tone mapping. Adjust the settings depending on how you want your HDR photo to look.
image enhancement techniques[edit]
 graylevel transformation functions
 linear (negative and identity transformations)
 logarithmic (log and inverselog transformations)
 powerlaw (nth power and nth root transformations).T
bit depth and bitrate[edit]
 8bit = 2^8 = 256 values. 256 to create an 8bit grayscale image
 16bit 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[edit]
 "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 bandlimited cos(x)" Inigo Quilez
color[edit]
"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[edit]
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 fourdimensional 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[edit]
The following example colors the origin in black, 1 in green, −1 in magenta, and a point at infinity in white:
There are a number of choices for the function . should be strictly monotonic and continuous. Another desirable property is 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
 and
 (with some parameter ). With , this corresponds to the stereographic projection onto the Riemann sphere.
A widespread choice which does not have this property is the function (with some parameter ) which for and is very close to .
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[edit]
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[edit]
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 is an integer, or any other equation.
Highlighting properties[edit]
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[edit]
 1bit color = binary image
 8bit color = 256 colors, usually from a fullyprogrammable palette ( VGA)
 24bit = True color
 64bit color = stores 16bit R, 16bit G, 16bit B and 16bit A. All integer, no floats
number format[edit]
Number type
 integer
 floating point number format
 Small_Float_Formats in OpenGl and Color formats in OpenGl
 Half in OpenEXR
store[edit]
Image formats to store RGB images with floating point pixels:^{[4]}
 you cannot use PNG ( png64 stores 16bit R, 16bit G, 16bit B and 16bit A. All integer, no floats), JPEG, TGA or GIF formats
 use either
processing[edit]
For processing such images:
 cannot use these with PIL/Pillow because it doesn't support 32bit float RGB files.
 use :
 OpenGl
 OpenCV
 scikitimage
 libvips
 matplotlib
Dynamic range[edit]
Dynamic rang types:
 image
 display
HDR and:
color variations[edit]
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[edit]
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 + (BA)*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 + (1r)*t; // where t varies between 0 and 1 newG = g + (1g)*t; // where t varies between 0 and 1 newB = b + (1b)*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 + (0r)*t; // where t varies between 0 and 1 newG = g + (0g)*t; // where t varies between 0 and 1 newB = b + (0b)*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[edit]
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/randompastelcolor/ 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/proceduralcoloralgorithmscolorvariations
* Mix randomlygenerated RGB colors with a specified
* base color. The mixing is performed taking into
* account a userspecified 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 randomlygenerated 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 userspecified color using given weight */
r = (1w) * r + w * red(base);
g = (1w) * g + w * green(base);
b = (1w) * b + w * blue(base);
return color(r, g, b);
}
color effects[edit]
color wheel[edit]
 https://theblog.adobe.com/thepowerofthepalettewhycoloriskeyindatavisualizationandhowtouseit/
 https://opendarkroom.com/colorwheelrendering/
 https://stackoverflow.com/questions/21490210/howtoplotacolourwheelbyusingggplot
 http://sape.inf.usi.ch/quickreference/ggplot2/colour
 https://stackoverflow.com/questions/10289279/convertinghsvcirclecodefromdelphitocsharp
 http://color.support/huecircle.html
 https://weallsew.com/colorwheelbasics/
 http://www.printernational.org/rgbversuscmyk.php
 https://github.com/jamesthurley/colorwheelrenderer
Intermediate Colour Values[edit]
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)]
color gradient[edit]
 https://fractalforums.org/programming/11/cellularcoloringofmandelbrotinsides/3264
 https://fractalforums.org/fractalmathematicsandnewtheories/28/entropymandelbrotcoloring/368
 https://www.fractalforums.com/programming/classicmandelbrotwithdistanceandgradientforcoloring/
 https://www.fractalforums.com/programming/classicmandelbrotwithdistanceandgradientforcoloring/
discrete[edit]
 https://mokole.com/palette.html
 https://www.vis4.net/blog/2011/12/avoidequidistanthsvcolors/
 https://stackoverflow.com/questions/470690/howtoautomaticallygeneratendistinctcolors
2D[edit]
Gimp[edit]
 flatpak run org.gimp.GIMP
Commons[edit]
 Category:Images_by_technical_criteria
 Commons:Maximum file size
 Commons:File captions
 Commons:Structured data
 Commons:Image_guidelines
 Commons:Quality_images
 https://journals.aas.org/graphicsguide/
 Commons:Graphics village pump
 https://commonshelper.toolforge.org/
 svg optimizers
Template[edit]
 Template:Multilingual_description
 Template:Information
 Template:Image generation
 Template:Created_with_Octave
 Rename=Move
nested squares[edit]
float SCALE, RAD; 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, widthi)/width; RAD = atan2(widthi, i*VECT); for (int j=0; j<16; j++) { scale(SCALE); rotate(RAD); fill(map(j, 0, 15, 64, 255)); rect(0, 0, width, height); } }
logpolar mapping[edit]
examples
 An Introduction to the LogPolar Mapping by Jorge Dias, Helder Araujo, 1996, Proceedings of 2nd Workshop on Cybernetic Vision , also from semanticscholar
 Logspherical Mapping in SDF Raymarching by Pierre Cusa
 https://arxiv.org/pdf/1509.06344.pdf
 Finding straight lines and circles in logpolar images by David Young, University of Sussex
 https://stackoverflow.com/questions/13211595/howcaniconvertcoordinatesonacircletocoordinatesonasquare
 http://squircular.blogspot.com/2015/09/mappingcircletosquare.html
 https://rghmatlabstuff.wordpress.com/2012/02/21/mappingsplanetozplane/
 https://dspcan.homestead.com/files/Ztran/zlap.htm
 https://sthoduka.github.io/imreg_fmt/docs/logpolartransform/
 https://stackoverflow.com/questions/7781416/findaninverselogtransformationofanimageinmatlab?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa
 https://www.foerstemann.name/dokuwiki/doku.php?id=log_z_mandelbrotzooms
 http://users.isr.ist.utl.pt/~alex/Projects/TemplateTracking/logpolar.htm
 Rapid Anisotropic Diffusion Using SpaceVariant Vision: complex log transformation for various values of the map parameter a. Note that decreasing a (moving from left to right) increases the representation of the foveal region in the log plane. Dark areas correspond to regions outside the domain of the mapping.
 https://www.researchgate.net/publication/257641946_A_novel_quantum_representation_for_logpolar_images_Quantum_Inf
 https://www.youtube.com/watch?v=v_0otkcYY18
 STRIATE CORTEX: ADVANCED PIXEL MANIPULATION by Amnon on August 21, 2011
mapping[edit]
 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]}
 exponential_mapping_with_kalles_fraktaler by Claude HeilandAllen
 Exponential map From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
 feigenbaum stretch by Claude HeilandAllen
 wolfram Exp function visualizations
 Interactive Decal Compositing with Discrete Exponential Maps
 Polar and LogPolar Transformations in scikitimage python
 escher droste math by B. de Smit and H. W. Lenstra Jr.
 FRACTALS IN POLAR COORDINATES by Mika Seppä
 mapping of represents the map of the complex plane. It maps the positive half plane to the complex plane minus the negative real axis.
notation[edit]
is meant to say is a map whose domain is $A$ and whose codomain is $B$. $A$ and $B$ are both sets. E.g.
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. $\text{sq}:8\mapsto 64$.
To do[edit]
 http://functions.wolfram.com/ElementaryFunctions/Exp/visualizations/4/ShowAll.html
 http://functions.wolfram.com/ElementaryFunctions/Log/visualizations/4/
 https://www.chebfun.org/examples/complex/ConformalVis.html
 http://paulbourke.net/fractals/septagon/?fbclid=IwAR0finOMTmkd18jN8eq0rC5x8PKfIY7Lxy6yIjPlLnbwoXcyRhg1aMavE
spiral[edit]
 on the parameter plane
 on the dynamic plne
 component of Julia set near fixed point fractalforums.org: shapingspirals
 arm consisted from many components ( renormalizabel ? ) like https://commons.wikimedia.org/wiki/File:Julia_set_for_f(z)_%3D_z%5E2%2B0.355534_0.337292*i.png
 external rays landing on the fixed point
 critical orbit
 component of Julia set near fixed point fractalforums.org: shapingspirals
Mandelbrot[edit]
 The Misiurewicz point with preperiod k = 23 and period n = 2 is near c = 0.7756837680090538+0.13646736829469008i https://www.youtube.com/watch?v=6DTBaNOZXlU center of spiral
 rays are wiggly, here is one of period 20, with external angle .(00000000001111111111)
dense[edit]
 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[edit]
 0. Assume a wellbehaved 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 disklike component, often larger for cardioidlike 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.
22legged ant in the mset[edit]
 centered on 0.72398340 + 0.28671980i, with magnifications ranging from 0.714 to 120,000
 Rolo Silver's newsletter "Amygdala" issue 7 page 1
 I was watching this video by Rudy Rucker https://www.youtube.com/watch?v=ICrNOTQBS8U where he found an interesting structure in the mset that he called "the 22 legged ant".
 https://fractalforums.org/noobscorner/76/ineedhelpfindingthe22leggedantinthemset/4065/msg35757#msg35757
$ mdescribe 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 eastsoutheast 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 southwest 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 3.7103e07 and is pointing eastsoutheast 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 northnortheast 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 1.7465e07 and is pointing eastsoutheast the atom domain has size 7.0286e05 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 9.7098e08 to the westsouthwest 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 4.491e08 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 westsouthwest 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 westsouthwest 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 southsouthwest of the input point the multiplier has radius 1.0354 and angle 0.45526 (in turns)  12p11 the strength is 2.0286e05 the center is at 0.724112682973573563 + 0.286456567676711460 i the center is 0.00029327 to the southsouthwest of the input point the multiplier has radius 1.4656 and angle 0.00789 (in turns)  2p116 the strength is 2.3938e14 the center is at 0.723874171053648929 + 0.286847598521453362 i the center is 0.00016812 to the northeast of the input point the multiplier has radius 41405 and angle 0.45166 (in turns)  3p116 the strength is 9.1963e15 the center is at 0.723892769678345482 + 0.286784362650608693 i the center is 0.00011128 to the northeast 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"
alg by Claude[edit]
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 circlelike components, and are about 4x the size of the component, cardioidlike 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).
Here's my inner loop and supporting functions: https://code.mathr.co.uk/mandelbrotgraphics/blob/49377e6bc8a034403b26b79ee1c53f7a6c328a21:/c/lib/m_d_compute.c#l75 https://code.mathr.co.uk/mandelbrotnumerics/blob/6a23adebed4f3793d2b64ef32ce1b7ccfec3230c:/c/lib/m_d_interior_de.c#l8 https://code.mathr.co.uk/mandelbrotnumerics/blob/6a23adebed4f3793d2b64ef32ce1b7ccfec3230c:/c/lib/m_d_attractor.c#l8
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 }
mandelbrotnumerics] / c / lib / m_d_interior_de.c
1 // mandelbrotnumerics  numerical algorithms related to the Mandelbrot set
2 // Copyright (C) 20152018 Claude HeilandAllen
3 // License GPL3+ http://www.gnu.org/licenses/gpl.html
4
5 #include <mandelbrotnumerics.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[edit]
 z^2 + c to z^2 + 1/c
Mandelbrot set inversion, 4 different methods in one image by Arneauxtje
First, there's the transformation from the cardioid of the body of the set to a circle. This is done in cspace (a+ib) as follows:
rho=sqrt(a*a+b*b)1/4 phi=arctan(b/a) anew=rho*(2*cos(phi)cos(2*phi))/3 bnew=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 > cc*t+t/c and t=0..1
 multiplication: c > c/(t*(c*c1)+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
structure[edit]
other images[edit]
Julia[edit]
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 goodquality picture of J(f).
other Julia[edit]
 θ = 1/(10 + 1/(10 + · · ·)) Consider a polynomial P : C → C of degree d ≥ 2, with an irrationnally indifferent fixed point at the origin: P(0) = 0, P′ (0) = e 2iπθ, θ ∈ R \ Q. ( from On the linearization of degree three polynomials and Douady’s conjecture by Arnaud Ch´eritat (joint work with Xavier Buff)
 (*Koch center 3 cycle Julia*)\[IndentingNewLine]f[z_]=(0.1870.421*I)z^3+(2.3320.239*I)z^2+(0.285+2.879*I)z+1; Roger Lee BagulaSelfSimilarity and Fractals An example of a "Koch" Julia: I was looking for an image for my new music album and found this unique Julia and redid the picture with different colors and higher resolution: https://www.wolframcloud.com/.../j_koch9p1example2000...
 z^n + c*z where n=6, c=0.95 and R=0.2 https://geometricolor.wordpress.com/2013/10/28/coloringthejuliaset/
 parabolic z 32*z^5
 z3 + z + .6 I https://www.marksmath.org/scholarship/Julia.pdf
 Filled Julia set K(F) for 0 = [ai, a2, a3,...], where an = e^n. ????
 https://www.math.univtoulouse.fr/~cheritat/GalII/galery.html
 On the Julia set of a typical quadratic polynomial with a Siegel disk By C. L. PETERSEN and S. ZAKERI
 1/(cos(z)  z2/2) we can find this Julia set: http://www.juliasets.dk/UFP.htm GB
610[edit]
https://fractalforums.org/fractalmathematicsandnewtheories/28/juliaandparameterspaceimagesofpolynomials/2786/msg16210#msg16210 marcm200 A very basic cubic Julia set p(z)=z^3+(0.0996093750.794921875i), 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 noncycle 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 notoften 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[edit]
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[edit]
 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 (superattracting)
 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
 z^3+ (0.105,0.7905); https://www.shadertoy.com/view/3llyzl
rational[edit]
Julia sets of rational maps ( not polynomials )
 NonEuclidean Dreamer: It's the Julia Set of z=(1.4+1.2i)(1/(4z)1/(z+2)), with the Break Off condition increasing throughout the Video.
 f(z) = z^3/(z^3 + 1) + c, c = 0.18 + 0.68 i
 http://www.3dmeier.de/index.html
 http://www.juliasets.dk/Ratio.htm
 1/(z + z3)
 https://arxiv.org/pdf/1304.3881.pdf
 1/(z^21)
 1/(z^3) + the primitive 8th root of unity : https://arxiv.org/pdf/2004.02797.pdf
 parabolic (z^3z)/(1+4z z^2)
 julia sets for z⁴+c/z and z²+c/z³ photoniclabs
 McMullen Maps
 The Julia sets, defined by the equation (\ref{julia}), can take all kinds of shapes, and a small change in $c$ can change the Julia set very greatly https://complexanalysis.com/content/julia_set.html
 lapin f[z]=(z^2/(c*z^2+1)) c=0.30922922123657415`0.03212392698439477 ( pc(12)
Christopher Williams[edit]
 usefuljslibrary by Christopher Williams: fractals  rational_maps (*MandelbrotwithSQRT(x^2+y^2)limitedmeasure*)(*byR.L.BAGULA22.Nov2007©
Family:
Examples:
 Julia set of 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  0.005z2
 f = z^2  0.01z^2
 f = z2  0.01z2
Phoenix formula
 z5  0.06iz2
Robert L. Devaney[edit]
 Singular perturbations of complex polynomials
Julia sets for fc(z) = z*z + c[edit]
 http://www.scideveloper.com/Fractals/period.html
 mating : One polynomial is at the end of the 1/6 external ray of the Mandelbrot set, and is z>z^2+i (where i^2=1). The other one is at the end of the 5/14 external ray and is z>z^2+c with c = 1.23922555538957 + 0.412602181602004*i
 Slow divergence in a Julia set The appearance of more diverged area (ie the purple ‘river’) in the zoom above suggests that this particular Julia set (a=−0.29609091+0.62491i)
 c = 0.280000000000000 +0.011300000000000 i period = 1 but critical orbit is interesting
 z^3 0.0400000000000000360.78*I, period 2 : z0= z = 0.128494244956895 0.427153520116896 i, z1= = 0.108213690768581 0.723219415137925 i
 z^20.2+0.75i, period 3
 https://de.wikipedia.org/wiki/Benutzer:GeorgJohann/Mathematik#Visualising_Julis_sets
 http://www.3dmeier.de/tut20/Julia2/Seite1.html
 expanding orbits of periodic points : http://virtualmathmuseum.org/Fractal/julia/index.html
 0.3305 + 0.06i^{[10]}
 0.253930 + 0.000480i.
 0.27310 + 0.006990i.
 Julia set c=0.7630690.094691i
 −0.200062 + 0.807120i.
 http://yozh.org/2011/03/14/mset006/
 0.30.49i
 0.75 0.1i ; ( outside Mandelbrot set )
 0.75890.0753i http://socialbiz.org/2010/03/28/generatingchaos/
 c = 0.2349 + i*0.545 fractaldesire
 c = 0.750950561621545 +0.021484356174671 i parabolic)
 http://www.multiwingspan.co.uk/vb10.php?page=frac8
 c = 0.12375 + 0.56508i
 c = 0.12 + .74 i
 c = 0.11 + 0.6557i
 c = 0.194 + 0.6557i
 c = 0.125 + 0i
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[edit]
 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[edit]
 on the parameter plane
 part of Mset 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 manyarmed spirals (at least with distance estimation colouring)
In period 1 component of Mandelbrot set :
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[edit]
Period  c  r  1r  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[edit]
"c=0.027*%i0.757" period = 1 z= 0.01345178808414596*%i0.5035840525848648 r = m(z) = 1.007527366616821 1r = 0.007527366616821407 t = turn(m(z)) = 0.4957496478171055 p/q = 1/2 p/qt = 0.004250352182894435 z= 1.5035840525848650.01345178808414596*%i r = m(z) = 3.007288448945452 1r = 2.007288448945452 t = turn(m(z)) = 0.9985761611087214 p/q = 1/2 p/qt = 0.4985761611087214 period = 2 z= (0.1022072682395012*%i)0.3679154600985363 r = m(z) = 0.977981594918841 1r = 0.02201840508115904 t = turn(m(z)) = 0.01761164373863864 p/q = 1/2 p/qt = 0.4823883562613613 z= 0.1022072682395012*%i0.6320845399014637 r = m(z) = 0.9779815949188409 1r = 0.02201840508115915 t = turn(m(z)) = 0.01761164373863864 p/q = 1/2 p/qt = 0.4823883562613613
c=0.01*%i0.752" period = 1 z= 0.0049949453016411*%i0.501011962705025 r = m(z) = 1.002073722342039 1r = 0.00207372234203973 t = turn(m(z)) = 0.498413323518715 p/q = 0 p/qt = 0.498413323518715 z= 1.5010119627050250.0049949453016411*%i r = m(z) = 3.002040547136002 1r = 2.002040547136002 t = turn(m(z)) = 0.9994703791038458 p/q = 0 p/qt = 0.9994703791038458 period = 2 z= (0.06402358560400053*%i)0.4219037804142045 r = m(z) = 0.9928061240745848 1r = 0.007193875925415205 t = turn(m(z)) = 0.006414063302849116 p/q = 0 p/qt = 0.006414063302849116 z= 0.06402358560400053*%i0.5780962195857955 r = m(z) = 0.9928061240745848 1r = 0.007193875925415205 t = turn(m(z)) = 0.006414063302849116 p/q = 0 p/qt = 0.006414063302849116 (%o296) "/home/a/Dokumenty/periodic/MaximaCAS/p1/p.mac" (%i297)
marcm200[edit]
the input point was 1.2029905319213867e+00 + +1.4635562896728516e01 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 + +1.21263e01 i a point in the attractor is 7.0694e01 + +6.06308e02 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 5.00000e01 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 northwest the nucleus is 2.50250e01 to the southeast 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 8.11962e01 + +5.85423e01 i a point in the attractor is +1.81557e01 + 1.07368e01 i  a period 4 circle with nucleus at 1.310703e+00 + +3.761582e37 i the component has size 1.17960e01 and is pointing west the atom domain has size 2.34844e01 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 northeast the nucleus is 1.81719e01 to the southwest 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 +3.16563e01 + 9.50682e01 i a point in the attractor is +1.815562e01 + 1.073687e01 i external angles of this component are: .(0110) .(1001)  a period 10 circle with nucleus at 1.2103996e+00 + +1.5287483e01 i the component has size 2.02739e02 and is pointing northwest the atom domain has size 4.09884e02 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 southeast the nucleus is 9.86884e03 to the northwest of the input point the input point is interior to this component at radius 9.79333e01 and angle 0.985187275828761422 (in turns) the multiplier is +9.75094e01 + 9.10160e02 i a point in the attractor is +7.0332348e02 + 8.243835e02 i external angles of this component are: .(0110010110) .(0110011001)
the input point was 1.2255649566650391e+00 + +1.0837745666503906e01 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 + +8.91595e02 i a point in the attractor is 7.15546e01 + +4.45805e02 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 5.00000e01 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 westnorthwest the nucleus is 2.50250e01 to the eastsoutheast 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 9.02260e01 + +4.33510e01 i a point in the attractor is +1.94019e01 + 7.80792e02 i  a period 4 circle with nucleus at 1.310703e+00 + +0e+00 i the component has size 1.17960e01 and is pointing west the atom domain has size 2.34844e01 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 northeast the nucleus is 1.37819e01 to the southwest 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 +6.26142e01 + 7.82277e01 i a point in the attractor is +1.940184e01 + 7.80797e02 i external angles of this component are: .(0110) .(1001)  a period 14 circle with nucleus at 1.2299714e+00 + +1.1067143e01 i the component has size 1.06543e02 and is pointing westnorthwest the atom domain has size 1.95731e02 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 southeast the nucleus is 4.96796e03 to the westnorthwest of the input point the input point is interior to this component at radius 9.51928e01 and angle 0.992666114460366900 (in turns) the multiplier is +9.50917e01 + 4.38495e02 i a point in the attractor is +7.5747371e02 + 5.129233e02 i external angles of this component are: .(01100110010110) .(01100110011001)
the input point was 1.2256811857223511e+00 + +1.0814088582992554e01 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 + +8.89616e02 i a point in the attractor is 7.15576e01 + +4.44813e02 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 5.00000e01 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 westnorthwest the nucleus is 2.50253e01 to the eastsoutheast 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 9.02725e01 + +4.32564e01 i a point in the attractor is +1.9408e01 + 7.79018e02 i  a period 4 circle with nucleus at 1.310703e+00 + +0e+00 i the component has size 1.17960e01 and is pointing west the atom domain has size 2.34844e01 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 northeast the nucleus is 1.37561e01 to the southwest 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 +6.27801e01 + 7.80972e01 i a point in the attractor is +1.940823e01 + 7.790208e02 i external angles of this component are: .(0110) .(1001)  a period 14 circle with nucleus at 1.2299714e+00 + +1.1067143e01 i the component has size 1.06543e02 and is pointing westnorthwest the atom domain has size 1.95731e02 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 southeast the nucleus is 4.98108e03 to the westnorthwest of the input point the input point is interior to this component at radius 9.55071e01 and angle 0.984062994677356362 (in turns) the multiplier is +9.50287e01 + 9.54765e02 i a point in the attractor is +6.2976569e02 + 5.7543144e02 i external angles of this component are: .(01100110010110) .(01100110011001)
the input point was 8.422698974609375e01 + 1.9476318359375e01 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 8.64495e01 to the eastnortheast 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 + 1.85625e01 i a point in the attractor is 5.49225e01 + 9.28116e02 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 5.00000e01 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 southeast the nucleus is 2.50622e01 to the northwest 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 +6.30920e01 + 7.79053e01 i a point in the attractor is 1.0771e01 + +2.48238e01 i  a period 14 circle with nucleus at 8.4076071e01 + 1.9927227e01 i the component has size 1.01164e02 and is pointing southeast the atom domain has size 1.66560e02 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 southsoutheast the nucleus is 4.75495e03 to the southsoutheast of the input point the input point is interior to this component at radius 9.52171e01 and angle 0.935491618649184731 (in turns) the multiplier is +8.75023e01 + 3.75452e01 i a point in the attractor is +4.338561e02 + +7.777828e02 i external angles of this component are: .(10101010100110) .(10101010101001)
basilica[edit]
 San Marco Fractal = Basilica : c =  3/4
 parabolic perturbation
 5/11 :
5/11[edit]
 The angle 681/2047 or p01010101001 has preperiod = 0 and period = 11
 The angle 682/2047 or p01010101010 has preperiod = 0 and period = 11.
perturbated[edit]
 https://www.math.auckland.ac.nz/~berndk/transfer/hko_julia_prep.pdf
 http://archive.bridgesmathart.org/2019/bridges2019371.pdf
 http://math.bu.edu/people/bob/papers/rabbit.pdf
 http://gallery.bridgesmathart.org/exhibitions/2015jointmathematicsmeetings/amburns : "Adding a term d/z(z^21) introduces three poles: z=0, z=1, z=1. The orbits of initial points near the poles rapidly diverge to ∞; for very "small" (real, positive) d, amazingly, the boundary of the set of points whose orbit escapes (the Julia Set) contains an infinite number of tiny decorations resembling the decorations on the original "Basilica"."
 "Quasisymmetry group of the basilica Julia set", Sergiy Merenkov, March 26, 2020 NYGT Seminar talk
disconnected[edit]
 0.235434463552950220.5198893148651992i by Christopher Williams ( 2014)
 c = 0.750357820200574 +0.047756163825227 i ,
 c=a=0.786763484252847i*0.145686561560582
 https://www.youtube.com/watch?v=1nuG6X7XTf8
 http://dhushara.com/DarkHeart/
 connected : c = 0.0940639823675156+0.654706418514252i 1 > 3 > 57
 c = 0.2683101296424870.487654536962509i near 1> 4 > 52
 c = vec2(0.754, 0.05*(abs(cos(0.1*iTime))+0.8)); https://www.shadertoy.com/view/MllGzB
 http://www.imajeenyus.com/mathematics/20121112_distance_estimates/distance_estimation_method_for_fractals.pdf
 c = −0.8 + 0.16i.
 c = 0.8 +0.156 i
 c = −1.2+0.156i. 1 > 2> 10 > 70
 0.285 0.485 1> 4> 28 >
3 petals[edit]
 C= 0.125 +0.649519 i
 A limit of maps where the critical orbit escapes along the 1/7 ray (http://www.math.stonybrook.edu/~jack/RGFtprint.pdf)
Siegel Disk[edit]
Julia set = Jordan curve Irrational recurrent cycles
 0.59...+i0.43...
 0.33...+i0.07...
 C= 0.408792866 0.577405 i
 c=0.3905410.586788i
Other[edit]
Feigenbaum: C= 1.4011552 +0.0 i
Tower: C= 1 + 0.0 i
Cauliflower: C= 0.25 +0.0 i
Dendrite[edit]
Critical point eventually periodic 0 > 2 > 2 (fixed).
C= i
c^3 + 2c^2 +2c +2 =0
3[edit]
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 nonuniqueness, 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[edit]
Circle: C= 0.0 +0.0 i
Segment: C= 2 +0.0 i
centers[edit]
 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
 38cycle 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/tuneb.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[edit]
 c = 0.284912968784722 +0.484816779093857 i period = 84
Superattracting per 3 (up to complex conjugate)[edit]
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)[edit]
C= 1.9408
C= 1.3107
C= 1.62541
C= 0.15652 +1.03225 i
C= 0.282271 +0.530061 i
Kokopelli[edit]
 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 134 .
 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)[edit]
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[edit]
C= 0.0384261 +0.985494 i
dense[edit]
VividThickSpiral[edit]
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
gradient:
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
}
PinkLabyrinth[edit]
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
gradient:
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[edit]
https://www.deviantart.com/dinkydauset/art/Densitynearthecardoid3729541183
 Magnification:
 2^35.769
 5.8552024369543422761426995117521 E10
 Coordinates:
 Re = 0.360999615968828800
 Im = 0.121129382033034400
Density near the cardoid 3 by DinkydauSet[edit]
https://fractalforums.org/imagethreads/25/mandelbrotsetvariousstructures/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 7.14008235131944E11 7.14008235674045E11)
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[edit]
https://plus.google.com/u/0/photos/115452635610736407329/albums/6124078542095960129/6124078748270945650 frond tail Misiurewicz point of the period27 bulb of the quintic Mandelbrot set (I don't have a number ATM, but you can find that)
Machinereadable_data[edit]
 Machinereadable_data in commons
 Template:Information
  handling metadata ]
 Commons:Exif
 Category:Language templates with no text displayed
 description : Other fields=SOUL WINDSURFER developmentThis W3Cunspecified vector image was created with a text editor by n.
SVG[edit]
<?xml version="1.0" encoding="utf8" standalone="no"?> <!DOCTYPE svg PUBLIC "//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
Links :
 Commons:Maximum_file_size
 Template:Vector_version_available
 svg translator
 svg check
 wikipedia : List_of_ISO_6391_codes
{{Valid SVG}}
 optimize
Bezier curve[edit]
 https://www.particleincell.com/2012/beziersplines/
 http://www.jacos.nl/jacos_html/spline/
 https://pomax.github.io/bezierinfo/
ps[edit]
"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[edit]
Other fields=SOUL WINDSURFER development
Source code
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 and category ( example : Fractals gallery and Fractals category )
 animation and video
 category and tag
 Commons:Maximum file size
 Commons:File_renaming
<gallery> </gallery> {{SUL Boxenwikt}}
[[1]]
compare with[edit]
==Compare with== <gallery caption="Sample gallery" widths="100px" heights="100px" perrow="6"> </gallery> </nowiki>
Change in your preferences : Show hidden categories
metadata[edit]
 https://www.mediawiki.org/wiki/Manual:File_metadata_handling
 https://commons.wikimedia.org/wiki/Commons:Exif
syntaxhighlight[edit]
{{Galeria Nazwa = Trzy krzywe w różnych skalach wielkość = 400 pozycja = right Plik:LinLinScale.svgSkala liniowoliniowa Plik:LinLogScale.svgSkala liniowologarytmiczna Plik:LogLinScale.svgSkala logarytmicznoliniowa Plik:LogLogScale.svgSkala logarytmicznologarytmiczna }} == 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[edit]
 ↑ colortrac glossary: greyscale
 ↑ May 2004. http://users.mai.liu.se/hanlu09/complex/domain_coloring.html Retrieved 13 December 2018.
 ↑ (September 2012). "Domain Coloring of Complex Functions: An ImplementationOriented Introduction". IEEE Computer Graphics and Applications 32 (5): 90–97. DOI:10.1109/MCG.2012.100. PMID 24806991.
 ↑ stackoverflow.com/questions/71856674/howtosaveimagesafternormalizingthepixels/71863957?noredirect=1
 ↑ coolors by Fabrizio Bianchi
 ↑ https://colors.artyclick.com/colornamesdictionary/
 ↑ randompastelcolor by Micro Digital Tools
 ↑ theinnerframe : exponentialfunction/, Playing with Infinity
 ↑ Computability of BrolinLyubich Measure by Ilia Binder, Mark Braverman, Cristobal Rojas, Michael Yampolsky
 ↑ fractal_ken An "escape time" fractal generated by homemade software using recurrence relation z(n) = z(n  1)^2 + 0.3305 + 0.06i
GeSHi[edit]
description:
<syntaxhighlight lang="c" enclose="none"> g; a=2; </syntaxhighlight>
source templetate[edit]
deprecated:
<syntaxhighlight lang="gnuplot">ssssssssssssss</syntaxhighlight>
see also[edit]
wikitable collapsible collapsed[edit]
Source code 

[edit]
C++ source code  click on the right to view 

mandelbrot.cpp: 
sssssssssssssssssssssssssssss 
prettytable[edit]
Mathematical Function Plot  

Description  Function displaying a cusp at (0,1) 
Equation  
Coordinate System  Cartesian 
X Range  4 .. 4 
Y Range  0 .. 3 
Derivative  
Points of Interest in this Range  
Minima  
Cusps  
Derivatives at Cusp  ,
