Fractals/Iterations in the complex plane/MandelbrotSetExterior
Colouring of exterior of Mandelbrot set can be :
- non-smooth = Escape Time = dwell
- Boolean/binary Escape Time Method ( bETM/M )
- discrete = Level Set Method = LSM/M = integer ETM = iETM/M
- Smooth :
One can also draw curves :
- external rays
- equipotential lines ( closed curves - quasi circles)
Similar projects:
- Mandelbrot Notebook by Claude Heiland-Allen
- different drawing techniques and algorithms by Arnaud Cheritat
- Linas Vepstas' Art Gallery:
Escape time or dwell
editHere for given point c on parameter plane one checks how critical point behaves on dynamical plane under forward iteration. If you change initial point you will get different result [5]
To draw given plane one needs to check/scan (all) its points. See here for more details ( optimisation) Read definitions first.
How to find the number of iterations required to escape the mandelbrot set ?
edit- math.stackexchange question: is-there-an-equation-for-the-number-of-iterations-required-to-escape-the-mandelb
- math.stackexchange question: a-way-to-determine-the-ideal-number-of-maximum-iterations-for-an-arbitrary-zoom?
Boolean escape time
editThis algorithm answers the question: “For which values of c will the Julia fractal, J(c), be line-like and for which dust-like?”[6]
Here complex plane consists of 2 sets : Mandelbrot set and its complement :
ASCI graphic ( on screen)
edit// http://mrl.nyu.edu/~perlin/
main(k){float i,j,r,x,y=-16;while(puts(""),y++<15)for(x
=0;x++<84;putchar(" .:-;!/>)|&IH%*#"[k&15]))for(i=k=r=0;
j=r*r-i*i-2+x/25,i=2*r*i+y/10,j*j+i*i<11&&k++<111;r=j);}
-- Haskell code by Ochronus
-- http://blog.mostof.it/mandelbrot-set-in-ruby-and-haskell/
import Data.Complex
mandelbrot a = iterate (\z -> z^2 + a) a !! 500
main = mapM_ putStrLn [[if magnitude (mandelbrot (x :+ y)) < 2 then '*' else ' '
| x <- [-2, -1.9685 .. 0.5]]
| y <- [1, 0.95 .. -1]]
; common lisp
(loop for y from -1.5 to 1.5 by 0.05 do
(loop for x from -2.5 to 0.5 by 0.025 do
(let* ((c (complex x y)) ; parameter
(z (complex 0 0))
(iMax 20) ; maximal number of iterations
(i 0)) ; iteration number
(loop while (< i iMax ) do
(setq z (+ (* z z) c)) ; iteration
(incf i)
(when (> (abs z) 2) (return i)))
; color of pixel
(if (= i iMax) (princ (code-char 42)) ; inside M
(princ (code-char 32))))) ; outside M
(format t "~%")) ; new line
Comparison programs in various languages [7][8]
Graphic file ( PPM )
editHere are various programs for creating pbm file [9]
C
editThis is complete code of C one file program.
- It makes a ppm file which consists an image. To see the file (image) use external application ( graphic viewer).
- Program consists of 3 loops:
- iY and iX, which are used to scan rectangle area of parameter plane
- iterations.
For each point of screen (iX,iY) it's complex value is computed c=cx+cy*i.
For each point c is computed iterations of critical point
It uses some speed_improvement. Instead of checking :
sqrt(Zx2+Zy2)<ER
it checks :
(Zx2+Zy2)<ER2 // ER2 = ER*ER
It gives the same result but is faster.
/*
c program:
--------------------------------
1. draws Mandelbrot set for Fc(z)=z*z +c
using Mandelbrot algorithm ( boolean escape time )
-------------------------------
2. technique of creating ppm file is based on the code of Claudio Rocchini
http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
create 24 bit color graphic file , portable pixmap file = PPM
see http://en.wikipedia.org/wiki/Portable_pixmap
to see the file use external application ( graphic viewer)
*/
#include <stdio.h>
#include <math.h>
int main()
{
/* screen ( integer) coordinate */
int iX,iY;
const int iXmax = 800;
const int iYmax = 800;
/* world ( double) coordinate = parameter plane*/
double Cx,Cy;
const double CxMin=-2.5;
const double CxMax=1.5;
const double CyMin=-2.0;
const double CyMax=2.0;
/* */
double PixelWidth=(CxMax-CxMin)/iXmax;
double PixelHeight=(CyMax-CyMin)/iYmax;
/* color component ( R or G or B) is coded from 0 to 255 */
/* it is 24 bit color RGB file */
const int MaxColorComponentValue=255;
FILE * fp;
char *filename="new1.ppm";
char *comment="# ";/* comment should start with # */
static unsigned char color[3];
/* Z=Zx+Zy*i ; Z0 = 0 */
double Zx, Zy;
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
/* */
int Iteration;
const int IterationMax=200;
/* bail-out value , radius of circle ; */
const double EscapeRadius=2;
double ER2=EscapeRadius*EscapeRadius;
/*create new file,give it a name and open it in binary mode */
fp= fopen(filename,"wb"); /* b - binary mode */
/*write ASCII header to the file*/
fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue);
/* compute and write image data bytes to the file*/
for(iY=0;iY<iYmax;iY++)
{
Cy=CyMin + iY*PixelHeight;
if (fabs(Cy)< PixelHeight/2) Cy=0.0; /* Main antenna */
for(iX=0;iX<iXmax;iX++)
{
Cx=CxMin + iX*PixelWidth;
/* initial value of orbit = critical point Z= 0 */
Zx=0.0;
Zy=0.0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
/* */
for (Iteration=0;Iteration<IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
/* compute pixel color (24 bit = 3 bytes) */
if (Iteration==IterationMax)
{ /* interior of Mandelbrot set = black */
color[0]=0;
color[1]=0;
color[2]=0;
}
else
{ /* exterior of Mandelbrot set = white */
color[0]=255; /* Red*/
color[1]=255; /* Green */
color[2]=255;/* Blue */
};
/*write color to the file*/
fwrite(color,1,3,fp);
}
}
fclose(fp);
return 0;
}
Integer escape time = LSM/M = dwell bands
edit-
Number of details is proportional to maximal number of iterations
-
Mandelbrot animation based on a static number of iterations per pixel. Here you can see why offset is sometimes used ( because - color gradient changes : for high MaxIteration disapears.
Here color is proportional to last iteration ( of final_n, final iteration).[11]
This is also called Level Set Method ( LSM )
C
editDifference between Mandelbrot algorithm and LSM/M is in only in part instruction, which computes pixel color of exterior of Mandelbrot set. In LSM/M is :
if (Iteration==IterationMax)
{ /* interior of Mandelbrot set = black */
color[0]=0;
color[1]=0;
color[2]=0;
}
/* exterior of Mandelbrot set = LSM */
else if ((Iteration%2)==0)
{ /* even number = black */
color[0]=0; /* Red */
color[1]=0; /* Green */
color[2]=0; /* Blue */
}
else
{/* odd number = white */
color[0]=255; /* Red */
color[1]=255; /* Green */
color[2]=255; /* Blue */
};
Here is C function whithout explicit complex numbers, only doubles:
int GiveEscapeTime(double C_x, double C_y, int iMax, double _ER2)
{
int i;
double Zx, Zy;
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
Zx=0.0; /* initial value of orbit = critical point Z= 0 */
Zy=0.0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
for (i=0;i<iMax && ((Zx2+Zy2)<_ER2);i++)
{
Zy=2*Zx*Zy + C_y;
Zx=Zx2-Zy2 +C_x;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
return i;
}
here a short code with complex numbers:
// https://gitlab.com/adammajewski/mandelbrot_wiki_ACh/blob/master/betm.c
int iterate(double complex C , int iMax)
{
int i;
double complex Z= 0.0; // initial value for iteration Z0
for(i=0;i<iMax;i++)
{
Z=Z*Z+C; // https://stackoverflow.com/questions/6418807/how-to-work-with-complex-numbers-in-c
if(cabs(Z)>EscapeRadius) break;
}
return i;
}
C++
editHere is C++ function which can be used to draw LSM/M :
int iterate_mandel(complex C , int imax, int bailout)
{
int i;
std::complex Z(0,0); // initial value for iteration Z0
for(i=0;i<=imax-1;i++)
{
Z=Z*Z+C; // overloading of operators
if(abs(Z)>bailout)break;
}
return i;
}
I think that it can't be coded simpler (it looks better than pseudocode), but it can be coded in other way which can be executed faster .
Here is faster code :
// based on cpp code by Geek3
inline int fractal(double cx, double cy, int max_iters)
// gives last iteration
{
double zx = 0, zy = 0;
if (zx * zx + zy * zy > 4) return(0); // it=0
for (int it = 1; it < max_iters; it++)
{ double zx_old = zx;
zx = zx * zx - zy * zy;
zy = 2 * zx_old * zy;
zx += cx;
zy += cy;
if (zx * zx + zy * zy > 4.0) return(it);
}
return(max_iters);
}
A touch more optimised :
// optimised from cpp code by Geek3
inline int fractal(double cReal, double cImg, int max_iters)
// gives last iteration
{
double zReal = 0, zImg = 0, zReal2 = 0, zImg2 = 0;
//iteration zero is always 0^2+0^2, it will never escape
for (int it = 1; it < max_iters; it++)
{ //because we have zReal^2 and zImg^2 pre-calculated
//we can calculate zImg first
//then we don't need to calculate/store the "old" zReal
zImg = (2 * zReal * zImg ) + cImg;
zReal = zReal2 - zImg2 + cReal;
// calculate next iteration: zReal^2 and zImg^2
// they are used twice so calculate them once
zReal2 = zReal * zReal;
zImg2 = zImg * zImg;
if (zReal2 + zImg2 > 4.0) return(it);
}
return(max_iters);
}
See also :
GLSL
editJava
edit//Java code by Josef Jelinek
// http://java.rubikscube.info/
int mandel(double cx, double cy) {
double zx = 0.0, zy = 0.0;
double zx2 = 0.0, zy2 = 0.0;
int iter = 0;
while (iter < iterMax && zx2 + zy2 < 4.0) {
zy = 2.0 * zx * zy + cy;
zx = zx2 - zy2 + cx;
zx2 = zx * zx;
zy2 = zy * zy;
iter++;
}
return iter;
}
Java Script
editHere is JavaScript function which does not give last iteration but LastIteration modulo maxCol. It makes colour cycling ( if maxCol < maxIt ).
function iterate(Cr,Ci) {
// JavaScript function by Evgeny Demidov
// http://www.ibiblio.org/e-notes/html5/fractals/mandelbrot.htm
var I=0, R=0, I2=0, R2=0, n=0;
if (R2+I2 > max) return 0;
do { I=(R+R)*I+Ci; R=R2-I2+Cr; R2=R*R; I2=I*I; n++;
} while ((R2+I2 < max) && (n < maxIt) );
if (n == maxIt) return maxCol; else return n % maxCol;
}
Above functions do not use explicit definition of complex number.
Khan Academy
editLisp program
editWhole Lisp program making ASCII graphic based on code by Frank Buss [12] [13]
; common lisp
(loop for y from -1.5 to 1.5 by 0.1 do
(loop for x from -2.5 to 0.5 by 0.04 do
(let* ((i 0)
(z (complex x y))
(c z))
(loop while (< (abs
(setq z (+ (* z z) c)))
2)
while (< (incf i) 32))
(princ (code-char (+ i 32))))) ; ASCII chars <= 32 contains non-printing characters
(format t "~%"))
filter mandelbrot (gradient coloration) c=ri:(xy/xy:[X,X]*1.5-xy:[0.5,0]); z=ri:[0,0]; # initial value z0 = 0 # iteration of z iter=0; while abs(z)<2 && iter<31 do z=z*z+c; # z(n+1) = fc(zn) iter=iter+1 end; coloration(iter/32) # color of pixel end
Pov-Ray
editPov-Ray has a built-in function mandel[14]
Wolfram Mathematica
editHere is code by Paul Nylander
Level Curves of escape time Method = LCM/M
editLemniscates are boundaries of Level Sets of escape time ( LSM/M ). They can be drawn using :
- edge detection of Level sets.
- Algorithm described in paper by M. Romera et al.[15] This method is fast and allows looking for high iterations.
- boundary trace[16]
- drawing curves , see explanation and source code. This method is very complicated for iterations > 5.
Decomposition of target set for Mandelbrot set drawing
editDecomposition is modification of escape time algorithm.
The target set is divided into parts (2 or more). Very large escape radius is used, for example ER = 12.
Binary decomposition of LSM/M
editHere target set on dynamic plane is divided into 2 parts (binary decomposition = 2-decomposition ):
- upper half ( white)
- lower half (black)
Division of target set induces decomposition of level sets into parts ( cells, subsets):
- which is colored white,
- which is colored black.
"The Level Sets and Field Lines are superimposed, creating a sort of grid, and the "squares" of the grid are filled with N-digit binary numbers giving the first N binary digits of the external angles of field lines passing through the square. (Alternately, only the Nth binary digit is used.) Each level set is divided into 2n squares. It is easy to "read" the external arguments of points in the boundary of the Mandelbrot Set using a binary decomposition." Robert P. Munafo
For binary decomposition use exp(pi) as escape radius, so that the boxes appear square (a tip from mrob).
External rays of angles (measured in turns):
can be seen as borders of subsets.
Difference between binary decomposition algorithm and Mandel or LSM/M is in only in part of instruction , which computes pixel color of exterior of Mandelbrot set. In binary decomposition is :
if (Iteration==IterationMax)
{ /* interior of Mandelbrot set = black */
color[0]=0;
color[1]=0;
color[2]=0;
}
/* exterior of Mandelbrot set = LSM */
else if (Zy>0)
{
color[0]=0; /* Red */
color[1]=0; /* Green */
color[2]=0; /* Blue */
}
else
{
color[0]=255; /* Red */
color[1]=255; /* Green */
color[2]=255; /* Blue */
};
also GLSL code from Fragmentarium :
#include "2D.frag"
#group Simple Mandelbrot
// maximal number of iterations
uniform int iMax; slider[1,100,1000] // increase iMax
// er2= er^2 wher er= escape radius = bailout
uniform float er2; slider[4.0,1000,10000] // increase er2
// compute color of pixel
vec3 color(vec2 c) {
vec2 z = vec2(0.0); // initial value
// iteration
for (int i = 0; i < iMax; i++) {
z = vec2(z.x*z.x-z.y*z.y,2*z.x*z.y) + c; // z= z^2+c
if (dot(z,z)> er2) // escape test
// exterior
if (z.x>0){ return vec3( 1.0);} // upper part of the target set
else return vec3(0.0); //lower part of the target set
}
return vec3(0.0); //interior
}
// zoomasm -- zoom video assembler
// (c) 2019,2020,2021,2022 Claude Heiland-Allen
// SPDX-License-Identifier: AGPL-3.0-only
// recommended KF bailout settings: linear smoothing, custom radius 25
vec3 colour(void)
{
if (getInterior())
{
return vec3(1.0, 0.0, 0.0);
}
bool decomp = getT() < 0.5;
return vec3(decomp ? 0.0 : 1.0);
}
Point c is plotting white or black if imaginary value of last iteration ( Zy) is positive or negative.[17]
nth-decomposition
editThis method is extension of binary decomposition.
The target set T = { z : |zn| > R } with a very large escape radius ( for example R = 12 ) is divided into more than 2 parts ( for example 8).[18]
Real Escape Time
editOther names of this method/algorithm are :
- the fully-renormalized fractional iteration count ( by Linas Vepstas in 1997)[19]
- smooth iteration count for generalized Mandelbrot sets ( by Inigo Quilez in 2016)[20]
- continuous iteration count for the Mandelbrot set
- Normalized Iteration Count Algorithm
- Continuous coloring
- smooth colour gradient
- fractional iterations
- fractional escape time
Here color of exterior of Mandelbrot set is proportional not to Last Iteration ( which is integer number) but to real number :
Other methods and speedups
Colouring formula in Ultrafractal :[21]
smooth iter = iter + 1 + ( log(log(bailout)-log(log(cabs(z))) )/log(2)
where :
- log(log(bailout) can be precalculated
theory
editDescription by Claude :
First description :
If R is large, the first z to escape satisfies (approximately)[22]
so taking logs
so taking logs again
so by algebra
when at escape is bigger, the smooth iteration count should be smaller, so this value needs to be subtracted from the integer iteration count
Alternatively this fraction can be used for interpolation, or used with arg(z) for exterior tiling / binary decomposition.
Second description[23]
pick a radius R > 2, then |Z| > R implies that |Z^2 + C| > |Z| and more generally that |Z| -> infinity, this gives R the name escape radius. proof is on math.stackexchange.com somewhere
now suppose R is large, and n is the first iteration where |Z_n| > R.
consider what happens when |Z_n| increases as you move the point C a bit further from the Mandelbrot set boundary.
eventually |Z_n| > R^2, but then |Z_{n-1}| > R, so the iteration count should be n - 1.
for smoothing, we want a value to add to n that is 0 when |Z_n| = R and -1 when |Z_n| = R^2.
taking logs, get log |Z| is between log(R) and 2 log(R)
taking logs again, get log log |Z| is between log log R and log log R + log 2
dividing by log 2, get log_2 log |Z| is between log_2 log R and log_2 log R + 1
subtracting log_2 log R gives (log_2 log |Z| - log_2 log R) is between 0 and 1
negating it gives a value between 0 and -1, as desired
so the smooth iteration count is
(replace 2 by P if you do Z^P + C)
see also http://linas.org/art-gallery/escape/escape.html which makes a value independent of R, but that is not so useful for some colouring algorithms (e.g. smooth part of escape count doesn't align with angle of final iterate)
C
editTo use log2 function add :
#include <math.h>
at the beginning of program.
if (Iteration==IterationMax)
{ /* interior of Mandelbrot set = black */
color[0]=0;
color[1]=0;
color[2]=0;
}
/* exterior of Mandelbrot set */
else GiveRainbowColor((double)(Iteration- log2(log2(sqrt(Zx2+Zy2))))/IterationMax,color);
where :
- Zx2 = Zx*Zx
- Zy2 = Zy*Zy
Here is another version by Tony Finch[24]
while (n++ < max &&
x2+y2 < inf) {
y = 2*x*y + b;
x = x2-y2 + a;
y2 = y*y;
x2 = x*x;
}
nu = n - log(log(x2+y2)/2)/ log(2);
based on equation [25]
C++
edit// based on cpp code by Geek3 from http://en.wikibooks.org/wiki/File:Mandelbrot_set_rainbow_colors.png
sqrxy = x * x + y * y;
double m = LastIteration + 1.5 - log2(log2(sqrxy));
java
edit /**
Smooth coloring algorithm
https://gitlab.com/shreyas.siravara/mandelbrot-with-smooth-coloring/blob/master/Mandelbrot.java
Mandelbrot with Smooth Coloring by Shreyas Siravara
*/
double nsmooth = (iterations + 1 - Math.log(Math.log(Zn.getMagnitude())) / Math.log(ESCAPE_RADIUS));
double smoothcolor = nsmooth / MAX_ITERATIONS;
if (iterations < MAX_ITERATIONS) {
int rgb = Color.HSBtoRGB((float) (0.99f + 1.9 * smoothcolor), 0.9f, 0.9f);
g2d.setColor(new Color(rgb));
} else {
g2d.setColor(Color.black.darker());
}
Matemathica
editHere is code by Paul Nylander. It uses different formula :
Python
editPython code using mpmath library[26]
def mandelbrot(z):
c = z
for i in xrange(ITERATIONS):
zprev = z
z = z*z + c
if abs(z) > ESCAPE_RADIUS:
return ctx.exp(1j*(i + 1 - ctx.log(ctx.log(abs(z)))/ctx.log(2)))
return 0
Distance estimation DEM/M
edit-
Exterior DEM/M
-
simple boundary with DEM/M
-
Boundary with DEM/M and Sobel filter
Variants :
- exterior DEM/M
- interior DEM/M
Complex potential
editSee also
editReferences
edit- ↑ Mathematics of Divergent Fractals by
- ↑ jussi harkonen : coloring-techniques
- ↑ wikipedia : Orbit trap
- ↑ Mandelbrot Orbit Trap Rendering! Programming How-To Video by DKM101
- ↑ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
- ↑ Julia fractals in PostScript by Kees van der Laan, EUROTEX 2012 & 6CM PROCEEDINGS 47
- ↑ Fractal Benchmark by Erik Wrenholt
- ↑ 12-minute Mandelbrot: fractals on a 50 year old IBM 1401 mainframe
- ↑ Computer Language Benchmarks Game
- ↑ example-code-from-presentation-ways-of-seeing-julia-sets by ed Burke
- ↑ Computing the Mandelbrot set by Andrew Williams
- ↑ LIsp Program by Frank Buss
- ↑ Mandelbrot Set ASCII art at Bill Clementson's blog
- ↑ mandel function from 2.5.11.14 Fractal Patterns at Pov-Ray docs
- ↑ Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
- ↑ http://www.metabit.org/~rfigura/figura-fractal/math.html boundary trace by Robert Figura
- ↑ http://web.archive.org/20010415125044/www.geocities.com/CapeCanaveral/Launchpad/5113/fr27.htm%7C An open letter to Dr. Meech from Joyce Haslam in FRACTAL REPORT 27
- ↑ mandelbrot set n-th-decomposition
- ↑ linas.org : Renormalizing the Mandelbrot Escape
- ↑ I Quilez : mset_smooth
- ↑ fractalforums : What range/precision for fractional escape counts for Mandelbrot/Julia sets?
- ↑ fractalforums : gradient-pallet-with-two-colors
- ↑ fractalforums.org : can-anyone-help-me-understand-smooth-coloring
- ↑ Making Mandelbrot Set Movies by Tony Finch
- ↑ Linas Vepstas. Renormalizing the mandelbrot escape.
- ↑ mpmath Python library