Fractals/book

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Book here means :

versions[edit]

Features[edit]

Plane description[edit]

Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle". Radius has no initial value.

// / code/bin/mandelbrot_render.c
  int retval = 1;
  int width = 1280;
  int height = 720;
  int maximum_iterations = 4096;
  long double escape_radius = 512;
  mpfr_t radius;
  mpfr_init2(radius, 53);

Algorithms[edit]

Images with full src code :

  • exterior :
  • boundary : distance estimation ( DEM/M)
  • interior :
    • Interior coordinates[4]
    • Atom domains[5]

All algorithms are described in the book : "How To Write A Book About The Mandelbrot Set" by Claude Heiland-Allen[6]

nucleus or center[edit]

See function mandelbrot_nucleus function from book/code/gui/main.c.[7] It uses Newton method.

Compare with the muatom() function in mightymandel/src/atom.c[8]

instal[edit]

dependencies[edit]

  • OpenGl
    • GL/glew.h ( libglew-dev )
  • Gtk ( GUI
 sudo apt install mesa-utils
 sudo apt install libglew-dev
 sudo apt install freeglut3-dev
 sudo apt-get install libgtk-3-dev

clone[edit]

First clone the official git repository :

 git clone https://code.mathr.co.uk/mandelbrot-book

or the old one ( deprecated):

 git clone http://code.mathr.co.uk/book.git

or you can use unofficial repository :

git clone https://gitlab.com/adammajewski/my-book

make[edit]

go to the program directory and install :
cd book # or my_book
cd code
make -C lib
make -C bin
make
cd gui
make
cd ..

recompile[edit]

First:

make clean

then make again

pdf[edit]

First

sudo apt-get install texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra
sudo apt-get install texlive-science

then to make book in pdf format :

  • go to book/book directory
  • build
cd book
make

or use unofficial version with full c code[9]

images[edit]

 https://code.mathr.co.uk/mandelbrot-book-images/

update[edit]

git[edit]

From console opened in the mandelbrot-numerics directory :

 git pull

If you made some local changes you can undu them :

 git checkout -f

then

 git pull

Now install again

run[edit]

There are 2 types of the last versions of book program:

  • console
  • gui

To run gui program:

# cd gui
./mandelbrot_gui

To run console program use bash scripts, see examples :

 ./examples/standardview.sh

Examples[edit]

  • for the console program are bash sh files
  • for gui program are text files

for console program[edit]

#!/bin/bash

# Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
# https://en.wikibooks.org/wiki/Fractals/Computer_graphic_techniques/2D/plane
#
# in examples directory :  chmod +x standardview.sh
# then go to the parent directory : cd ..
# run run these example from the parent directory :
# ./examples/standardview.sh
#
# result : 
# ./render using double
# rgba 1.000000 1.000000 1.000000 1.000000
# Image  standard.png  is saved
# info text file  .txt is saved

       
view="-0.75 0 1.5" # "standard" view of parameter plane : center_re, center_im, radius

#
filename="standard"
pngfilename=$filename".png"

# Heredoc
./render $view && ./colour > out.ppm && ./annotate out.ppm  $pngfilename <<EOF
rgba 1 1 1 1
EOF

echo "Image " $pngfilename " is saved"

# info text file 
echo "view = re(center) im(center) radius = " $view > $filename".txt"
echo "info text file " $filname".txt is saved"

Extended use :

# Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
# https://en.wikibooks.org/wiki/Fractals/Computer_graphic_techniques/2D/plane
#
# in examples directory :  chmod +x standardview2.sh
# then go to the parent directory : cd ..
# run run these example from the parent directory :
# ./examples/ian.sh
#
# result : 
# ./render using double
# rgba 1.000000 1.000000 1.000000 1.000000
# Image  standard.png  is saved
# info text file  .txt is saved

# from file :  / code/bin/mandelbrot_render.c
#      int width = 1280;
#   int height = 720;

# standard view of parameter plane : center_re, center_im, radius
# plane description : view = center and radius 
center_re="-1.86057396001587505"
center_im="-0.00000093437424500"
center="$center_re $center_im"
radius="3.5884751097983668e-09" #$(echo 1.0/$zoom | bc -l) # real radius
view="$center $radius"

# inbits are proportional to zoom 
# inbits=$((54+$((4*$zoom))))

# image file names 
filename="ian" # filename stem : stem="standard"
ppmfilename=$filename".ppm"
pngfilename=$filename".png"

# escape radius
er="512"

# image size in pixels
w="800"
h="600"
# maximum iterations
n="1500"
# interior rendering
i="1"

# Heredoc
# ./render $view && ./colour "$stem" > "$stem.ppm"
#./render $view "$er" "$filename" "$w" "$h" "$n" "$i"  && ./colour "$filename" > "$ppmfilename" && ./annotate "$ppmfilename"  $pngfilename <<EOF
#rgba 1 1 1 1
#EOF

echo "Image " $pngfilename " is saved"

# info text file 
textfilename=$filename".txt"

ratio=$(echo $w/$h | bc -l)
# http://stackoverflow.com/questions/12882611/how-to-get-bc-to-handle-numbers-in-scientific-aka-exponential-notation
# bash do not use floating point 
rim=`echo ${radius} | sed -e 's/[eE]+*/\\*10\\^/'` # conver e to 10
rre=$(echo $ratio*$rim | bc -l) # real radius
cu=$(echo $center_im+$rim | bc -l)
cd=$(echo $center_im-$rim | bc -l)
cl=$(echo $center_re+$rre | bc -l)
cr=$(echo $center_re-$rre | bc -l)
rim="("$rim")" # add () because precedence of operators 
zoom=$(echo 1/$rim | bc -l) # zoom = 1/radius
z=$(echo "$zoom" | sed 's/e/*10^/g;s/ /*/' | bc)
echo "part of parameter complex plane " > $textfilename
echo "center of image c = " $center >> $textfilename
echo "radius = (image height/2) = " $rim >> $textfilename
echo "radius = (image height/2) = " $radius >> $textfilename
echo "up corner = center_im+radius = cu =" $cu >> $textfilename
echo "down corner = center_im-radius = cd =" $cd >> $textfilename
echo "left corner = center_re+ratio*radius = cl =" $cl >> $textfilename
echo "right corner = center_im-ratio*radius = cr =" $cr >> $textfilename
echo "image ratio = width/height =" $ratio >> $textfilename
echo "zoom level = fractint mag = "$zoom >>$textfilename
echo "ratio = image width/height =  " $ratio >> $textfilename
echo "info text file " $textfilename "is saved"

helper programs[edit]

period[edit]

usage :

./mandelbrot_box_period cx cy radius maxperiod

example :

./mandelbrot_box_period 3.06095924635699068e-01 2.37427672737983361e-02 2.0000000000000001e-10 1000

next example :

./mandelbrot_box_period -0.220857943592514  -0.650752006989254 0.0000020 2000
596

If nucleus in not inside rectangle defind by above parameters then returns 0

Compute nucleus[edit]

usage:

./mandelbrot_nucleu bits cx cy period maxiters

example :

./mandelbrot_nucleus 100 3.06095924635699068e-01 2.37427672737983361e-02 68 1000
angles[edit]
mandelbrot_external_angles[edit]
external rays on the root point

Compute angles of rays landing on the root point,[10] usage:

./mandelbrot_external_angles bits cx cy period

example :

./mandelbrot_external_angles 100 3.06095924635699068e-01 2.37427672737983361e-02 68

New :

 ./mandelbrot_external_angles 53 -3.9089629378291085e-01 5.8676031775674931e-01 89

result :

.(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001)
.(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010)

Another :

./mandelbrot_external_angles 100 -1.115234269232491  0.252761823831669 24
.(010110010110010110011001)
.(010110010110011001010110)
mandelbrot_describe_external_angle[edit]
External rays

G Pastor gave an example of external rays for which the resolution of the IEEE 754 is not sufficient:[11]

( period 3, lands on root point of period 3 component c3 = -0.125000000000000 +0.649519052838329i )

One can analyze these angles using program by Claude Heiland-Allen :

./bin/mandelbrot_describe_external_angle ".(001)"
binary: .(001)
decimal: 1/7
preperiod: 0
period: 3

 
./bin/mandelbrot_describe_external_angle 
".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)"
binary: 
.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)
decimal: 
33877456965431938318210482471113262183356704085033125021829876006886584214655562/237142198758023568227473377297792835283496928595231875152809132048206089502588927
preperiod: 0
period: 267

./bin/mandelbrot_describe_external_angle 
".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001)"
binary: 
.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001)
decimal: 
33877456965431938318210482471113262183356704085033125021829876006886584214655569/237142198758023568227473377297792835283496928595231875152809132048206089502588927
preperiod: 0
period: 267

./bin/mandelbrot_describe_external_angle 
".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)"
binary: 
.(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)
decimal: 
67754913930863876636420964942226524366713408170066250043659752013773168429311121/474284397516047136454946754595585670566993857190463750305618264096412179005177855
preperiod: 0
period: 268

 
./bin/mandelbrot_describe_external_angle 
".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)"
binary: 
.(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)
decimal: 
67754913930863876636420964942226524366713408170066250043659752013773168429311122/474284397516047136454946754595585670566993857190463750305618264096412179005177855
preperiod: 0
period: 268

Landing points of above rays are roots with angled internal addresses ( description by Claude Heiland-Allen) :

  • the upper one will be 1 -> 1/3 -> 3 -> 1/(period/3) -> period because it's the nearest bulb to the lower root cusp of 1/3 bulb and child bulbs of 1/3 bulb have periods 3 * denominator(internal angle) ie, 1 -> 1/3 -> 3 -> 1/89 -> 267
  • the lower one will be 1 -> floor(period/3)/period -> period because it's the nearest bulb below the 1/3 cusp ie, 1 -> 89/268 -> 268
  • the middle ray .(001) lands at the root of 1 -> 1/3 -> 3, from the cusp on the lower side (which is on the right in a standard unrotated view)

for gui program[edit]

size 2000 2000
view 53 3.0609592463186358e-01 2.3742767274521188e-02 7.6870929325598169e-11
text 57 3.060959246472445584e-01 2.374276727158354376e-02 272
text 56 3.06095924633099439e-01 2.3742767284688944e-02 204
text 56 3.06095924629285095e-01 2.37427672645622342e-02 204
text 54 3.06095924643046857e-01 2.374276727237906e-02 136
text 54 3.06095924629536442e-01 2.37427672749394494e-02 68
ray_in 2200 .(00000000000100000000000000110000000000000000000000000100000000000001)
ray_in 2200 .(00000000000100000000000000101111111111111100000000000010000000000010)

To understand loading parameter files see deserialize function :

static int deserialize(const char *filename) {
  FILE *in = fopen(filename, "rb");
  if (in) {
    while (G.anno) {
      GtkTreeIter iter;
      gpointer *thing;
      gtk_tree_model_get_iter_first(GTK_TREE_MODEL(G.annostore), &iter);
      if (gtk_tree_model_iter_next(GTK_TREE_MODEL(G.annostore), &iter)) {
        gtk_tree_model_get(GTK_TREE_MODEL(G.annostore), &iter, 1, &thing, -1);
        if (thing) {
          delete_annotation((struct annotation *) thing);
          gtk_tree_store_remove(G.annostore, &iter);
        }
      }
    }
    char *line = 0;
    size_t linelen = 0;
    ssize_t readlen = 0;
    while (-1 != (readlen = getline(&line, &linelen, in))) {
      if (readlen && line[readlen-1] == '\n') {
        line[readlen - 1] = 0;
      }
      if (0 == strncmp(line, "size ", 5)) {
        int w = 0, h = 0;
        sscanf(line + 5, "%d %d\n", &w, &h);
        // resize_image(w, h); // FIXME TODO make this work...
      } else if (0 == strncmp(line, "view ", 5)) {
        int p = 53;
        int set_result;
        char *xs = malloc(readlen);
        char *ys = malloc(readlen);
        char *rs = malloc(readlen);
        sscanf(line + 5, "%d %s %s %s", &p, xs, ys, rs);
        struct view *v = view_new();
        mpfr_set_prec(v->cx, p);
        mpfr_set_prec(v->cy, p);
        set_result = mpfr_set_str(v->cx, xs, 0, GMP_RNDN);
        set_result = set_result + mpfr_set_str(v->cy, ys, 0, GMP_RNDN);
        set_result = set_result + mpfr_set_str(v->radius, rs, 0, GMP_RNDN);
        free(xs);
        free(ys);
        free(rs);
        if (set_result < 0) { printf("view line not valid, check chars\n"); return 1; }
        start_render(v);
      } else if (0 == strncmp(line, "ray_out ", 8)) {
        int p = 53;
        int set_result;
        char *xs = malloc(readlen);
        char *ys = malloc(readlen);
        sscanf(line + 8, "%d %s %s", &p, xs, ys);
        mpfr_t cx, cy;
        mpfr_init2(cx, p);
        mpfr_init2(cy, p);
        set_result = mpfr_set_str(cx, xs, 0, GMP_RNDN);
        set_result = set_result + mpfr_set_str(cy, ys, 0, GMP_RNDN);
        free(xs);
        free(ys);
        double x = 0, y = 0;
        param_to_screen(&x, &y, cx, cy);
        mpfr_clear(cx);
        mpfr_clear(cy);
        if (set_result < 0) { printf("ray_out line not valid, check chars\n"); return 1; }
        start_ray_out(x, y);
      } else if (0 == strncmp(line, "ray_in ", 7)) {
        int depth = 1000;
        char *as = malloc(readlen);
        sscanf(line + 7, "%d %s", &depth, as);
        struct mandelbrot_binary_angle *angle = mandelbrot_binary_angle_from_string(as);
        struct mandelbrot_binary_angle *angle2 = mandelbrot_binary_angle_canonicalize(angle);
        mandelbrot_binary_angle_delete(angle);
        start_ray_in(angle2, depth);
        free(as);
      } else if (0 == strncmp(line, "text ", 5)) {
        int p = 53;
        int set_result;
        char *xs = malloc(readlen);
        char *ys = malloc(readlen);
        char *ss = malloc(readlen);
        sscanf(line + 5, "%d %s %s %s", &p, xs, ys, ss);
        struct annotation *anno = calloc(1, sizeof(struct annotation));
        anno->tag = annotation_text;
        anno->label = ss;
        mpfr_init2(anno->u.text.x, p);
        mpfr_init2(anno->u.text.y, p);
        set_result = mpfr_set_str(anno->u.text.x, xs, 0, GMP_RNDN);
        set_result = set_result + mpfr_set_str(anno->u.text.y, ys, 0, GMP_RNDN);
        free(xs);
        free(ys);
        if (set_result < 0) {
          free(ss);
          mpfr_clear(anno->u.text.x);
          mpfr_clear(anno->u.text.y);
          printf("text line not valid, check chars \n");
          return 1;
        }
        add_annotation(anno);
      }
    }
    free(line);
    fclose(in);
    return 0;
  } else {
    return 1;
  }
}

ray-out[edit]

Input format should be of the form:

 .pre(period)

eg:

  .010(110)

Gallery[edit]

commons category: Fractals created with mandelbrot-book program

use[edit]

In the gui program box means that user have to mark rectangle ( like nucleus) not to click ( like bond)

Dictionary[edit]

bulb[edit]

  • component
  • Mu-Atom

Bond[edit]

"Bond is a " point where two mu-atoms meet. The parent "binds to" the child through the bond, and the two are said to be adjacent. This use of the word 'bond' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature." 
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.

In the book program result of computing bond is the internal angle ( = combinatorial rotation number ) of the parent components

dwell[edit]

"Dwell is a colloquial name for the Representation Function called Escape-Iterations." From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.

"The dwell bands are the regions of solid color that appear in a standard view of the Mandelbrot Set; they are also called level sets. All points in a dwell band have the same dwell. They are separated from one another by the "contour lines" or lemniscates."

hub[edit]

  • branch type Misiurewicz point

nucleus[edit]

Nucleus of a Mu-Atom = center of hyperbolic component of Mandelbrot set

"The unique point within any mu-atom which has the property of belonging to its own limit cycle. This point is called the superstable point.
This use of the word 'nucleus' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature.
If you set the polynomial formula for a lemniscate ZN equal to zero and solve for C (to get the roots of the polynomial), 
the roots are the nuclei of the mu-atoms of period N, plus any mu-atoms of periods that divide evenly into N. 
This procedure has been used numerically by Jay Hill to find all mu-atoms for periods up to about 16." 
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.   

Finding nucleus

A nucleus n of period p is a point of parameter plane

which satisfies :

where :

Claude is using Newton's method in one variable to compute c=n

Partials[edit]

  • good guesses for the period of hyperbolic component (enclosed by atom domain) of c are the partials, namely the values of p for which |fpc(0)| reach a new minimum.
  • the partials, namely the values of p for which |fpc(0)| reach a new minimum. It is related with the interior distance[12]
  • "partials" are the iterations p where |z_p| is smaller than all previous |z_n|. They are good candidates for possible periods.
  • "Partials" is my term for the set of n with 0 < n such that |z_n| < |z_m| for all 0 < m < n, where z_{n+1} = z_n^2 + c, z_0 = 0. This is related to "atom period domain" colouring: http://mrob.com/pub/muency/atomdomain.html
  • A different way is to check that z_p is near zero, rather than z_{p+1} is near c. They have the same effect in the end, but it's a little easier to calculate.

A few examples:

-9.82053364278e-2 + 0.87751161636 i is the center of a bulb with period 30 and approximate size 8.7e-4 its partials are 1 3 6 12 30

-1.40107054826248 + 1.68078322683058e-2 i is the center of a cardioid with period 25 and approximate size 2.9e-7 its partials are 1 2 4 8 25

-6.30751837180080329933379814864882594423413629277790243935409e-02 + 9.97813152226579778761450011018468925066022924931316287706002e-01 i is the center of a cardioid with period 660 and approximate size 1e-51 its partials are 1 3 4 5 12 34 133 660

-1.949583466095265215576424927006606703613013182307914337344552997126238598475224082315026579e+00 + -7.76868505224924928703440073040924718407938044210292978384443422263629366944037056031909997e-06 i is the center of a cardioid with period 1820 and approximate size 3.5e-83 its partials are 1 2 3 4 9 25 83 359 1820

// gcc -std=c99 -Wall -Wextra -pedantic -O3 -o partials partials.c -lm
// ./partials re im
// http://www.fractalforums.com/help-and-support/period-detection-for-dummies/?topicseen

#include <complex.h>
#include <stdio.h>
#include <stdlib.h>

int main(int argc, char **argv)
{
  if (argc != 3)
  {
    fprintf(stderr, "usage: %s re im\n", argv[0]);
    return 1;
  }
  double re = atof(argv[1]);
  double im = atof(argv[2]);
  double _Complex c = re + im * I;
  double _Complex z = 0;
  double minimum = 1.0 / 0.0; // infninity
  printf("   n    z                                                  |z|                      min\n");
  for (int n = 1; n < 100; ++n)
  {
    z = z * z + c;
    double absz = cabs(z);
    printf("%4d    %+.18f + %+.18f i    %+.18f", n, creal(z), cimag(z), absz);
    if (absz < minimum)
    {
      minimum = absz;
      printf("    ****");
    }
    printf("\n");
  }
  return 0;
}

Example output :

 ./partials -0.509 0.576
   n    z                                                  |z|                      min
   1    -0.509000000000000008 + +0.575999999999999956 i    +0.768672231838772757    ****
   2    -0.581694999999999962 + -0.010368000000000044 i    +0.581787391105204277    ****
   3    -0.170738422399000056 + +0.588062027520000030 i    +0.612346762132562117
   4    -0.825665339327633863 + +0.375190434296955644 i    +0.906912958643195766
   5    +0.031955390579078591 + -0.043563474492556487 i    +0.054027060783695097    ****
   6    -0.509876629322802200 + +0.573215824315217226 i    +0.767170488467169953
   7    -0.577602204115791773 + -0.008538704752669046 i    +0.577665314588191370
   8    -0.175448603279432458 + +0.585863949370871162 i    +0.611570747800398218
   9    -0.821454354779731055 + +0.370421976742216996 i    +0.901110258425790955
  10    +0.028574816132972747 + -0.032569491802020845 i    +0.043327726841770761    ****
  11    -0.509244251679208726 + +0.574138665520425695 i    +0.767440496138881434
  12    -0.579305499377257949 + -0.008753630166097426 i    +0.579371631726838143
  13    -0.173481764432350638 + +0.586142052189469798 i    +0.611276065240121014
  14    -0.822466582754321607 + +0.372630085156343605 i    +0.902942113377815159
  15    +0.028598099383947528 + -0.036951585539979570 i    +0.046725485147749588
  16    -0.509547568385544269 + +0.573886509768666397 i    +0.767453223683425945
  17    -0.578707001646840746 + -0.008844951163781811 i    +0.578774590765840591
  18    -0.174176439406013128 + +0.586237270335409733 i    +0.611564852795244307
  19    -0.822336705086155639 + +0.371782559211755903 i    +0.902474336402979138
  20    +0.029015385197912136 + -0.035460889501387816 i    +0.045818852696383125

final ...[edit]

  • final_n = last iteration = n when z escapes
  • final_z = last z
  • final angle = angle of last z


 int final_n; // last iteration 
 complex float final_z; // 
 
 if (cabs2(z) > escape_radius2) {
      final_n = n;
      final_z = z; }

spoke[edit]

spoke = branch of the tree , arm of dendritic structure

wucleus[edit]

Periodic point z = wucleus

A wucleus w of a point c within a hyperbolic component of period p satisfies :

 


where

 


so w is a point from dynamic plane ( z-plane)

How to contribute ?[edit]

git add book.pdf book.tex code/*.c
git commit -m "description"
git format-patch origin/master

and send patch to Claude by e-mail.

gcc render.c -E -o 26render.c
remove some code manually
add include

References[edit]

  1. c program by Claude Heiland-Allen
  2. Claude Heiland-Allen - blog
  3. An algorithm to draw external rays of the Mandelbrot set by Tomoki KAWAHIRA
  4. Interior coordinates in the Mandelbrot set
  5. Modified Atom Domains
  6. Mandelbrot Notebook
  7. book/code/gui/main.c
  8. mightymandel/src/atom.c code
  9. unofficial version ( with full c code ) of book "How to write a book about the Mandelbrot set" by by Claude Heiland-Allen
  10. Automatically finding external angles by Claude Heiland-Allen
  11. A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera,1 G. Pastor, A. B. Orue,1 A. Martin, M.-F. Danca,and F. Montoya
  12. Practical interior distance rendering by Claude Heiland-Allen