How to program a fractal

Question

I do not have any experience with programming fractals  Of course I ve seen the famous Mandelbrot images and such   Can you provide me with simple algorithms for fractals   Programming language doesn t matter really  but I m most familiar with actionscript  C   Java   I know that if I google fractals  I get a lot of  complicated  information but I would like to start with a simple algorithm and play with it   Suggestions to improve on the basic algorithm are also welcome  like how to make them in those lovely colors and such

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I would start with something simple  like a Koch Snowflake   It s a simple process of taking a line and transforming it  then repeating the process recursively until it looks neat-o   Something super simple like taking 2 points  a line  and adding a 3rd point  making a corner   then repeating on each new section that s created   fractal p0  p1       Pmid   midpoint p0 p1    moved some distance perpendicular to p0 or p1      fractal p0 Pmid       fractal Pmid  p1

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Well  simple and graphically appealing don t really go hand in hand  If you re serious about programming fractals  I suggest reading up on iterated function systems and the advances that have been made in rendering them   http   flam3 com flame draves pdf

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There is a great book called Chaos and Fractals that has simple example code at the end of each chapter that implements some fractal or other example  A long time ago when I read that book  I converted each sample program  in some Basic dialect  into a Java applet that runs on a web page  The applets are here  http   hewgill com chaos-and-fractals   One of the samples is a simple Mandelbrot implementation

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There is a great book called Chaos and Fractals that has simple example code at the end of each chapter that implements some fractal or other example  A long time ago when I read that book  I converted each sample program  in some Basic dialect  into a Java applet that runs on a web page  The applets are here  http   hewgill com chaos-and-fractals   One of the samples is a simple Mandelbrot implementation

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The mandelbrot set is generated by repeatedly evaluating a function until it overflows  some defined limit   then checking how long it took you to overflow    Pseudocode   MAX COUNT   64    if we haven t escaped to infinity after 64 iterations                     then we re inside the mandelbrot set     foreach  x-pixel    foreach  y-pixel      calculate x y as mathematical coordinates from your pixel coordinates     value    x  y      count   0     while value absolutevalue  lt  1 billion and count  lt  MAX COUNT         value   value   value    x  y          count   count   1         the following should really be one statement  but I split it for clarity     if count    MAX COUNT          pixel at  x-pixel  y-pixel    BLACK     else          pixel at  x-pixel  y-pixel    colors count     some color map     Notes   value is a complex number  a complex number  a bi  is squared to give  aa-b b 2 abi   You ll have to use a complex type  or include that calculation in your loop

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You should indeed start with the Mandelbrot set  and understand what it really is   The idea behind it is relatively simple  You start with a function of complex variable     f z    z2   C   where z is a complex variable and C is a complex constant  Now you iterate it starting from z   0  i e  you compute z1   f 0   z2   f z1   z3   f z2  and so on  The set of those constants C for which the sequence z1  z2  z3      is bounded  i e  it does not go to infinity  is the Mandelbrot set  the black set in the figure on the Wikipedia page    In practice  to draw the Mandelbrot set you should    Choose a rectangle in the complex plane  say  from point -2-2i to point 2 2i   Cover the rectangle with a suitable rectangular grid of points  say  400x400 points   which will be mapped to pixels on your monitor  For each point pixel  let C be that point  compute  say  20 terms of the corresponding iterated sequence z1  z2  z3      and check whether it  goes to infinity   In practice you can check  while iterating   if the absolute value of one of the 20 terms is greater than 2  if one of the terms does  the subsequent terms are guaranteed to be unbounded   If some z k does  the sequence  goes to infinity   otherwise  you can consider it as bounded  If the sequence corresponding to a certain point C is bounded  draw the corresponding pixel on the picture in black  for it belongs to the Mandelbrot set   Otherwise  draw it in another color  If you want to have fun and produce pretty plots  draw it in different colors depending on the magnitude of abs 20th term     The astounding fact about fractals is how we can obtain a tremendously complex set  in particular  the frontier of the Mandelbrot set  from easy and apparently innocuous requirements   Enjoy

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Programming the Mandelbrot is easy  My quick-n-dirty code is below  not guaranteed to be bug-free  but a good outline    Here s the outline  The Mandelbrot-set lies in the Complex-grid completely within a circle with radius 2   So  start by scanning every point in that rectangular area  Each point represents a Complex number  x   yi   Iterate that complex number    new value     old-value  2    original-value  while keeping track of two things   1   the number of iterations  2   the distance of  new-value  from the origin   If you reach the Maximum number of iterations  you re done  If the distance from the origin is greater than 2  you re done   When done  color the original pixel depending on the number of iterations you ve done  Then move on to the next pixel   public void MBrot         float epsilon   0 0001     The step size across the X and Y axis     float x      float y      int maxIterations   10     increasing this will give you a more detailed fractal     int maxColors   256     Change as appropriate for your display       Complex Z      Complex C      int iterations      for x -2  x lt  2  x   epsilon                for y -2  y lt  2  y   epsilon                        iterations   0              C   new Complex x  y               Z   new Complex 0 0               while Complex Abs Z   lt  2  amp  amp  iterations  lt  maxIterations                                Z   Z Z   C                  iterations                              Screen Plot x y  iterations   maxColors     depending on the number of iterations  color a pixel                      Some details left out are   1   Learn exactly what the Square of a Complex number is and how to calculate it   2   Figure out how to translate the  -2 2  rectangular region to screen coordinates

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Well  simple and graphically appealing don t really go hand in hand  If you re serious about programming fractals  I suggest reading up on iterated function systems and the advances that have been made in rendering them   http   flam3 com flame draves pdf

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The Sierpinski triangle and the Koch curve are special types of flame fractals  Flame fractals are a very generalized type of Iterated function system  since it uses non-linear functions    An algorithm for IFS es are as follows   Start with a random point   Repeat the following many times  a million at least  depending on final image size     Apply one of N predefined transformations  matrix transformations or similar  to the point  An example would be that multiply each coordinate with 0 5  Plot the new point on the screen    If the point is outside the screen  choose randomly a new one inside the screen instead   If you want nice colors  let the color depend on the last used transformation

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The mandelbrot set is generated by repeatedly evaluating a function until it overflows  some defined limit   then checking how long it took you to overflow    Pseudocode   MAX COUNT   64    if we haven t escaped to infinity after 64 iterations                     then we re inside the mandelbrot set     foreach  x-pixel    foreach  y-pixel      calculate x y as mathematical coordinates from your pixel coordinates     value    x  y      count   0     while value absolutevalue  lt  1 billion and count  lt  MAX COUNT         value   value   value    x  y          count   count   1         the following should really be one statement  but I split it for clarity     if count    MAX COUNT          pixel at  x-pixel  y-pixel    BLACK     else          pixel at  x-pixel  y-pixel    colors count     some color map     Notes   value is a complex number  a complex number  a bi  is squared to give  aa-b b 2 abi   You ll have to use a complex type  or include that calculation in your loop

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Programming the Mandelbrot is easy  My quick-n-dirty code is below  not guaranteed to be bug-free  but a good outline    Here s the outline  The Mandelbrot-set lies in the Complex-grid completely within a circle with radius 2   So  start by scanning every point in that rectangular area  Each point represents a Complex number  x   yi   Iterate that complex number    new value     old-value  2    original-value  while keeping track of two things   1   the number of iterations  2   the distance of  new-value  from the origin   If you reach the Maximum number of iterations  you re done  If the distance from the origin is greater than 2  you re done   When done  color the original pixel depending on the number of iterations you ve done  Then move on to the next pixel   public void MBrot         float epsilon   0 0001     The step size across the X and Y axis     float x      float y      int maxIterations   10     increasing this will give you a more detailed fractal     int maxColors   256     Change as appropriate for your display       Complex Z      Complex C      int iterations      for x -2  x lt  2  x   epsilon                for y -2  y lt  2  y   epsilon                        iterations   0              C   new Complex x  y               Z   new Complex 0 0               while Complex Abs Z   lt  2  amp  amp  iterations  lt  maxIterations                                Z   Z Z   C                  iterations                              Screen Plot x y  iterations   maxColors     depending on the number of iterations  color a pixel                      Some details left out are   1   Learn exactly what the Square of a Complex number is and how to calculate it   2   Figure out how to translate the  -2 2  rectangular region to screen coordinates

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You should indeed start with the Mandelbrot set  and understand what it really is   The idea behind it is relatively simple  You start with a function of complex variable     f z    z2   C   where z is a complex variable and C is a complex constant  Now you iterate it starting from z   0  i e  you compute z1   f 0   z2   f z1   z3   f z2  and so on  The set of those constants C for which the sequence z1  z2  z3      is bounded  i e  it does not go to infinity  is the Mandelbrot set  the black set in the figure on the Wikipedia page    In practice  to draw the Mandelbrot set you should    Choose a rectangle in the complex plane  say  from point -2-2i to point 2 2i   Cover the rectangle with a suitable rectangular grid of points  say  400x400 points   which will be mapped to pixels on your monitor  For each point pixel  let C be that point  compute  say  20 terms of the corresponding iterated sequence z1  z2  z3      and check whether it  goes to infinity   In practice you can check  while iterating   if the absolute value of one of the 20 terms is greater than 2  if one of the terms does  the subsequent terms are guaranteed to be unbounded   If some z k does  the sequence  goes to infinity   otherwise  you can consider it as bounded  If the sequence corresponding to a certain point C is bounded  draw the corresponding pixel on the picture in black  for it belongs to the Mandelbrot set   Otherwise  draw it in another color  If you want to have fun and produce pretty plots  draw it in different colors depending on the magnitude of abs 20th term     The astounding fact about fractals is how we can obtain a tremendously complex set  in particular  the frontier of the Mandelbrot set  from easy and apparently innocuous requirements   Enjoy

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Programming the Mandelbrot is easy  My quick-n-dirty code is below  not guaranteed to be bug-free  but a good outline    Here s the outline  The Mandelbrot-set lies in the Complex-grid completely within a circle with radius 2   So  start by scanning every point in that rectangular area  Each point represents a Complex number  x   yi   Iterate that complex number    new value     old-value  2    original-value  while keeping track of two things   1   the number of iterations  2   the distance of  new-value  from the origin   If you reach the Maximum number of iterations  you re done  If the distance from the origin is greater than 2  you re done   When done  color the original pixel depending on the number of iterations you ve done  Then move on to the next pixel   public void MBrot         float epsilon   0 0001     The step size across the X and Y axis     float x      float y      int maxIterations   10     increasing this will give you a more detailed fractal     int maxColors   256     Change as appropriate for your display       Complex Z      Complex C      int iterations      for x -2  x lt  2  x   epsilon                for y -2  y lt  2  y   epsilon                        iterations   0              C   new Complex x  y               Z   new Complex 0 0               while Complex Abs Z   lt  2  amp  amp  iterations  lt  maxIterations                                Z   Z Z   C                  iterations                              Screen Plot x y  iterations   maxColors     depending on the number of iterations  color a pixel                      Some details left out are   1   Learn exactly what the Square of a Complex number is and how to calculate it   2   Figure out how to translate the  -2 2  rectangular region to screen coordinates

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Here is a codepen that I wrote for the Mandelbrot fractal using plain javascript and HTML   Hopefully it is easy to understand the code   The most complicated part is scale and translate the coordinate systems   Also complicated is making the rainbow palette   function mandel x y      var a 0  var b 0    for  i   0  i lt 250    i           Complex z   z 2   c     var t   a a - b b      b   2 a b      a   t      a   a   x      b   b   y      var m   a a   b b      if  m  gt  10   return i        return 250

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I would start with something simple  like a Koch Snowflake   It s a simple process of taking a line and transforming it  then repeating the process recursively until it looks neat-o   Something super simple like taking 2 points  a line  and adding a 3rd point  making a corner   then repeating on each new section that s created   fractal p0  p1       Pmid   midpoint p0 p1    moved some distance perpendicular to p0 or p1      fractal p0 Pmid       fractal Pmid  p1

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Well  simple and graphically appealing don t really go hand in hand  If you re serious about programming fractals  I suggest reading up on iterated function systems and the advances that have been made in rendering them   http   flam3 com flame draves pdf

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Sometimes I program fractals for fun and as a challenge  You can find them here  The code is written in Javascript using the P5 js library and can be read directly from the HTML source code   For those I have seen the algorithms are quite simple  just find the core element and then repeat it over and over  I do it with recursive functions  but can be done differently

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If complex numbers give you a headache  there is a broad range of fractals that can be formulated using an L-system  This requires a couple of layers interacting  but each is interesting in it own right   First you need a turtle  Forward  Back  Left  Right  Pen-up  Pen-down  There are lots of fun shapes to be made with turtle graphics using turtle geometry even without an L-system driving it  Search for  LOGO graphics  or  Turtle graphics   A full LOGO system is in fact a Lisp programming environment using an unparenthesized Cambridge Polish syntax  But you don t have to go nearly that far to get some pretty pictures using the turtle concept   Then you need a layer to execute an L-system  L-systems are related to Post-systems and Semi-Thue systems  and like virii  they straddle the border of Turing Completeness  The concept is string-rewriting  It can be implemented as a macro-expansion or a procedure set with extra controls to bound the recursion  If using macro-expansion  as in the example below   you will still need a procedure set to map symbols to turtle commands and a procedure to iterate through the string or array to run the encoded turtle program  For a bounded-recursion procedure set  eg    you embed the turtle commands in the procedures and either add recursion-level checks to each procedure or factor it out to a handler function   Here s an example of a Pythagoras  Tree in postscript using macro-expansion and a very abbreviated set of turtle commands  For some examples in python and mathematica  see my code golf challenge

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There is a great book called Chaos and Fractals that has simple example code at the end of each chapter that implements some fractal or other example  A long time ago when I read that book  I converted each sample program  in some Basic dialect  into a Java applet that runs on a web page  The applets are here  http   hewgill com chaos-and-fractals   One of the samples is a simple Mandelbrot implementation

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There is a great book called Chaos and Fractals that has simple example code at the end of each chapter that implements some fractal or other example  A long time ago when I read that book  I converted each sample program  in some Basic dialect  into a Java applet that runs on a web page  The applets are here  http   hewgill com chaos-and-fractals   One of the samples is a simple Mandelbrot implementation

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Programming the Mandelbrot is easy  My quick-n-dirty code is below  not guaranteed to be bug-free  but a good outline    Here s the outline  The Mandelbrot-set lies in the Complex-grid completely within a circle with radius 2   So  start by scanning every point in that rectangular area  Each point represents a Complex number  x   yi   Iterate that complex number    new value     old-value  2    original-value  while keeping track of two things   1   the number of iterations  2   the distance of  new-value  from the origin   If you reach the Maximum number of iterations  you re done  If the distance from the origin is greater than 2  you re done   When done  color the original pixel depending on the number of iterations you ve done  Then move on to the next pixel   public void MBrot         float epsilon   0 0001     The step size across the X and Y axis     float x      float y      int maxIterations   10     increasing this will give you a more detailed fractal     int maxColors   256     Change as appropriate for your display       Complex Z      Complex C      int iterations      for x -2  x lt  2  x   epsilon                for y -2  y lt  2  y   epsilon                        iterations   0              C   new Complex x  y               Z   new Complex 0 0               while Complex Abs Z   lt  2  amp  amp  iterations  lt  maxIterations                                Z   Z Z   C                  iterations                              Screen Plot x y  iterations   maxColors     depending on the number of iterations  color a pixel                      Some details left out are   1   Learn exactly what the Square of a Complex number is and how to calculate it   2   Figure out how to translate the  -2 2  rectangular region to screen coordinates

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People above are using finding midpoints for sierpinski and Koch  I d much more recommend copying shapes  scaling them  and then translating them to achieve the  fractal  effect  Pseudo-code in Java for sierpinski would look something like this   public ShapeObject transform ShapeObject originalCurve                Make a copy of the original curve         Scale x and y to half of the original         make a copy of the copied shape  and translate it to the right so it touches the first copied shape         make a third shape that is a copy of the first copy  and translate it halfway between the first and second shape and translate it up         Group the 3 new shapes into one         return the new shape

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I think you might not see fractals as an algorithm or something to program  Fractals is a concept  It is a mathematical concept of detailed pattern repeating itself   Therefore you can create a fractal in many ways  using different approaches  as shown in the image below     Choose an approach and then investigate how to implement it  These four examples were implemented using Marvin Framework  The source codes are available here

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The Sierpinski triangle and the Koch curve are special types of flame fractals  Flame fractals are a very generalized type of Iterated function system  since it uses non-linear functions    An algorithm for IFS es are as follows   Start with a random point   Repeat the following many times  a million at least  depending on final image size     Apply one of N predefined transformations  matrix transformations or similar  to the point  An example would be that multiply each coordinate with 0 5  Plot the new point on the screen    If the point is outside the screen  choose randomly a new one inside the screen instead   If you want nice colors  let the color depend on the last used transformation

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Here s a simple and easy to understand code in Java for mandelbrot and other fractal examples  http   code google com p gaima wiki VLFImages  Just download the BuildFractal jar to test it in Java and run with command   java -Xmx1500M -jar BuildFractal jar 1000 1000 default MANDELBROT  The source code is also free to download explore edit expand

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Sometimes I program fractals for fun and as a challenge  You can find them here  The code is written in Javascript using the P5 js library and can be read directly from the HTML source code   For those I have seen the algorithms are quite simple  just find the core element and then repeat it over and over  I do it with recursive functions  but can be done differently

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The mandelbrot set is generated by repeatedly evaluating a function until it overflows  some defined limit   then checking how long it took you to overflow    Pseudocode   MAX COUNT   64    if we haven t escaped to infinity after 64 iterations                     then we re inside the mandelbrot set     foreach  x-pixel    foreach  y-pixel      calculate x y as mathematical coordinates from your pixel coordinates     value    x  y      count   0     while value absolutevalue  lt  1 billion and count  lt  MAX COUNT         value   value   value    x  y          count   count   1         the following should really be one statement  but I split it for clarity     if count    MAX COUNT          pixel at  x-pixel  y-pixel    BLACK     else          pixel at  x-pixel  y-pixel    colors count     some color map     Notes   value is a complex number  a complex number  a bi  is squared to give  aa-b b 2 abi   You ll have to use a complex type  or include that calculation in your loop

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Here s a simple and easy to understand code in Java for mandelbrot and other fractal examples  http   code google com p gaima wiki VLFImages  Just download the BuildFractal jar to test it in Java and run with command   java -Xmx1500M -jar BuildFractal jar 1000 1000 default MANDELBROT  The source code is also free to download explore edit expand

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Well  simple and graphically appealing don t really go hand in hand  If you re serious about programming fractals  I suggest reading up on iterated function systems and the advances that have been made in rendering them   http   flam3 com flame draves pdf

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I would start with something simple  like a Koch Snowflake   It s a simple process of taking a line and transforming it  then repeating the process recursively until it looks neat-o   Something super simple like taking 2 points  a line  and adding a 3rd point  making a corner   then repeating on each new section that s created   fractal p0  p1       Pmid   midpoint p0 p1    moved some distance perpendicular to p0 or p1      fractal p0 Pmid       fractal Pmid  p1

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Another excellent fractal to learn is the Sierpinski Triangle Fractal   Basically  draw three corners of a triangle  an equilateral is preferred  but any triangle will work   then start a point P at one of those corners   Move P halfway to any of the 3 corners at random  and draw a point there   Again move P halfway towards any random corner  draw  and repeat   You d think the random motion would create a random result  but it really doesn t   Reference  http   en wikipedia org wiki Sierpinski triangle

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I think you might not see fractals as an algorithm or something to program  Fractals is a concept  It is a mathematical concept of detailed pattern repeating itself   Therefore you can create a fractal in many ways  using different approaches  as shown in the image below     Choose an approach and then investigate how to implement it  These four examples were implemented using Marvin Framework  The source codes are available here

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Another excellent fractal to learn is the Sierpinski Triangle Fractal   Basically  draw three corners of a triangle  an equilateral is preferred  but any triangle will work   then start a point P at one of those corners   Move P halfway to any of the 3 corners at random  and draw a point there   Again move P halfway towards any random corner  draw  and repeat   You d think the random motion would create a random result  but it really doesn t   Reference  http   en wikipedia org wiki Sierpinski triangle

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Another excellent fractal to learn is the Sierpinski Triangle Fractal   Basically  draw three corners of a triangle  an equilateral is preferred  but any triangle will work   then start a point P at one of those corners   Move P halfway to any of the 3 corners at random  and draw a point there   Again move P halfway towards any random corner  draw  and repeat   You d think the random motion would create a random result  but it really doesn t   Reference  http   en wikipedia org wiki Sierpinski triangle

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You should indeed start with the Mandelbrot set  and understand what it really is   The idea behind it is relatively simple  You start with a function of complex variable     f z    z2   C   where z is a complex variable and C is a complex constant  Now you iterate it starting from z   0  i e  you compute z1   f 0   z2   f z1   z3   f z2  and so on  The set of those constants C for which the sequence z1  z2  z3      is bounded  i e  it does not go to infinity  is the Mandelbrot set  the black set in the figure on the Wikipedia page    In practice  to draw the Mandelbrot set you should    Choose a rectangle in the complex plane  say  from point -2-2i to point 2 2i   Cover the rectangle with a suitable rectangular grid of points  say  400x400 points   which will be mapped to pixels on your monitor  For each point pixel  let C be that point  compute  say  20 terms of the corresponding iterated sequence z1  z2  z3      and check whether it  goes to infinity   In practice you can check  while iterating   if the absolute value of one of the 20 terms is greater than 2  if one of the terms does  the subsequent terms are guaranteed to be unbounded   If some z k does  the sequence  goes to infinity   otherwise  you can consider it as bounded  If the sequence corresponding to a certain point C is bounded  draw the corresponding pixel on the picture in black  for it belongs to the Mandelbrot set   Otherwise  draw it in another color  If you want to have fun and produce pretty plots  draw it in different colors depending on the magnitude of abs 20th term     The astounding fact about fractals is how we can obtain a tremendously complex set  in particular  the frontier of the Mandelbrot set  from easy and apparently innocuous requirements   Enjoy

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You should indeed start with the Mandelbrot set  and understand what it really is   The idea behind it is relatively simple  You start with a function of complex variable     f z    z2   C   where z is a complex variable and C is a complex constant  Now you iterate it starting from z   0  i e  you compute z1   f 0   z2   f z1   z3   f z2  and so on  The set of those constants C for which the sequence z1  z2  z3      is bounded  i e  it does not go to infinity  is the Mandelbrot set  the black set in the figure on the Wikipedia page    In practice  to draw the Mandelbrot set you should    Choose a rectangle in the complex plane  say  from point -2-2i to point 2 2i   Cover the rectangle with a suitable rectangular grid of points  say  400x400 points   which will be mapped to pixels on your monitor  For each point pixel  let C be that point  compute  say  20 terms of the corresponding iterated sequence z1  z2  z3      and check whether it  goes to infinity   In practice you can check  while iterating   if the absolute value of one of the 20 terms is greater than 2  if one of the terms does  the subsequent terms are guaranteed to be unbounded   If some z k does  the sequence  goes to infinity   otherwise  you can consider it as bounded  If the sequence corresponding to a certain point C is bounded  draw the corresponding pixel on the picture in black  for it belongs to the Mandelbrot set   Otherwise  draw it in another color  If you want to have fun and produce pretty plots  draw it in different colors depending on the magnitude of abs 20th term     The astounding fact about fractals is how we can obtain a tremendously complex set  in particular  the frontier of the Mandelbrot set  from easy and apparently innocuous requirements   Enjoy

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I would start with something simple  like a Koch Snowflake   It s a simple process of taking a line and transforming it  then repeating the process recursively until it looks neat-o   Something super simple like taking 2 points  a line  and adding a 3rd point  making a corner   then repeating on each new section that s created   fractal p0  p1       Pmid   midpoint p0 p1    moved some distance perpendicular to p0 or p1      fractal p0 Pmid       fractal Pmid  p1

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If complex numbers give you a headache  there is a broad range of fractals that can be formulated using an L-system  This requires a couple of layers interacting  but each is interesting in it own right   First you need a turtle  Forward  Back  Left  Right  Pen-up  Pen-down  There are lots of fun shapes to be made with turtle graphics using turtle geometry even without an L-system driving it  Search for  LOGO graphics  or  Turtle graphics   A full LOGO system is in fact a Lisp programming environment using an unparenthesized Cambridge Polish syntax  But you don t have to go nearly that far to get some pretty pictures using the turtle concept   Then you need a layer to execute an L-system  L-systems are related to Post-systems and Semi-Thue systems  and like virii  they straddle the border of Turing Completeness  The concept is string-rewriting  It can be implemented as a macro-expansion or a procedure set with extra controls to bound the recursion  If using macro-expansion  as in the example below   you will still need a procedure set to map symbols to turtle commands and a procedure to iterate through the string or array to run the encoded turtle program  For a bounded-recursion procedure set  eg    you embed the turtle commands in the procedures and either add recursion-level checks to each procedure or factor it out to a handler function   Here s an example of a Pythagoras  Tree in postscript using macro-expansion and a very abbreviated set of turtle commands  For some examples in python and mathematica  see my code golf challenge

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Here is a codepen that I wrote for the Mandelbrot fractal using plain javascript and HTML   Hopefully it is easy to understand the code   The most complicated part is scale and translate the coordinate systems   Also complicated is making the rainbow palette   function mandel x y      var a 0  var b 0    for  i   0  i lt 250    i           Complex z   z 2   c     var t   a a - b b      b   2 a b      a   t      a   a   x      b   b   y      var m   a a   b b      if  m  gt  10   return i        return 250

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People above are using finding midpoints for sierpinski and Koch  I d much more recommend copying shapes  scaling them  and then translating them to achieve the  fractal  effect  Pseudo-code in Java for sierpinski would look something like this   public ShapeObject transform ShapeObject originalCurve                Make a copy of the original curve         Scale x and y to half of the original         make a copy of the copied shape  and translate it to the right so it touches the first copied shape         make a third shape that is a copy of the first copy  and translate it halfway between the first and second shape and translate it up         Group the 3 new shapes into one         return the new shape

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Another excellent fractal to learn is the Sierpinski Triangle Fractal   Basically  draw three corners of a triangle  an equilateral is preferred  but any triangle will work   then start a point P at one of those corners   Move P halfway to any of the 3 corners at random  and draw a point there   Again move P halfway towards any random corner  draw  and repeat   You d think the random motion would create a random result  but it really doesn t   Reference  http   en wikipedia org wiki Sierpinski triangle

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The mandelbrot set is generated by repeatedly evaluating a function until it overflows  some defined limit   then checking how long it took you to overflow    Pseudocode   MAX COUNT   64    if we haven t escaped to infinity after 64 iterations                     then we re inside the mandelbrot set     foreach  x-pixel    foreach  y-pixel      calculate x y as mathematical coordinates from your pixel coordinates     value    x  y      count   0     while value absolutevalue  lt  1 billion and count  lt  MAX COUNT         value   value   value    x  y          count   count   1         the following should really be one statement  but I split it for clarity     if count    MAX COUNT          pixel at  x-pixel  y-pixel    BLACK     else          pixel at  x-pixel  y-pixel    colors count     some color map     Notes   value is a complex number  a complex number  a bi  is squared to give  aa-b b 2 abi   You ll have to use a complex type  or include that calculation in your loop

[theory] How to program a fractal?

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