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<title>Make your own 'prime factorization' diagram | John Graham-Cumming</title>

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<h2 class="date-header" class="headerTxt4">Monday, April 30, 2012</h2>

Make your own 'prime factorization' diagram<img src="http://blog.jgc.org/cdn-cgi/cl/197ea504ddebf89f3a5e6bfcb64899d380b1abe4.gif"/>

The<a href="http://sonderbooks.com/blog/?p=843"> Prime Factorization Sweater </a>is a lovely idea and I thought it would be fun to reproduce the same idea electronically so that I could print out a poster version for home. <br/>

Enter <a href="http://processing.org/">Processing</a>. <br/>

With it I've developed a small program that produces a diagram of the first 100 numbers and for each number there's a circle broken up into arcs. &nbsp;Each arc is a prime factor. &nbsp;As in the original sweater each factor gets a unique color (assigning unique colors is rather complex and I ended up using the <a href="http://en.wikipedia.org/wiki/Color_difference#CMC_l:c_.281984.29">color difference method based on CMC l:c</a>&nbsp;and a nice online tool that <a href="http://phrogz.net/css/distinct-colors.html">does the work</a> for you).<br/>

Here's the finished product. &nbsp;The top left corner is the number 1 and the numbers read right to left. &nbsp;So the first red circle is a prime number (2), the second the next number (3, which is prime) and so on.<br/>

There's also an option to print the numbers involved.<br/>

And here's the source code (it's yours under GPLv2):</div>

<pre class="prettyprint" style="font-size: 11pt; line-height: 11pt;">// Prime factorization diagram

//

// Copyright (c) 2012 John Graham-Cumming

//

// Draws a picture of the numbers 1 to 100 as a square.  Each number

// is represented by a circle broken up into arcs of equal size.  Each

// arc represents a prime factor of the number such that all the arcs

// together form the original number.  Each prime is given a unique

// color.

//

// Inspired by the prime factorization sweater:

// http://sonderbooks.com/blog/?p=843

 

// 25 distinct RGB colors generated using the generating service

// on http://phrogz.net/css/distinct-colors.html.  This tries to

// find visually distinct colors that humans will be able to

// distinguish.

 

int[] colors = { #ff0000, #e5b073, #73ff40, #23698c, #312040, #a62929,

  #fff240, #00cc88, #80b3ff, #cc00a3, #e6b4ac, #5c6633, #6cd9d2, #00004d,

  #ff0066, #cc5200, #e1ffbf, #1a3331, #1f00e6, #b3597d, #593000, #338000,

  #00ccff, #986cd9, #401016 };

 

// This is the size of the square on which the prime factorization is

// drawn.  This can be made much larger than the screen for saving a large

// file using saveFrame()

 

int side = 1024;

 

// When set to true the factors are printed inside the circles, otherwise

// just colors are used

 

boolean number = false;

 

// I do everything in setup() because there's no animation and therefore

// no reason to have code in draw()

 

void setup() {

    size(side, side);

    background(255);

    stroke(255);

    strokeWeight(2);

    smooth();

    textAlign(CENTER, CENTER);

 

    int[] primes = new int[100];

    int space = side/10;

    int used = 0;

 

    for (int c = 0; c &lt; 10; c++ ) {

        for (int r = 0; r &lt; 10; r++ ) {

          int n = r+1 + c*10;

          if ( n != 1 ) {

           

            // Factorize the number into its prime factors and determine

            // if it was itself prime.  If so, then assign the next color

            // available to this number for future use.

           

            int[] f = factor(n);

           

            if ( f.length == 1 ) {

              primes[n] = used;

              used += 1;              

            }

           

            float start_angle = PI / 4;

            float angle = start_angle;

            float per_factor = (2 * PI) / f.length;

            float x = space*(r+0.5);

            float y = space*(c+0.5);

           

            // First draw the arcs themselves with the right colors

            // based on the prime factor each represents.  Then draw

            // white lines across the circle that break up the arcs

            // and (if necessary) add the number itself.

           

            for ( int i = 0; i &lt; f.length; i++ ) {

              fill(colors[primes[f[i]]]);

              arc(x, y, space, space, angle, angle+per_factor);

              angle += per_factor;

            }

           

            if ( f.length &gt; 1 ) {

              angle = start_angle;

              for ( int i = 0; i &lt; f.length; i++ ) {

                line( x, y, x + space/2 * cos(angle), y + space/2.0 * sin(angle) );

                line( x, y, x + space/2 * cos(angle+per_factor),

                  y + space/2.0 * sin(angle+per_factor) );

               

                if ( number ) {

                  opposite(colors[primes[f[i]]]);

                  text(f[i], x + space/4*cos(angle+per_factor/2),

                    y + space/4*sin(angle+per_factor/2));

                }

               

                angle += per_factor;

              }

            } else {

              if ( number ) {

                opposite(colors[primes[n]]);

                text(n, x, y);

              }

            }

          } else {

            if ( number ) {

              opposite(#FFFFFF);

              text(1, space/2, space/2);

            }

          }

        }

    }

   

    saveFrame();

}

 

// factor: return an ordered list (smallest to largest) of the

// prime factors of an integer.

int[] factor(int n) {

  int factors[] = new int[0];

 

  int d = 2;

 

  while (n &gt; 1) {

    while (n % d == 0) {

      factors = append(factors, d);

      n /= d;

    }

    d += 1;

  }

   

  return factors;

}

 

// opposite: set the current fill color to the 'opposite' of

// the passed in color so that it can be seen on top of it. It does this

// by calculating the Euclidean distance of the color from white (255, 255, 255)

// and having a cut off the determines whether to display text in white or black.

void opposite(int c) {

  float distance = sqrt(red(c)*red(c)+green(c)*green(c)+blue(c)*blue(c));

 

  if ( distance &lt; 250 ) {

    fill(255);

  } else {

    fill(0);

  }

}

<p><i>If you enjoyed this blog post, you might enjoy my travel book for people interested in science and technology: <a href="http://amzn.to/cNCL33">The Geek Atlas</a>. Signed copies of The Geek Atlas are <a href="http://blog.jgc.org/2011/01/how-to-buy-signed-copies-of-geek-atlas.html">available</a>. Looking for a new job? Try <a href="http://jobs.usethesource.com/">UseTheSource</a>.</i></p>

<em>posted by John Graham-Cumming at <a href="http://blog.jgc.org/2012/04/make-your-own-prime-factorization.html" title="permanent link">13:36</a> <a href="http://blog.jgc.org/2012/04/make-your-own-prime-factorization.html" title="permanent link">Permalink</a> </em>

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