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#!/usr/bin/env perl# -*- cperl -*-

BEGIN { $^W = 1 }

use strict;use diagnostics;use warnings;

use Carp;use English;use Math::Trig;use GD;use constant PI => 4*atan(1);

# June 30, 2007# Allan Peda## Ulam prime number spiral# http://mathworld.wolfram.com/PrimeSpiral.html# http://en.wikipedia.org/wiki/Ulam_spiral#

# we iterate over a grid, but we want to follow a spiral.# Direction is cartesian based x being horizontal motion,# and y being vertical. Up and right are postive motion.

unless(defined($ARGV[0])){die "Please provide a png file to save to.";

}

my $pngfile = 'ulam_spiral.png';if($ARGV[0] =~ m/\.png$/i){

$pngfile = $ARGV[0];} else {

print "Supplied file not suffixed with .png defaulting to \"$pngfile\".\n";

}

# how many turns to make in the spiralmy $num_corners = 1600;my @turning_points = mkturns($num_corners);# which directions do we followmy $max = $turning_points[$#turning_points];print "Generating map for first $max primes.\n";my ($x_width, $y_width) = (0,0);$x_width = $y_width = int(sqrt($max+0.5));

print "Dimensions are: $x_width x $y_width\n";my %primes = %{genprimes($max)};

my $im = new GD::Image($x_width, $y_width);my $white = $im->colorAllocate(255,255,255);my $black = $im->colorAllocate(0,0,0);my $origin_color = $im->colorAllocate(255,0,0);my $position=$turning_points[0];my ($x, $y) = (0,0);my ($delx,$dely) = (1,0); # start out moving to the right

sub gdx { return( shift()+($x_width/2) ) }sub gdy { return( ($y_width/2)-shift()) }

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$im->setPixel(gdx($x),gdy($y),$origin_color);

foreach my $next_turn (@turning_points) {for (my $i=$position;$isetPixel(gdx($x),gdy($y),$black);

}}$position = $next_turn+1;

}

open(PNGFH, ">".$pngfile)

or die "Could not open image file \"$pngfile\".";print PNGFH $im->png;close PNGFH;

# an array of run lengths up to each turn# equation was guessed at after inspection of coordinates# from a hand drawn cartesian spiralsub mkturns{

my $ulim = shift;my $len = 0;my @rv;foreach my $i (1..$ulim) {

my $delta = ($i%2?$i+1:$i)/2;

$len = $len+$delta;push( @rv, $len);

}return @rv;

}

# generates prime numbers as hash keyssub genprimes {

my $max = shift;my $sieve = '';my %primes = ( 2 => 1 );

GUESS: for (my $guess = 2; $guess

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my %memo = ();sub nextDirection {

my ($x, $y) = (shift, shift);my $nextoffset = undef;if( defined($memo{$x}{$y})){

$nextoffset = $memo{$x}{$y};} else {

my $acos = acos($x);

my $asin = asin($y);if( $asin < 0){

$asin+=(PI*2);}my $offset = int(2*($acos>$asin?$acos:$asin)/PI);# hard coded count of four directions in spiral (right, up, left down)$nextoffset = $memo{$x}{$y} = ($offset+1)%4;

}return to_xy($nextoffset);

}}

# uses memoized trig function

{my %memo = ();sub to_xy {

defined($memo{$_[0]}) and return @{$memo{$_[0]}};$memo{$_[0]}= [int(cos($_[0]*PI/2)),int(sin($_[0]*PI/2))];return @{$memo{$_[0]}};

}}