in reply to ulam's spiral too slow
After seeing the difference in outputs between the OP code and liverpole's version, and then actually looking up the references at wikipedia, I realized that there were other problems besides execution speed: way too many spots were being drawn in black, which meant way too many numbers were being categorized as primes I felt compelled to figure out who was wrong; I have updated my earlier reply accordingly.
I also noticed that the OP code actually draws all the points (from 1 to $max), both the supposed primes and the supposed nonprimes. But you really only need to draw one set, and just leave the other set as "background color".
Then, I thought it would be helpful to get a sense of what took longest  computing or drawing  so I decided to refactor the code as follows:
 run through all the desired numeric range before doing any drawning, keeping track of the x/y coords as we go;
 each time we find a prime number, store that as both a hash key and as an item in an ascending array; use the hash value to store the x/y coords where the dot should be drawn for that prime;
 once all the primes have been found, set up the Tk::Canvas, then run through the sorted list of prime values (which are hash keys), and draw a dot for each one.
As it turns out, the computing takes about twice as long as the drawing (4 sec vs. 2 sec for a max number of 150,000, which pretty well fills the given grid); but bear in mind the compute step loops over all integers in the given range, while the drawing step only loops over the primes.
And luckily, the resulting canvas looks a lot more like the pictures I saw over at wikipedia (it still might not be entirely correct  but you could try dumping a portion of the %primecoord hash structure to make sure the values are as intended).
#!/usr/local/bin/perl
use strict;
use Tk;
die "Usage: $0 max_number\n" # use the commandline, Luke!
if ( @ARGV != 1 or $ARGV[0] !~ /^\d+/ );
my $lastnumber = shift;
my ( $mincoord, $midcoord, $maxcoord ) = ( 50, 250, 450 );
plot_grid( compute_primes( $lastnumber ));
MainLoop;
{ # closure for building prime number hash structure
my @sorted_primes = ( 1 ); # primes in ascending order
my %primecoord = ( 1 => [$midcoord, $midcoord] );
# HoA: keys are primes, values are x/y coords in grid
sub ck_prime {
my ( $num, $x, $y ) = @_;
my $end = int( sqrt( $num ));
my $keep = 1;
for my $prime ( @sorted_primes ) {
if ( 0 == $num % $prime ) {
$keep = 0;
last;
}
elsif ( $prime > $end ) {
last;
}
}
if ( $keep ) {
$primecoord{$num} = [ $x, $y ];
push @sorted_primes, $num;
}
}
sub compute_primes
{
# list of directions for incrementing x, y coordinates:
my %vector = ( 0 => [ 1, 0 ], # rightward
1 => [ 0, 1 ], # downward
2 => [ 1, 0 ], # leftward
3 => [ 0, 1 ], # upward
);
my $x = my $y = 250; # starting position
my $direction = 0; # changes when we've gone $pathlen steps
my $traveled = 0; # no. of points drawn in current direction
my $pathlen = 1; # increments on every second direction chang
+e
my $changes = 0; # no. of dir. changes since last $pathlen in
+crement
my $number = 1;
my $bgn = time;
while ( ++$number <= $lastnumber )
{
$x += $vector{$direction}[0];
$y += $vector{$direction}[1];
last if ( $x < $mincoord or $x > $maxcoord );
if ( ++$traveled == $pathlen ) {
$direction = ++$direction % 4;
$traveled = 0;
if ( ++$changes % 2 == 0 ) {
$changes = 0;
$pathlen++;
}
}
ck_prime( $number, $x, $y );
}
my $end = time;
warn "Checked $lastnumber integers in ", $end  $bgn, " sec\n"
+;
return( \@sorted_primes, \%primecoord );
}
}
sub plot_grid
{
my ( $primes, $coords ) = @_;
my $mw = MainWindow>new;
my $c = $mw>Canvas(width => $maxcoord+50, height => $maxcoord+5
+0,
background => 'white' )>pack;
$c>createLine( $mincoord, $midcoord, $maxcoord, $midcoord );
$c>createText( $mincoord30, $midcoord, fill => 'blue', text =>
+ 'X');
$c>createLine( $midcoord, $mincoord, $midcoord, $maxcoord );
$c>createText( $midcoord, $mincoord30, fill => 'blue', text =>
+ 'Y');
my $bgn = time;
for my $prime ( @$primes ) {
my ( $x, $y ) = @{$$coords{$prime}};
$c>createText( $x, $y, fill => 'black', text => "\xb7" );
}
my $end = time;
warn "Plotted ", scalar @$primes, " primes in ", $end  $bgn, " se
+c\n";
}
Note that I used "\xb7" (centered dot) as the character  seemed like "." (period) was offcenter.
(updated to correct the comments about the "%vector" hash, to reflect the fact that coords in Tk::Canvas put 0,0 at the upper left corner)
