Here's my perl solution, which is more or less the translation of the J solution:

#!perl

use warnings;
use strict;
use List::Util "max";
use Math::Complex; # what for?

my $n = int($ARGV[-1] || 5);
my($i, $j, $k, $x, $y, @l, @a, @v, @r, $wd, $fmt);
for $i (0 .. $n - 1) {
        $x = $i - ($n - 1)/2;
        for $j (0 .. $n - 1) {
                $y = $j - ($n - 1)/2;
                $k = $i * $n + $j;
                $l[$k] = -max(abs($x), abs($y));
                $a[$k] = -arg(cplx(1, -1) * cplx($x, $y));
        }
}
@v = sort { $l[$a] <=> $l[$b] || $a[$a] <=> $a[$b] } 0 .. $n**2 - 1;
@r[@v] = 1 .. $n**2;
$wd = length($n**2);
$fmt = join(" ", ("%${wd}d") x $n) . "\n";
for $i (0 .. $n - 1) {
        printf $fmt, @r[$i * $n .. $i * $n + $n - 1];
}

__END__
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