Slight variations on what other have done. This dies if your polygon is not closed. 
#!/usr/bin/perl
use strict;
my @seg = (
    [ 1, 1, 2, 1 ],
    [ 5, 5, 1, 5 ],
    [ 2, 3, 5, 3 ],
    [ 5, 3, 5, 5 ],
    [ 2, 1, 2, 3 ],
    [ 1, 5, 1, 1 ]);

my %starts=map{"@$_[0,1]",$_} @seg;
my @sorted_segs = my $cur = $seg[0];
while (1) {
  my $next = $starts{"@$cur[2,3]"};
  die "your polygon is broken\n" if !$next;
  $next==$seg[0] ? last : push @sorted_segs,$next;
  $cur = $next;
}

print "@$_\n" for @sorted_segs;

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In reply to Re: reordering segments to form a polygon by fletcher_the_dog
in thread reordering segments to form a polygon by dada

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