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    $dy = -$dy;
    ($dx,$dy) = ($dy - $realpart*$dx,
                 $dx + $realpart*$dy);
  }
  # print 3*$xmin/$len+.001," / 3\n";
  # print 6*$xmax/$len+.001," / 6\n";
  # print 3*$ymin/$len+.001," / 3\n";
  # print 3*$ymax/$len+.001," / 3\n";
  exit 0;

  sub to_bin {
    my ($n) = @_;
    return ($n < 0 ? '-' : '') . sprintf('%b', abs($n));
  }
}

{
  # min/max hypot for level
  $|=1;
  my $realpart = 2;
  my $norm = $realpart**2 + 1;
  my $path = Math::PlanePath::ComplexMinus->new (realpart => $realpart);
  my $prev_min = 1;
  my $prev_max = 1;
  for (my $level = 1; $level < 25; $level++) {
    my $n_start = $norm**($level-1);
    my $n_end = $norm**$level;

    my $min_hypot = POSIX::DBL_MAX();
    my $min_x = 0;
    my $min_y = 0;
    my $min_pos = '';

    my $max_hypot = 0;
    my $max_x = 0;
    my $max_y = 0;
    my $max_pos = '';

    print "level $level  n=$n_start .. $n_end\n";

    foreach my $n ($n_start .. $n_end) {
      my ($x,$y) = $path->n_to_xy($n);
      my $h = $x*$x + $y*$y;

      if ($h < $min_hypot) {
        $min_hypot = $h;
        $min_pos = "$x,$y";
      }
      if ($h > $max_hypot) {
        $max_hypot = $h;
        $max_pos = "$x,$y";
      }
    }
    # print "$min_hypot,";

    # print "  min $min_hypot   at $min_x,$min_y\n";
    # print "  max $max_hypot   at $max_x,$max_y\n";
    {
      my $factor = $min_hypot / $prev_min;
      print "  min r^2 $min_hypot 0b".sprintf('%b',$min_hypot)."   at $min_pos  factor $factor\n";
      print "  cf formula ", 2**($level-7), "\n";
    }
    # {
    #   my $factor = $max_hypot / $prev_max;
    #   print "  max r^2 $max_hypot 0b".sprintf('%b',$max_hypot)."   at $max_pos  factor $factor\n";
    # }
    $prev_min = $min_hypot;
    $prev_max = $max_hypot;
  }
  exit 0;
}

{
  # covered inner rect
  # depends on which coord extended first
  require Math::BaseCnv;
  $|=1;
  my $realpart = 1;
  my $norm = $realpart**2 + 1;
  my $path = Math::PlanePath::ComplexMinus->new (realpart => $realpart);
  my %seen;
  my $xmin = 0;
  my $xmax = 0;
  my $ymin = 0;
  my $ymax = 0;
  for (my $level = 1; $level < 25; $level++) {
    my $n_start = $norm**($level-1);
    my $n_end = $norm**$level - 1;

    foreach my $n ($n_start .. $n_end) {
      my ($x,$y) = $path->n_to_xy($n);
      $seen{"$x,$y"} = 1;
      $xmin = min ($xmin, $x);
      $xmax = max ($xmax, $x);
      $ymin = min ($ymin, $y);
      $ymax = max ($ymax, $y);
    }
    my $x1 = 0;
    my $y1 = 0;
    my $x2 = 0;
    my $y2 = 0;
    for (;;) {
      my $more = 0;
      {
        my $x = $x1-1;
        my $good = 1;
        foreach my $y ($y1 .. $y2) {
          if (! $seen{"$x,$y"}) {
            $good = 0;
            last;
          }
        }
        if ($good) {
          $more = 1;
          $x1 = $x;
        }
      }
      {
        my $x = $x2+1;
        my $good = 1;
        foreach my $y ($y1 .. $y2) {