libmath-planepath-perl 117-1 / usr / share / perl5 / Math / PlanePath / SquareSpiral.pm

# Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde

# This file is part of Math-PlanePath.
#
# Math-PlanePath is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 3, or (at your option) any later
# version.
#
# Math-PlanePath is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.


# http://d4maths.lowtech.org/mirage/ulam.htm
# http://d4maths.lowtech.org/mirage/img/ulam.gif
#     sample gif of primes made by APL or something
#
# http://www.sciencenews.org/view/generic/id/2696/title/Prime_Spirals
#     Ulam's spiral of primes
#
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/primeSpiral.htm
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/triangleUlam.htm
#     Pulchritudinous Primes of Ulam spiral.

# http://mathworld.wolfram.com/PrimeSpiral.html
#
# Mark C. Chu-Carroll "The Surprises Never Eend: The Ulam Spiral of Primes"
# http://scienceblogs.com/goodmath/2010/06/the_surprises_never_eend_the_u.php
#
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/index.html
# including image highlighting the lines

# S. M. Ellerstein, The square spiral, J. Recreational
# Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
#
# Stein, M. and Ulam, S. M. "An Observation on the
# Distribution of Primes." Amer. Math. Monthly 74, 43-44,
# 1967.
#
# Stein, M. L.; Ulam, S. M.; and Wells, M. B. "A Visual
# Display of Some Properties of the Distribution of Primes."
# Amer. Math. Monthly 71, 516-520, 1964.

# cf sides alternately prime and fibonacci
# A160790 corner N
# A160791 side lengths, alternately integer and triangular adding that integer
# A160792 corner N
# A160793 side lengths, alternately integer and sum primes
# A160794 corner N
# A160795 side lengths, alternately primes and fibonaccis


package Math::PlanePath::SquareSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;

use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
use Math::PlanePath::Base::NSEW;
@ISA = ('Math::PlanePath::Base::NSEW',
        'Math::PlanePath');

use Math::PlanePath::Base::Generic
  'round_nearest';

# uncomment this to run the ### lines
#use Smart::Comments '###';


# Note: this shared by other paths
use constant parameter_info_array =>
  [
   { name        => 'wider',
     display     => 'Wider',
     type        => 'integer',
     minimum     => 0,
     default     => 0,
     width       => 3,
     description => 'Wider path.',
   },
   Math::PlanePath::Base::Generic::parameter_info_nstart1(),
  ];

use constant xy_is_visited => 1;

#    2w+4 -- 2w+3 ----- w+2
#      |                 |
#    2w+5      0------- w+1
#      |     
#    2w+6 ---
#                  ^
#                 X=0
#
sub x_negative_at_n {
  my ($self) = @_;
  return $self->n_start + ($self->{'wider'} ? 0 : 4);
}
sub y_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 2*$self->{'wider'} + 6;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
  my ($self) = @_;
  return $self->n_start + 2*$self->{'wider'} + 4;
}


#------------------------------------------------------------------------------

sub new {
  my $self = shift->SUPER::new (@_);

  # parameters
  $self->{'wider'} ||= 0;  # default
  if (! defined $self->{'n_start'}) {
    $self->{'n_start'} = $self->default_n_start;
  }

  return $self;
}

# wider==0
# base from bottom-right corner
#   d = [ 1,  2,  3,  4 ]
#   N = [ 2, 10, 26, 50 ]
#   N = (4 d^2 - 4 d + 2)
#   d = 1/2 + sqrt(1/4 * $n + -4/16)
#
# wider==1
# base from bottom-right corner
#   d = [ 1,  2,  3,  4 ]
#   N = [ 3, 13, 31, 57 ]
#   N = (4 d^2 - 2 d + 1)
#   d = 1/4 + sqrt(1/4 * $n + -3/16)
#
# wider==2
# base from bottom-right corner
#   d = [ 1,  2,  3, 4 ]
#   N = [ 4, 16, 36, 64 ]
#   N = (4 d^2)
#   d = 0 + sqrt(1/4 * $n + 0)
#
# wider==3
# base from bottom-right corner
#   d = [ 1,  2,  3 ]
#   N = [ 5, 19, 41 ]
#   N = (4 d^2 + 2 d - 1)
#   d = -1/4 + sqrt(1/4 * $n + 5/16)
#
# N = 4*d^2 + (-4+2*w)*d + (2-w)
#   = 4*$d*$d + (-4+2*$w)*$d + (2-$w)
# d = 1/2-w/4 + sqrt(1/4*$n + b^2-4ac)
# (b^2-4ac)/(2a)^2 = [ (2w-4)^2 - 4*4*(2-w) ] / 64
#                  = [ 4w^2 - 16w + 16 - 32 + 16w ] / 64
#                  = [ 4w^2 - 16 ] / 64
#                  = [ w^2 - 4 ] / 16
# d = 1/2-w/4 + sqrt(1/4*$n + (w^2 - 4) / 16)
#   = 1/4 * (2-w + sqrt(4*$n + w^2 - 4))
#   = 0.25 * (2-$w + sqrt(4*$n + $w*$w - 4))
#
# then offset the base by +4*$d+$w-1 for top left corner for +/- remainder
# rem = $n - (4*$d*$d + (-4+2*$w)*$d + (2-$w) + 4*$d + $w - 1)
#     = $n - (4*$d*$d + (-4+2*$w)*$d + 2 - $w + 4*$d + $w - 1)
#     = $n - (4*$d*$d + (-4+2*$w)*$d + 1 - $w + 4*$d + $w)
#     = $n - (4*$d*$d + (-4+2*$w)*$d + 1 + 4*$d)
#     = $n - (4*$d*$d + (2*$w)*$d + 1)
#     = $n - ((4*$d + 2*$w)*$d + 1)
#

sub n_to_xy {
  my ($self, $n) = @_;
  #### SquareSpiral n_to_xy: $n

  $n = $n - $self->{'n_start'};  # starting $n==0, warn if $n==undef
  if ($n < 0) {
    #### before n_start ...
    return;
  }

  my $w = $self->{'wider'};
  my $w_right = int($w/2);
  my $w_left = $w - $w_right;
  if ($n <= $w+1) {
    #### centre horizontal
    # n=0 at w_left
    # x = $n - int(($w+1)/2)
    #   = $n - int(($w+1)/2)
    return ($n - $w_left,  # n=0 at w_left
            0);
  }

  my $d = int ((2-$w + sqrt(int(4*$n) + $w*$w)) / 4);
  #### d frac: ((2-$w + sqrt(int(4*$n) + $w*$w)) / 4)
  #### $d

  #### base: 4*$d*$d + (-4+2*$w)*$d + (2-$w)
  $n -= ((4*$d + 2*$w)*$d);
  #### remainder: $n

  if ($n >= 0) {
    if ($n <= 2*$d) {
      ### left vertical
      return (-$d - $w_left,
              -$n + $d);
    } else {
      ### bottom horizontal
      return ($n - $w_left - 3*$d,
              -$d);
    }
  } else {
    if ($n >= -2*$d-$w) {
      ### top horizontal
      return (-$n - $d - $w_left,
              $d);
    } else {
      ### right vertical
      return ($d + $w_right,
              $n + 3*$d + $w);
    }
  }
}

sub xy_to_n {
  my ($self, $x, $y) = @_;

  my $w = $self->{'wider'};
  my $w_right = int($w/2);
  my $w_left = $w - $w_right;
  $x = round_nearest ($x);
  $y = round_nearest ($y);
  ### xy_to_n: "x=$x, y=$y"
  ### $w_left
  ### $w_right

  my $d;
  if (($d = $x - $w_right) > abs($y)) {
    ### right vertical
    ### $d
    #
    # base bottom right per above
    ### BR: 4*$d*$d + (-4+2*$w)*$d + (2-$w)
    # then +$d-1 for the y=0 point
    # N_Y0  = 4*$d*$d + (-4+2*$w)*$d + (2-$w) + $d-1
    #       = 4*$d*$d + (-3+2*$w)*$d + (2-$w) + -1
    #       = 4*$d*$d + (-3+2*$w)*$d +  1-$w
    ### N_Y0: (4*$d + -3 + 2*$w)*$d + 1-$w
    #
    return (4*$d + -3 + 2*$w)*$d - $w + $y + $self->{'n_start'};
  }

  if (($d = -$x - $w_left) > abs($y)) {
    ### left vertical
    ### $d
    #
    # top left per above
    ### TL: 4*$d*$d + (2*$w)*$d + 1
    # then +$d for the y=0 point
    # N_Y0  = 4*$d*$d + (2*$w)*$d + 1 + $d
    #       = 4*$d*$d + (1 + 2*$w)*$d + 1
    ### N_Y0: (4*$d + 1 + 2*$w)*$d + 1
    #
    return (4*$d + 1 + 2*$w)*$d - $y + $self->{'n_start'};
  }

  $d = abs($y);
  if ($y > 0) {
    ### top horizontal
    ### $d
    #
    # top left per above
    ### TL: 4*$d*$d + (2*$w)*$d + 1
    # then -($d+$w_left) for the x=0 point
    # N_X0  = 4*$d*$d + (2*$w)*$d + 1 + -($d+$w_left)
    #       = 4*$d*$d + (-1 + 2*$w)*$d + 1 - $w_left
    ### N_Y0: (4*$d - 1 + 2*$w)*$d + 1 - $w_left
    #
    return (4*$d - 1 + 2*$w)*$d - $w_left - $x + $self->{'n_start'};
  }

  ### bottom horizontal, and centre y=0
  ### $d
  #
  # top left per above
  ### TL: 4*$d*$d + (2*$w)*$d + 1
  # then +2*$d to bottom left, +$d+$w_left for the x=0 point
  # N_X0  = 4*$d*$d + (2*$w)*$d + 1 + 2*$d + $d+$w_left)
  #       = 4*$d*$d + (3 + 2*$w)*$d + 1 + $w_left
  ### N_Y0: (4*$d + 3 + 2*$w)*$d + 1 + $w_left
  #
  return (4*$d + 3 + 2*$w)*$d + $w_left + $x + $self->{'n_start'};
}

# hi is exact but lo is not
# not exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;

  $x1 = round_nearest ($x1);
  $y1 = round_nearest ($y1);
  $x2 = round_nearest ($x2);
  $y2 = round_nearest ($y2);

  # ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
  return ($self->{'n_start'},
          max ($self->xy_to_n($x1,$y1),
               $self->xy_to_n($x2,$y1),
               $self->xy_to_n($x1,$y2),
               $self->xy_to_n($x2,$y2)));

  # my $w = $self->{'wider'};
  # my $w_right = int($w/2);
  # my $w_left = $w - $w_right;
  #
  # my $d = 1 + max (abs($y1),
  #                  abs($y2),
  #                  $x1 - $w_right, -$x1 - $w_left,
  #                  $x2 - $w_right, -$x2 - $w_left,
  #                  1);
  # ### $d
  # ### is: $d*$d
  #
  # # ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
  # return (1,
  #         (4*$d - 4 + 2*$w)*$d + 2);  # bottom-right
}


# [ 1, 2, 3,  4,  5 ],
# [ 1, 3, 7, 13, 21 ]
# N = (d^2 - d + 1)
#   = ($d**2 - $d + 1)
#   = (($d - 1)*$d + 1)
# d = 1/2 + sqrt(1 * $n + -3/4)
#   = (1 + sqrt(4*$n - 3)) / 2
#
# wider=3
# [ 2, 3,  4,  5 ],
# [ 6, 13, 22, 33 ]
# N = (d^2 + 2 d - 2)
#   = ($d**2 + 2*$d - 2)
#   = (($d + 2)*$d - 2)
# d = -1 + sqrt(1 * $n + 3)
#
# wider=5
# [ 2, 3,  4,  5 ],
# [ 8, 17, 28, 41 ]
# N = (d^2 + 4 d - 4)
#   = ($d**2 + 4*$d - 4)
#   = (($d + 4)*$d - 4)
# d = -2 + sqrt(1 * $n + 8)
#
# wider=7
# [ 2, 3,  4,  5 ],
# [ 10, 21, 34, 49 ]
# N = (d^2 + 6 d - 6)
#   = ($d**2 + 6*$d - 6)
#   = (($d + 6)*$d - 6)
# d = -3 + sqrt(1 * $n + 15)
#
#
# N = (d^2 + (w-1)*d + 1-w)
# d = (1-w)/2 + sqrt($n + (w^2 + 2w - 3)/4)
#   = (1-w + sqrt(4*$n + (w-3)(w+1))) / 2
#
# extra subtract d+w-1
# Nbase = (d^2 + (w-1)*d + 1-w) + d+w-1
#       = d^2 + w*d

sub n_to_dxdy {
  my ($self, $n) = @_;
  ### n_to_dxdy(): $n

  $n = $n - $self->{'n_start'};  # starting $n==0, warn if $n==undef
  if ($n < 0) {
    #### before n_start ...
    return;
  }

  my $w = $self->{'wider'};
  my $d = int((1-$w + sqrt(int(4*$n) + ($w+2)*$w+1)) / 2);

  my $int = int($n);
  $n -= $int;  # fraction 0 <= $n < 1
  $int -= ($d+$w)*$d-1;

  ### $d
  ### $w
  ### $n
  ### $int

  my ($dx, $dy);
  if ($int <= 0) {
    if ($int < 0) {
      ### horizontal ...
      $dx = 1;
      $dy = 0;
    } else {
      ### corner horiz to vert ...
      $dx = 1-$n;
      $dy = $n;
    }
  } else {
    if ($int < $d) {
      ### vertical ...
      $dx = 0;
      $dy = 1;
    } else {
      ### corner vert to horiz ...
      $dx = -$n;
      $dy = 1-$n;
    }
  }

  unless ($d % 2) {
    ### rotate +180 for even d ...
    $dx = -$dx;
    $dy = -$dy;
  }

  ### result: "$dx, $dy"
  return ($dx,$dy);
}



# old bit:
#
# wider==0
# base from two-way diagonal top-right and bottom-left
# s even for top-right diagonal doing top leftwards then left downwards
# s odd for bottom-left diagonal doing bottom rightwards then right pupwards
#   s = [ 0,  1,   2,   3,   4,   5,   6 ]
#   N = [ 1,  1,   3,   7,  13,  21,  31 ]
#         +0  +2  +4  +6  +8  +10
#            2   2   2   2   2
#
#   n = (($d - 1)*$d + 1)
#   s = 1/2 + sqrt(1 * $n + -3/4)
#     = .5 + sqrt ($n - .75)
#
#

#------------------------------------------------------------------------------

sub _NOTDOCUMENTED_n_to_figure_boundary {
  my ($self, $n) = @_;
  ### _NOTDOCUMENTED_n_to_figure_boundary(): $n

  # adjust to N=1 at origin X=0,Y=0
  $n = $n - $self->{'n_start'} + 1;

  if ($n < 1) {
    return undef;
  }

  my $wider = $self->{'wider'};
  if ($n <= $wider) {
    # single block row
    # +---+-----+----+
    # | 1 | ... | $n |  boundary = 2*N + 2
    # +---+-----+----+
    return 2*$n + 2;
  }

  my $d = int((sqrt(int(4*$n) + $wider*$wider - 2) - $wider) / 2);
  ### $d
  ### $wider
  ### cmp: $d*($d+1+$wider) + $wider + 1

  if ($n > $d*($d+1+$wider)) {
    $wider++;
    ### increment for +2 after turn ...
  }
  return 4*$d + 2*$wider + 2;
}

#------------------------------------------------------------------------------
1;
__END__


=for stopwords Stanislaw Ulam pronic PlanePath Ryde Math-PlanePath Ulam's Honaker's decagonal OEIS Nbase sqrt BigRat Nrem wl wr Nsig incrementing

=head1 NAME

Math::PlanePath::SquareSpiral -- integer points drawn around a square (or rectangle)

=head1 SYNOPSIS

 use Math::PlanePath::SquareSpiral;
 my $path = Math::PlanePath::SquareSpiral->new;
 my ($x, $y) = $path->n_to_xy (123);

=head1 DESCRIPTION

This path makes a square spiral,

=cut

# math-image --path=SquareSpiral --all --output=numbers_dash --size=40x16

=pod

    37--36--35--34--33--32--31              3
     |                       |
    38  17--16--15--14--13  30              2
     |   |               |   |
    39  18   5---4---3  12  29              1
     |   |   |       |   |   |
    40  19   6   1---2  11  28  ...    <- Y=0
     |   |   |           |   |   |
    41  20   7---8---9--10  27  52         -1
     |   |                   |   |
    42  21--22--23--24--25--26  51         -2
     |                           |
    43--44--45--46--47--48--49--50         -3

                 ^
    -3  -2  -1  X=0  1   2   3   4

See F<examples/square-numbers.pl> in the sources for a simple program
printing these numbers.

This path is well known from Stanislaw Ulam finding interesting straight
lines when plotting the prime numbers on it.  The cover of Scientific
American March 1964 featured this spiral,

=over

L<http://www.nature.com/scientificamerican/journal/v210/n3/covers/index.html>

L<http://oeis.org/A143861/a143861.jpg>

=back

See F<examples/ulam-spiral-xpm.pl> in the sources for a standalone program,
or see L<math-image> using this C<SquareSpiral> to draw this pattern and
more.

=head2 Straight Lines

X<Square numbers>The perfect squares 1,4,9,16,25 fall on two diagonals with
the even perfect squares going to the upper left and the odd squares to the
lower right.  The X<Pronic numbers>pronic numbers 2,6,12,20,30,42 etc k^2+k
half way between the squares fall on similar diagonals to the upper right
and lower left.  The decagonal numbers 10,27,52,85 etc 4*k^2-3*k go
horizontally to the right at Y=-1.

In general straight lines and diagonals are 4*k^2 + b*k + c.  b=0 is the
even perfect squares up to the left, then incrementing b is an eighth turn
anti-clockwise, or clockwise if negative.  So b=1 is horizontal West, b=2
diagonally down South-West, b=3 down South, etc.

Honaker's prime-generating polynomial 4*k^2 + 4*k + 59 goes down to the
right, after the first 30 or so values loop around a bit.

=head2 Wider

An optional C<wider> parameter makes the path wider, becoming a rectangle
spiral instead of a square.  For example

    $path = Math::PlanePath::SquareSpiral->new (wider => 3);

gives

    29--28--27--26--25--24--23--22        2
     |                           |
    30  11--10-- 9-- 8-- 7-- 6  21        1
     |   |                   |   |
    31  12   1-- 2-- 3-- 4-- 5  20   <- Y=0
     |   |                       |
    32  13--14--15--16--17--18--19       -1
     |
    33--34--35--36-...                   -2

                     ^
    -4  -3  -2  -1  X=0  1   2   3

The centre horizontal 1 to 2 is extended by C<wider> many further places,
then the path loops around that shape.  The starting point 1 is shifted to
the left by ceil(wider/2) places to keep the spiral centred on the origin
X=0,Y=0.

Widening doesn't change the nature of the straight lines which arise, it
just rotates them around.  For example in this wider=3 example the perfect
squares are still on diagonals, but the even squares go towards the bottom
left (instead of top left when wider=0) and the odd squares to the top right
(instead of the bottom right).

Each loop is still 8 longer than the previous, as the widening is basically
a constant amount in each loop.

=head2 N Start

The default is to number points starting N=1 as shown above.  An optional
C<n_start> can give a different start with the same shape.  For example to
start at 0,

=cut

# math-image --path=SquareSpiral,n_start=0 --all --output=numbers_dash --size=35x16

=pod

    n_start => 0

    16-15-14-13-12 ...
     |           |  | 
    17  4--3--2 11 28 
     |  |     |  |  | 
    18  5  0--1 10 27 
     |  |        |  | 
    19  6--7--8--9 26 
     |              | 
    20-21-22-23-24-25 

The only effect is to push the N values around by a constant amount.  It
might help match coordinates with something else zero-based.

=head2 Corners

Other spirals can be formed by cutting the corners of the square so as to go
around faster.  See the following modules,

    Corners Cut    Class
    -----------    -----
         1        HeptSpiralSkewed
         2        HexSpiralSkewed
         3        PentSpiralSkewed
         4        DiamondSpiral

The C<PyramidSpiral> is a re-shaped C<SquareSpiral> looping at the same
rate.  It shifts corners but doesn't cut them.

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4

=item C<$path = Math::PlanePath::SquareSpiral-E<gt>new ()>

=item C<$path = Math::PlanePath::SquareSpiral-E<gt>new (wider =E<gt> $integer, n_start =E<gt> $n)>

Create and return a new square spiral object.  An optional C<wider>
parameter widens the spiral path, it defaults to 0 which is no widening.

=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>

Return the X,Y coordinates of point number C<$n> on the path.

For C<$n E<lt> 1> the return is an empty list, as the path starts at 1.

=item C<$n = $path-E<gt>xy_to_n ($x,$y)>

Return the point number for coordinates C<$x,$y>.  C<$x> and C<$y> are
each rounded to the nearest integer, which has the effect of treating each N
in the path as centred in a square of side 1, so the entire plane is
covered.

=back

=head1 FORMULAS

=head2 N to X,Y

There's a few ways to break an N into a side and offset into the side.  One
convenient way is to treat a loop as starting at the bottom right corner, so
N=2,10,26,50,etc,  If the first at N=2 is reckoned loop number d=1 then

    Nbase = 4*d^2 - 4*d + 2

For example d=3 is Nbase=4*3^2-4*3+2=26 at X=3,Y=-2.  The biggest d with
Nbase E<lt>= N can be found by inverting with the usual quadratic formula

    d = floor (1/2 + sqrt(N/4 - 1/4))

For Perl it's good to keep the sqrt argument an integer (when a UV integer
is bigger than an NV float, and for BigRat accuracy), so rearranging

    d = floor ((1+sqrt(N-1)) / 2)

So Nbase from this d leaves a remainder which is an offset into the loop

    Nrem = N - Nbase
         = N - (4*d^2 - 4*d + 2)

The loop starts at X=d,Y=d-1 and has sides length 2d, 2d+1, 2d+1 and 2d+2,

             2d      
         +------------+        <- Y=d
         |            |
    2d   |            |  2d-1
         |     .      |
         |            |
         |            + X=d,Y=-d+1
         |
         +---------------+     <- Y=-d
             2d+1

         ^
       X=-d

The X,Y for an Nrem is then

     side      Nrem range            X,Y result
     ----      ----------            ----------
    right           Nrem <= 2d-1     X = d
                                     Y = -d+1+Nrem
    top     2d-1 <= Nrem <= 4d-1     X = d-(Nrem-(2d-1)) = 3d-1-Nrem
                                     Y = d
    left    4d-1 <= Nrem <= 6d-1     X = -d
                                     Y = d-(Nrem-(4d-1)) = 5d-1-Nrem
    bottom  6d-1 <= Nrem             X = -d+(Nrem-(6d-1)) = -7d+1+Nrem
                                     Y = -d

The corners Nrem=2d-1, Nrem=4d-1 and Nrem=6d-1 get the same result from the
two sides that meet so it doesn't matter if the high comparison is "E<lt>"
or "E<lt>=".

The bottom edge runs through to Nrem E<lt> 8d, but there's no need to
check that since d=floor(sqrt()) above ensures Nrem is within the loop.

A small simplification can be had by subtracting an extra 4d-1 from Nrem to
make negatives for the right and top sides and positives for the left and
bottom.

    Nsig = N - Nbase - (4d-1)
         = N - (4*d^2 - 4*d + 2) - (4d-1)
         = N - (4*d^2 + 1)

     side      Nsig range            X,Y result
     ----      ----------            ----------
    right           Nsig <= -2d      X = d
                                     Y = d+(Nsig+2d) = 3d+Nsig
    top      -2d <= Nsig <= 0        X = -d-Nsig
                                     Y = d
    left       0 <= Nsig <= 2d       X = -d
                                     Y = d-Nsig
    bottom    2d <= Nsig             X = -d+1+(Nsig-(2d+1)) = Nsig-3d
                                     Y = -d

=head2 N to X,Y with Wider

With the C<wider> parameter stretching the spiral loops the formulas above
become

    Nbase = 4*d^2 + (-4+2w)*d + 2-w

    d = floor ((2-w + sqrt(4N + w^2 - 4)) / 4)

Notice for Nbase the w is a term 2*w*d, being an extra 2*w for each loop.

The left offset ceil(w/2) described above (L</Wider>) for the N=1 starting
position is written here as wl, and the other half wr arises too,

    wl = ceil(w/2)
    wr = floor(w/2) = w - wl

The horizontal lengths increase by w, and positions shift by wl or wr, but
the verticals are unchanged.

             2d+w      
         +------------+        <- Y=d
         |            |
    2d   |            |  2d-1
         |     .      |
         |            |
         |            + X=d+wr,Y=-d+1
         |
         +---------------+     <- Y=-d
             2d+1+w

         ^
       X=-d-wl

The Nsig formulas then have w, wl or wr variously inserted.  In all cases if
w=wl=wr=0 then they simplify to the plain versions.

    Nsig = N - Nbase - (4d-1+w)
         = N - ((4d + 2w)*d + 1)

     side      Nsig range            X,Y result
     ----      ----------            ----------
    right         Nsig <= -(2d+w)    X = d+wr
                                     Y = d+(Nsig+2d+w) = 3d+w+Nsig
    top      -(2d+w) <= Nsig <= 0    X = -d-wl-Nsig
                                     Y = d
    left       0 <= Nsig <= 2d       X = -d-wl
                                     Y = d-Nsig
    bottom    2d <= Nsig             X = -d+1-wl+(Nsig-(2d+1)) = Nsig-wl-3d
                                     Y = -d

=head2 Rectangle to N Range

Within each row the minimum N is on the X=Y diagonal and N values increases
monotonically as X moves away to the left or right.  Similarly in each
column there's a minimum N on the X=-Y opposite diagonal, or X=-Y+1 diagonal
when X negative, and N increases monotonically as Y moves away from there up
or down.  When widerE<gt>0 the location of the minimum changes, but N is
still monotonic moving away from the minimum.

On that basis the maximum N in a rectangle is at one of the four corners,

              |
    x1,y2 M---|----M x2,y2      corner candidates
          |   |    |            for maximum N
       -------O---------
          |   |    |
          |   |    |
    x1,y1 M---|----M x1,y1
              |

=head1 OEIS

This path is in Sloane's Online Encyclopedia of Integer Sequences in various
forms.  Summary at

=over

L<http://oeis.org/A068225/a068225.html>

=back

And various sequences,

=over

L<http://oeis.org/A174344> (etc),
L<https://oeis.org/wiki/Ulam's_spiral>

=back

    wider=0 (the default)
      A174344    X coordinate
      A214526    abs(X)+abs(Y) "Manhattan" distance

      A079813    abs(dY), being k 0s followed by k 1s
      A063826    direction 1=right,2=up,3=left,4=down

      A027709    boundary length of N unit squares
      A078633    grid sticks to make N unit squares

      A033638    N turn positions (extra initial 1, 1)
      A172979    N turn positions which are primes too

      A054552    N values on X axis (East)
      A054556    N values on Y axis (North)
      A054567    N values on negative X axis (West)
      A033951    N values on negative Y axis (South)
      A054554    N values on X=Y diagonal (NE)
      A054569    N values on negative X=Y diagonal (SW)
      A053755    N values on X=-Y opp diagonal X<=0 (NW)
      A016754    N values on X=-Y opp diagonal X>=0 (SE)
      A200975    N values on all four diagonals

      A137928    N values on X=-Y+1 opposite diagonal
      A002061    N values on X=Y diagonal pos and neg
      A016814    (4k+1)^2, every second N on south-east diagonal

      A143856    N values on ENE slope dX=2,dY=1
      A143861    N values on NNE slope dX=1,dY=2
      A215470    N prime and >=4 primes among its 8 neighbours

      A214664    X coordinate of prime N (Ulam's spiral)
      A214665    Y coordinate of prime N (Ulam's spiral)
      A214666    -X  \ reckoning spiral starting West
      A214667    -Y  /

      A053999    prime[N] on X=-Y opp diagonal X>=0 (SE)
      A054551    prime[N] on the X axis (E)
      A054553    prime[N] on the X=Y diagonal (NE)
      A054555    prime[N] on the Y axis (N)
      A054564    prime[N] on X=-Y opp diagonal X<=0 (NW)
      A054566    prime[N] on negative X axis (W)

      A090925    permutation N at rotate +90
      A090928    permutation N at rotate +180
      A090929    permutation N at rotate +270
      A090930    permutation N at clockwise spiralling
      A020703    permutation N at rotate +90 and go clockwise
      A090861    permutation N at rotate +180 and go clockwise
      A090915    permutation N at rotate +270 and go clockwise
      A185413    permutation N at 1-X,Y
                   being rotate +180, offset X+1, clockwise

      A068225    permutation N to the N to its right, X+1,Y
      A121496     run lengths of consecutive N in that permutation
      A068226    permutation N to the N to its left, X-1,Y
      A020703    permutation N at transpose Y,X
                   (clockwise <-> anti-clockwise)

      A033952    digits on negative Y axis
      A033953    digits on negative Y axis, starting 0
      A033988    digits on negative X axis, starting 0
      A033989    digits on Y axis, starting 0
      A033990    digits on X axis, starting 0

      A062410    total sum previous row or column

    wider=1
      A069894    N on South-West diagonal

The following have "offset 0" in the OEIS and therefore are based on
starting from N=0.

    n_start=0
      A180714    X+Y coordinate sum
      A053615    abs(X-Y), runs n to 0 to n, distance to nearest pronic

      A001107    N on X axis
      A033991    N on Y axis
      A033954    N on negative Y axis, second 10-gonals
      A002939    N on X=Y diagonal North-East
      A016742    N on North-West diagonal, 4*k^2
      A002943    N on South-West diagonal
      A156859    N on Y axis positive and negative

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::PyramidSpiral>

L<Math::PlanePath::DiamondSpiral>,
L<Math::PlanePath::PentSpiralSkewed>,
L<Math::PlanePath::HexSpiralSkewed>,
L<Math::PlanePath::HeptSpiralSkewed>

L<Math::PlanePath::CretanLabyrinth>

L<Math::NumSeq::SpiroFibonacci>

X11 cursor font "box spiral" cursor which is this style (but going
clockwise).

=head1 HOME PAGE

L<http://user42.tuxfamily.org/math-planepath/index.html>

=head1 LICENSE

Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.

Math-PlanePath is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for
more details.

You should have received a copy of the GNU General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.

=cut