Provided by: libmath-planepath-perl_129-1_all bug

NAME

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

SYNOPSIS

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

DESCRIPTION

       This path makes a square spiral,

           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 examples/square-numbers.pl for a simple program printing these numbers.

   Ulam Spiral
       This path is well known from Stanislaw Ulam finding interesting straight lines when
       plotting the prime numbers on it.

           Stein, Ulam and Wells, "A Visual Display of Some Properties of the Distribution of
           Primes", American Mathematical Monthly, volume 71, number 5, May 1964, pages 516-520.
           <http://www.jstor.org/stable/2312588>

       The cover of Scientific American March 1964 featured this spiral,

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

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

       See examples/ulam-spiral-xpm.pl for a standalone program, or see math-image using this
       "SquareSpiral" to draw this pattern and more.

       Stein, Ulam and Wells above also considered primes on the Math::PlanePath::Corner path,
       and on a half-plane like two corners.

   Straight Lines
       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 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.

   Wider
       An optional "wider" parameter makes the path wider, becoming a rectangle spiral instead of
       a square.  For example

           wider => 3

           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 "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, since the widening is a constant amount in
       each loop.

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

           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.

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

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::SquareSpiral->new ()"
       "$path = Math::PlanePath::SquareSpiral->new (wider => $integer, n_start => $n)"
           Create and return a new square spiral object.  An optional "wider" parameter widens
           the spiral path, it defaults to 0 which is no widening.

       "($x,$y) = $path->n_to_xy ($n)"
           Return the X,Y coordinates of point number $n on the path.

           For "$n < 1" the return is an empty list, as the path starts at 1.

       "$n = $path->xy_to_n ($x,$y)"
           Return the point number for coordinates "$x,$y".  $x and $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.

FORMULAS

   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 turn, so N=2,10,26,50,etc, If the first
       loop at N=2 is reckoned loop number d=1 then the loop starts at

           Nbase = 4*d^2 - 4*d + 2
                 = 2,10,26,50,... for d=1,2,3,4,...
                          (A069894 but it going from d=0)

       For example d=3 is Nbase=4*3^2-4*3+2=26 at X=3,Y=-2.  The biggest d with Nbase <= 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 rearrange to

           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 "<" or "<=".

       The bottom edge runs through to Nrem < 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

       This calculation can be found in

           Ronald L. Graham, Donald E. Knuth, Oren Patashnik, "Concrete Mathematics", Addison-
           Wesley, 1989, chapter 3 "Integer Functions", exercise 40 page 99, answer page 498.

       They start the spiral from 0, and first step North so their x is -Y here.  Their formula
       for x(n) tests a floor(2*sqrt(N)) to decide whether on a horizontal and so whether to
       apply the equivalent of Nrem to the result.

   N to X,Y with Wider
       With the "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 ("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

   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 wider>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
                     |

OEIS

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

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

       And various sequences,

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

           wider=0 (the default)
             A174344    X coordinate
             A274923    Y coordinate
             A268038    negative Y
             A296030    X,Y pairs
             A214526    abs(X)+abs(Y) "Manhattan" distance

             A079813    abs(dY), being k 0s followed by k 1s
             A055086    direction (total turn)
             A000267    direction + 1
             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

             A240025    turn 1=left,0=straight (extra initial 1)

             A033638    N turn positions (extra initial 1, 1)
             A172979    N turn positions which are primes too
             A242601    X and Y location of origin then each turn,
                          X=A242601(n+1), Y=A242601(n)
             A080037    N straight-ahead (except initial 2)
             A248333    num straight points among the first N
             A083479    num non-turn points among the first N
                          (straight and the origin)

             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
             A317186    N on Y axis positive and negative
             A267682    N on Y axis positive and negative (origin twice)
             A265400    N inner neighbour

             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 at X+1,Y
             A121496     run lengths of consecutive N in this permutation
             A068226    permutation N at X-1,Y
             A334752    permutation N at X,Y+1
             A334751    permutation N at X,Y-1
             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" S and therefore start 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

SEE ALSO

       Math::PlanePath, Math::PlanePath::PyramidSpiral

       Math::PlanePath::DiamondSpiral, Math::PlanePath::PentSpiralSkewed,
       Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HeptSpiralSkewed

       Math::PlanePath::CretanLabyrinth

       Math::NumSeq::SpiroFibonacci

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

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/>.