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,
37363534333231 3   38 1716151413 30 2     39 18 543 12 29 1       40 19 6 12 11 28 ... < Y=0       41 20 78910 27 52 1     42 212223242526 51 2   4344454647484950 3 ^ 3 2 1 X=0 1 2 3 4
See examples/squarenumbers.pl in the sources 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 516520. <http://www.jstor.org/stable/2312588>
The cover of Scientific American March 1964 featured this spiral,
See examples/ulamspiralxpm.pl in the sources for a standalone program, or see mathimage using this "SquareSpiral" to draw this pattern and more.
Stein, Ulam and Wells also considered primes on the Math::PlanePath::Corner path, and on a halfplane 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^23*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 anticlockwise, or clockwise if negative. So b=1 is horizontal West, b=2 diagonally down SouthWest, b=3 down South, etc.
Honaker's primegenerating 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 2928272625242322 2   30 1110 9 8 7 6 21 1     31 12 1 2 3 4 5 20 < Y=0    32 13141516171819 1  33343536... 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, as the widening is basically 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 1615141312 ...    17 432 11 28      18 5 01 10 27     19 6789 26   202122232425
The only effect is to push the N values around by a constant amount. It might help match coordinates with something else zerobased.
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 reshaped "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*3^24*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 rearranging to
d = floor ((1+sqrt(N1)) / 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=d1 and has sides length 2d, 2d+1, 2d+1 and 2d+2,
2d ++ < Y=d   2d   2d1  .     + 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 <= 2d1 X = d Y = d+1+Nrem top 2d1 <= Nrem <= 4d1 X = d(Nrem(2d1)) = 3d1Nrem Y = d left 4d1 <= Nrem <= 6d1 X = d Y = d(Nrem(4d1)) = 5d1Nrem bottom 6d1 <= Nrem X = d+(Nrem(6d1)) = 7d+1+Nrem Y = d
The corners Nrem=2d1, Nrem=4d1 and Nrem=6d1 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 4d1 from Nrem to make negatives for the right and top sides and positives for the left and bottom.
Nsig = N  Nbase  (4d1) = N  (4*d^2  4*d + 2)  (4d1) = 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 = dNsig Y = d left 0 <= Nsig <= 2d X = d Y = dNsig bottom 2d <= Nsig X = d+1+(Nsig(2d+1)) = Nsig3d Y = d
This calculation can be found as an exercise in Graham, Knuth and Patashnik ``Concrete Mathematics'', chapter 3 ``Integer Functions'', exercise 40, page 99. They start the spiral from 0, and vertically 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 + 2w d = floor ((2w + 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   2d1  .     + X=d+wr,Y=d+1  ++ < Y=d 2d+1+w ^ X=dwl
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  (4d1+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 = dwlNsig Y = d left 0 <= Nsig <= 2d X = dwl Y = dNsig bottom 2d <= Nsig X = d+1wl+(Nsig(2d+1)) = Nsigwl3d 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 MM x2,y2 corner candidates    for maximum N O       x1,y1 MM x1,y1 
OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences in various forms. Summary at
And various sequences,
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 southeast 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 1X,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, X1,Y A020703 permutation N at transpose Y,X (clockwise <> anticlockwise) 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 SouthWest 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(XY), 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 10gonals A002939 N on X=Y diagonal NorthEast A016742 N on NorthWest diagonal, 4*k^2 A002943 N on SouthWest diagonal A156859 N on Y axis positive and negative
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.