Math::PlanePath::SquareSpiral(3) 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 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 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>

See examples/ulam-spiral-xpm.pl in the sources for a standalone program, or see math-image 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 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, 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
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.

## 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 rearranging 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 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 + 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
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
```

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