Ulam Spiral

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,

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.

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

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
              |