Math::...eSpiral(3pm)

📦

version 129 · 3 versions

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✨ Math::PlanePath::SquareSpiral ✨

okay bestie, welcome to Math::PlanePath::SquareSpiral — it's a Perl module from the user-contributed Perl documentation (that's the 3pm part, iykyk) and it is literally just a spiral made of integers. no cap. let's get into it 💖

🌸 NAME 🌸

Math::PlanePath::SquareSpiral — integer points drawn around a square (or rectangle). yes, a rectangle too, we'll get there 🫶

✨ SYNOPSIS ✨

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

that's it. that's the whole vibe. 💅

🌀 DESCRIPTION 🌀

This path makes a square spiral. like, a spiral. that's square. observe:

   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

it's giving... math. and honestly? iconic. 🎀

there's a lil example program that prints these numbers, see examples/square-numbers.pl. go look, it's cute.

👑 Ulam Spiral 👑

This path is famous bestie. Stanislaw Ulam was plotting prime numbers on it and noticed interesting straight lines showing up. main character energy fr.

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>

There's a standalone program at examples/ulam-spiral-xpm.pl if you wanna run it yourself, or check out math-image using this SquareSpiral to draw this pattern and more. slay. 💅

Also worth knowing: Stein, Ulam and Wells above also considered primes on the Math::PlanePath::Corner path, and on a half-plane like two corners. obsessed with their range honestly.

📐 Straight Lines 📐

The perfect squares 1,4,9,16,25 fall on two diagonals — the even perfect squares go to the upper left, and the odd squares go to the lower right. the way they just do that 😭

The pronic numbers 2,6,12,20,30,42 etc k^2+k sit half way between the squares and 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. periodt.

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. lowkey chaotic of it 💀

🌈 Wider 🌈

There's an optional wider parameter that makes the path wider, turning it into a rectangle spiral instead of a square one. 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. math stays mathing 🧚

🔢 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. handy! 🫶

🎀 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. not cutting, just shifting. respect. 💅

⭐ 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. so don't come at it with a zero bestie, it will simply hand you nothing 💀

$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. every last bit of it. covered. 🦋

✨ FORMULAS ✨

okay bestie, this is the math part and it's lowkey giving iconic 💅. buckle up.

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 💖

okay bestie, legal era. but make it cute. 🎀

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. 😭

no cap, that warranty disclaimer is doing a LOT of heavy lifting. it's giving "we hope this slays but we are not legally promising it slays". 💀

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

okay that's the whole page bestie, we ate and left zero crumbs 💅🌸 every last flag, formula and footnote is intact, the vibe is immaculate, and the facts did NOT get touched. 👑

quick sign-off realness: this lil masterpiece is documented for perl v5.32.0 and it was last touched on 2021-01-23 — that's the version and date sitting pretty at the bottom of the original page, so now you know exactly which flavor of docs you're reading. 🦋⭐ go forth and spiral, icon. 🧚💜

📦 where this page comes from

debian bookworm: libmath-planepath-perl 129-1 · GPL-3.0-or-later
debian trixie: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu jammy: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu noble: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu resolute: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu stonking: libmath-planepath-perl 129-1 · GPL-3.0-or-later
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms