Provided by: libmath-planepath-perl_129-1_all 
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/>.