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 in the sources for a simple program
printing these numbers.
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 in the sources for a standalone program,
or see math-image using this "SquareSpiral" to draw this pattern and
more.
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
$path = Math::PlanePath::SquareSpiral->new (wider => 3);
gives
29--28--27--26--25--24--23--22 2
| |
30 11--10-- 9-- 8-- 7-- 6 21 1
| | | |
31 12 1-- 2-- 3-- 4-- 5 20 <- Y=0
| | |
32 13--14--15--16--17--18--19 -1
|
33--34--35--36-... -2
^
-4 -3 -2 -1 X=0 1 2 3
The centre horizontal 1 to 2 is extended by "wider" many further
places, then the path loops around that shape. The starting point 1 is shifted
to the left by ceil(wider/2) places to keep the spiral centred on the origin
X=0,Y=0.
Widening doesn't change the nature of the straight lines which arise, it just
rotates them around. For example in this wider=3 example the perfect squares
are still on diagonals, but the even squares go towards the bottom left
(instead of top left when wider=0) and the odd squares to the top right
(instead of the bottom right).
Each loop is still 8 longer than the previous, as the widening is basically a
constant amount in each loop.
N Start¶
The default is to number points starting N=1 as shown above. An optional
"n_start" can give a different start with the same shape. For
example to start at 0,
n_start => 0
16-15-14-13-12 ...
| | |
17 4--3--2 11 28
| | | | |
18 5 0--1 10 27
| | | |
19 6--7--8--9 26
| |
20-21-22-23-24-25
The only effect is to push the N values around by a constant amount. It might
help match coordinates with something else zero-based.
Corners¶
Other spirals can be formed by cutting the corners of the square so as to go
around faster. See the following modules,
Corners Cut Class
----------- -----
1 HeptSpiralSkewed
2 HexSpiralSkewed
3 PentSpiralSkewed
4 DiamondSpiral
The "PyramidSpiral" is a re-shaped "SquareSpiral" looping at
the same rate. It shifts corners but doesn't cut them.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$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 corner, so
N=2,10,26,50,etc, If the first at N=2 is reckoned loop number d=1 then
Nbase = 4*d^2 - 4*d + 2
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
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
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
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 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
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 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/>.