NAME
Math::PlanePath::SquareSpiral -- integer points drawn around a square
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 -1
| | |
42 21--22--23--24--25--26 -2
|
43--44--45--46--47 ...
^
-3 -2 -1 x=0 1 2 3
This is quite well known from Stanislaw Ulam finding interesting straight lines plotting the prime numbers on it. See examples/ulam-spiral-xpm.pl in the sources for a program generating that, or see the author's math-image
program using this SquareSpiral to draw Ulam's pattern and more.
The perfect squares 1,4,9,16,25 fall on diagonals to the lower right and upper left, one term on each alternately. The pronic numbers 2,6,12,20,30,42 etc half way between the squares similarly fall on diagonals to the upper right and lower left.
In general straight lines in this SquareSpiral and other stepped spirals (meaning everything except the VogelFloret) are quadratics a*k^2+b*k+c, with a=step/2 where step is how much longer each loop takes than the preceding, which is 8 in the case of the SquareSpiral. There are various interesting properties of primes in quadratic progressions like this and some quadratics seem to have more primes than others.
Other spirals can be formed by cutting the corners of the square for faster looping. The current paths doing so include
corners cut class
----------- -----
2 HexSpiralSkewed
3 PentSpiralSkewed
4 DiamondSpiral
And see the PyramidSpiral for a re-shaped SquareSpiral.
FUNCTIONS
$path = Math::PlanePath::SquareSpiral->new ()
-
Create and return a new square spiral object.
($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, it being considered 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.
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiral
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Math-PlanePath is Copyright 2010 Kevin Ryde
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/>.