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 path is quite well known from Stanislaw Ulam finding an interesting pattern plotting the prime numbers on it. See examples/ulam-spiral-xpm.pl in the sources for a program generating that, or see math-image using this SquareSpiral to draw Ulam's pattern and much 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.

FUNCTIONS

$path = Math::PlanePath::SquareSpiral->new (key=>value, ...)

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::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/>.