Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral
use Math::PlanePath::HexSpiral; my $path = Math::PlanePath::HexSpiral->new; my ($x, $y) = $path->n_to_xy (123);
This path makes a hexagonal spiral, with points spread out horizontally to fit on a square grid.
28 -- 27 -- 26 -- 25 3 / \ 29 13 -- 12 -- 11 24 2 / / \ \ 30 14 4 --- 3 10 23 1 / / / \ \ \ 31 15 5 1 --- 2 9 22 <- Y=0 \ \ \ / / 32 16 6 --- 7 --- 8 21 -1 \ \ / 33 17 -- 18 -- 19 -- 20 -2 \ 34 -- 35 ... -3 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is at X=2,Y=0. The diagonals are just 1 across, so n=3 is at X=1,Y=1. Each alternate row is offset from the one above or below. The result is a triangular lattice per "Triangular Lattice" in Math::PlanePath.
The octagonal numbers 8,21,40,65, etc 3*k^2-2*k fall on a horizontal straight line at Y=-1. In general straight lines are 3*k^2 + b*k + c. A plain 3*k^2 goes diagonally up to the left, then b is a 1/6 turn anti-clockwise, or clockwise if negative. So b=1 goes horizontally to the left, b=2 diagonally down to the left, b=3 diagonally down to the right, etc.
An optional wider parameter makes the path wider, stretched along the top and bottom horizontals. For example
wider
$path = Math::PlanePath::HexSpiral->new (wider => 2);
gives
... 36----35 3 \ 21----20----19----18----17 34 2 / \ \ 22 8---- 7---- 6---- 5 16 33 1 / / \ \ \ 23 9 1---- 2---- 3---- 4 15 32 <- Y=0 \ \ / / 24 10----11----12----13----14 31 -1 \ / 25----26----27----28---29----30 -2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The centre horizontal from N=1 is extended by wider many extra places, then the path loops around that shape. The starting point N=1 is shifted to the left by wider many places to keep the spiral centred on the origin X=0,Y=0. Each horizontal gap is still 2.
Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::HexSpiral->new ()
$path = Math::PlanePath::HexSpiral->new (wider => $w)
Create and return a new hex spiral object. An optional wider parameter widens the path, it defaults to 0 which is no widening.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n on the path.
$n
For $n < 1 the return is an empty list, it being considered the path starts at 1.
$n < 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 a square of side 1.
$x,$y
$x
$y
Only every second square in the plane has an N, being those where X,Y both odd or both even. If $x,$y is a position without an N, ie. one of X,Y odd the other even, then the return is undef.
undef
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A063178 (etc) A063178 total sum previous row or diagonal
Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012 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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.