SYNOPSIS
use Math::PlanePath::HexSpiral;
my $path = Math::PlanePath::HexSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
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^22*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 anticlockwise, 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.
Wider
An optional "wider" parameter makes the path wider, stretched along the top and bottom horizontals. For example
$path = Math::PlanePath::HexSpiral>new (wider => 2);
gives
... 3635 3 \ 2120191817 34 2 / \ \ 22 8 7 6 5 16 33 1 / / \ \ \ 23 9 1 2 3 4 15 32 < Y=0 \ \ / / 24 1011121314 31 1 \ / 252627282930 2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 7 6 5 4 3 2 1 X=0 1 2 3 4 5 6 7
The centre horizontal from N=1 is extended by "wider" many extra places, then the path loops around that shape. The starting point N=1 is shifted to the left by wider many places to keep the spiral centred on the origin X=0,Y=0. Each horizontal gap is still 2.
Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start with the same shape etc. For example to start at 0,
n_start => 0 27 26 25 24 3 28 12 11 10 23 2 29 13 3 2 9 22 1 30 14 4 0 1 8 21 < Y=0 31 15 5 6 7 20 ... 1 32 16 17 18 19 38 2 33 34 35 36 37 3 ^ 6 5 4 3 2 1 X=0 1 2 3 4 5 6
In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1).
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$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 $n on the path.
For "$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
$n in the path as a square of side 1.
Only every second square in the plane has an N, being those where X,Y both odd or both even. If "$x,$y" is a position without an N, ie. one of X,Y odd the other even, then the return is "undef".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
 <http://oeis.org/A056105> (etc)
A056105 N on X axis A056106 N on X=Y diagonal A056107 N on NorthWest diagonal A056108 N on negative X axis A056109 N on SouthWest diagonal A003215 N on SouthEast diagonal A063178 total sum N previous row or diagonal A135711 boundary length of N hexagons A135708 grid sticks of N hexagons n_start=0 A000567 N on X axis, octagonal numbers A049451 N on X negative axis A049450 N on X=Y diagonal northeast A033428 N on northwest diagonal, 3*k^2 A045944 N on southwest diagonal, octagonal numbers second kind A063436 N on WSW slope dX=3,dY=1 A028896 N on southeast diagonal
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.