Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral
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^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.
Wider
An optional "wider" parameter makes the path wider, stretched along the top and bottom
horizontals. For example
$path = Math::PlanePath::HexSpiral->new (wider => 2);
gives
... 36----35 3
\
21----20----19----18----17 34 2
/ \ \
22 8---- 7---- 6---- 5 16 33 1
/ / \ \ \
23 9 1---- 2---- 3---- 4 15 32 <- Y=0
\ \ / /
24 10----11----12----13----14 31 -1
\ /
25----26----27----28---29----30 -2
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The centre horizontal from N=1 is extended by "wider" many extra places, then the path
loops around that shape. The starting point N=1 is shifted to the left by wider many
places to keep the spiral centred on the origin X=0,Y=0. Each horizontal gap is still 2.
Each loop is still 6 longer than the previous, since the widening is basically a constant
amount added into each loop.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start with the same shape etc. For example to start at 0,
n_start => 0
27 26 25 24 3
28 12 11 10 23 2
29 13 3 2 9 22 1
30 14 4 0 1 8 21 <- Y=0
31 15 5 6 7 20 ... -1
32 16 17 18 19 38 -2
33 34 35 36 37 -3
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1).
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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 North-West diagonal
A056108 N on negative X axis
A056109 N on South-West diagonal
A003215 N on South-East 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 north-east
A033428 N on north-west diagonal, 3*k^2
A045944 N on south-west diagonal, octagonal numbers second kind
A063436 N on WSW slope dX=-3,dY=-1
A028896 N on south-east diagonal
SEE ALSO
Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms,
Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.