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^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 isundef
.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A056105 (etc)
A328818 X coordinate
A307012 Y coordinate
A307011 (XY)/2
A307013 (X+Y)/2
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
A001399 N positions of turns (extra initial 1)
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
SEE ALSO
Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This 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/>.