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
$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.
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).
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
.
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
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, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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