/ 
Math-PlanePath-126
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::HexSpiral

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

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/>.