Reference

Math::PlanePath::HexSpiral(3pm)

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)

    A328818    X coordinate
    A307012    Y coordinate
    A307011    (X-Y)/2
    A307013    (X+Y)/2
    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
      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 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, 2018, 2019, 2020 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/>.

History