Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::HexSpiral

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

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 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/>.

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