NAME

Math::PlanePath::HexArms -- six spiral arms

SYNOPSIS

use Math::PlanePath::HexArms;
my $path = Math::PlanePath::HexArms->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path follows six spiral arms, each advancing successively,

                               ...--66                      5
                                      \
         67----61----55----49----43    60                   4
        /                         \      \
     ...    38----32----26----20    37    54                3
           /                    \     \     \
         44    21----15---- 9    14    31    48   ...       2
        /     /              \      \    \     \     \
      50    27    10---- 4     3     8    25    42    65    1
      /    /     /                 /     /     /     /
   56    33    16     5     1     2    19    36    59    <-Y=0
  /     /     /     /        \        /     /     /
62    39    22    11     6     7----13    30    53         -1
  \     \     \     \     \              /     /
  ...    45    28    17    12----18----24    47            -2
           \     \     \                    /
            51    34    23----29----35----41   ...         -3
              \     \                          /
               57    40----46----52----58----64            -4
                 \
                  63--...                                  -5

 ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9

The X,Y points are integers using every second position to give a triangular lattice, per "Triangular Lattice" in Math::PlanePath.

Each arm is N=6*k+rem for a remainder rem=0,1,2,3,4,5, so sequences related to multiples of 6 or with a modulo 6 pattern may fall on particular arms.

Abundant Numbers

The "abundant" numbers are those N with sum of proper divisors > N. For example 12 is abundant because it's divisible by 1,2,3,4,6 and their sum is 16. All multiples of 6 starting from 12 are abundant. Plotting the abundant numbers on the path gives the 6*k arm and some other points in between,

            * * * * * * * * * * * *   *   *   ...
           *                       *           *
          *   *   *           *     *   *       *
         *                           *           *
        *           *                 *           *
       *                           *   *           *
      *           * * * * * *           *       *   *
     *           *           *   *       *           *
    *   *   *   *         *   *           *       *   *
   *           *               *   *   *   *           *
  *   *   *   *                 *           *   *       *
 *           *   *             *   *       *           *
*       *   *                 *           *           *
 *           *           * * *           *           *
  *           *                 *       *           *
   *   *       *   *   *           *   *           *
    *           *                     *   *       *
     *           *       *           *           *
      *   *       *                 *   *   *   *
       *           * * * * * * * * *           *
        *   *                         *       *
         *         *       *                 *
          *   *                         *   *
           *         *       *       *     *
            *                             *
             * * * * * * * * * * * * * * *

There's blank arms either side of the 6*k because 6*k+1 and 6*k-1 are not abundant until some fairly big values. The first abundant 6*k+1 might be 5,391,411,025, and the first 6*k-1 might be 26,957,055,125.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::HexArms->new ()

Create and return a new square spiral object.

($x,$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, as the path starts at 1.

Fractional $n gives a point on the line between $n and $n+6, that $n+6 being the next on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation.

Descriptive Methods

$arms = $path->arms_count()

Return 6.

SEE ALSO

Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 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/>.