Reference

Math::PlanePath::HexArms(3pm)

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/>.

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