Provided by: libmath-planepath-perl_113-1_all bug

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.

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 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/>.