Math::PlanePath::HexArms(3) 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.

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