Math::PlanePath::DiamondArms(3) four spiral arms

## SYNOPSIS

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

## DESCRIPTION

This path follows four spiral arms, each advancing successively in a diamond pattern,

```                 25   ...                    4
29  14  21  36                  3
33  18   7  10  17  32              2
... 22  11   4   3   6  13  28          1
26  15   8   1   2   9  24 ...      <- Y=0
30  19  12   5  20  35             -1
34  23  16  31                 -2
...   27                     -3
^
-3  -2  -1  X=0  1   2   3   4
```

Each arm makes a spiral widening out by 4 each time around, thus leaving room for four such arms. Each arm loop is 64 longer than the preceding loop. For example N=13 to N=85 below is 84-13=72 points, and the next loop N=85 to N=221 is 221-85=136 which is an extra 64, ie. 72+64=136.

```                 25          ...
/  \           \
29  . 21  .  .  . 93
/        \           \
33  .  .  . 17  .  .  . 89
/              \           \
37  .  .  .  .  . 13  .  .  . 85
/                 /           /
41  .  .  .  1  .  9  .  .  . 81
\           \  /           /
45  .  .  .  5  .  .  . 77
\                    /
49  .  .  .  .  . 73
\              /
53  .  .  . 69
\        /
57  . 65
\  /
61
```

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

The starts of each arm N=1,2,3,4 are at X=0 or 1 and Y=0 or 1,

```               ..
\
4    3  ..          Y=1
/        /
..  1    2           <- Y=0
\
..
^    ^
X=0  X=1
```

They could be centred around the origin by taking X-1/2,Y-1/2 so for example N=1 would be at -1/2,-1/2. But the it's done as N=1 at 0,0 to stay in integers.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::DiamondArms->new ()"
Create and return a new path 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+4", that "\$n+4" being the next point 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 4.

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