 Math::PlanePath::PentSpiral(3) integer points in a pentagonal shape

## SYNOPSIS

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

## DESCRIPTION

This path makes a pentagonal (five-sided) spiral with points spread out to fit on a square grid.

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

Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The lower diagonals are 1 across and 1 down, so n=17 is at x=4,y=-2 and n=18 is x=5,y=-1. But the upper angles go 2 across and 1 up, so n=20 is x=4,y=1 then n=21 is x=2,y=2.

The effect is to make the sides equal length, except for a kink at the lower right corner. Only every second square in the plane is used. In the top half (y>=0) those points line up, in the lower half (y<0) they're offset on alternate rows.

## N Start

The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, in the same pattern. For example to start at 0,

```    n_start => 0            38
39    21    37
...
40    22     9    20    36    57
41    23    10     2     8    19    35    56
42    24    11     3     0     1     7    18    34    55
43    25    12     4     5     6    17    33    54
44    26    13    14    15    16    32    53
45    27    28    29    30    31    52
46    47    48    49    50    51
```

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::PentSpiral->new ()"
"\$path = Math::PlanePath::PentSpiral->new (n_start => \$n)"
Create and return a new pentagon spiral object.
"\$n = \$path->xy_to_n (\$x,\$y)"
Return the point number for coordinates "\$x,\$y". \$x and \$y are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1.

## N to X,Y

It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal,

```                      21                loop d=3
--    --
22          20
--                --
23                      19
--                            --
24                 0                18
\                                /
25          .                 17
\                          /
26    13----14----15----16
\
.
```

The SW diagonal is N=0,4,13,27,46,etc which is

```    N = (5d-7)*d/2 + 1           # starting d=1 first loop
```

This can be inverted to get d from N

```    d = floor( (sqrt(40*N + 9) + 7) / 10 )
```

Each side is length d, except the lower right diagonal slope which is d-1. For the very first loop that lower right is length 0.

## OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A140066> (etc)

```    n_start=1 (the default)
A192136    N on X axis, (5*n^2 - 3*n + 2)/2
A140066    N on Y axis
A116668    N on X negative axis
A005891    N on South-East diagonal, centred pentagonals
A134238    N on South-West diagonal
n_start=0
A000566    N on X axis, heptagonal numbers
A005476    N on Y axis
A028895    N on South-East diagonal
```

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