and 1 contributors

# NAME

Math::PlanePath::OctagramSpiral -- integer points drawn around an octagram

# SYNOPSIS

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

# DESCRIPTION

This path makes a spiral around an octagram (8-pointed star),

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

^
-4 -3 -2 -1 X=0 1  2  3  4  5 ...``````

Each loop is 16 longer than the previous. The 18-gonal numbers 18,51,100,etc fall on the horizontal at Y=-1.

The inner corners like 23, 31, 39, 47 are similar to the `SquareSpiral` path, but instead of going directly between them the octagram takes a detour out to make the points of the star. Those excursions make each loops 8 longer (1 per excursion), hence a step of 16 here as compared to 8 for the `SquareSpiral`.

## 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

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

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::OctagramSpiral->new ()`

Create and return a new octagram 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, it being considered the path starts at 1.

`\$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 N in the path as centred in a square of side 1, so the entire plane is covered.

# FORMULAS

## X,Y to N

The symmetry of the octagram can be used by rotating a given X,Y back to the first star excursion such as N=19 to N=23. If Y is negative then rotate back by 180 degrees, then if X is negative rotate back by 90, and if Y>=X then by a further 45 degrees. Each such rotation, if needed, is counted as a multiple of the side-length to be added to the final N. For example at N=19 the side length is 2. Rotating by 180 degrees is 8 side lengths, by 90 degrees 4 sides, and by 45 degrees is 2 sides.

# OEIS

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

``````    n_start=1 (the default)
A125201    N on X axis, from X=1 onwards, 18-gonals + 1
A194268    N on diagonal South-East

n_start=0
A051870    N on X axis, 18-gonal numbers
A139273    N on Y axis
A139275    N on X negative axis
A139277    N on Y negative axis
A139272    N on diagonal X=Y
A139274    N on diagonal North-West
A139276    N on diagonal South-West
A139278    N on diagonal South-East, second 18-gonals``````

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