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

       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".

   NStart
       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

X,YtoN
       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.

       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.

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

       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.

perl v5.32.0                                       2021-01-23               Math::PlanePath::OctagramSpiral(3pm)

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

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

           <http://oeis.org/A125201> (etc)

           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

       Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral

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