NAME

       Math::PlanePath::PentSpiral -- 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.

FORMULAS

   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

SEE ALSO

       Math::PlanePath, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiral

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013, 2014, 2015 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/>.