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

HOME PAGE

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

LICENSE

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/>.

2021-01-23

perl v5.32.0