NAME

       Math::PlanePath::PyramidSpiral -- integer points drawn around a pyramid

SYNOPSIS

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

DESCRIPTION

       This path makes a pyramid shaped spiral,

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

                              ^
           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

       The perfect squares 1,4,9,16 fall one before the bottom left corner of each loop, and the
       pronic numbers 2,6,12,20,30,etc are the vertical upwards from X=1,Y=0.

   Square Spiral
       This spiral goes around at the same rate as the "SquareSpiral".  It's as if two corners
       are cut off (like the "DiamondSpiral") and two others extended (like the
       "OctagramSpiral").  The net effect is the same looping rate but the points pushed around a
       bit.

       Taking points up to a perfect square shows the similarity.  The two triangular cut-off
       corners marked by "."s are matched by the two triangular extensions.

                   +--------------------+   7x7 square
                   | .  .  . 31  .  .  .|
                   | .  . 32 13 30  .  .|
                   | . 33 14  3 12 29  .|
                   |34 15  4  1  2 11 28|
                 35|16  5  6  7  8  9 10|27
              36 17|18 19 20 21 22 23 24|25 26
           37 38 39|40 41 42 43 44 45 46|47 48 49
                   +--------------------+

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

                       12         n_start => 0
                      /  \
                    13  2 11
                   /  /  \  \
                 14  3  0--1 10
                /  /           \
              15  4--5--6--7--8--9
             /
           16-17-18-19-20-21-22-...

FUNCTIONS

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

       "$path = Math::PlanePath::PyramidSpiral->new ()"
       "$path = Math::PlanePath::PyramidSpiral->new (n_start => $n)"
           Create and return a new pyramid 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 N in the path as centred in a
           square of side 1, so the entire plane is covered.

OEIS

       This path is in Sloane's Online Encyclopedia of Integer Sequences as

           <http://oeis.org/A053615> (etc)

           n_start=1 (the default)
             A053615    abs(X), distance to next pronic, but starts n=0
             A054552    N on X axis, 4n^2 - 3n + 1
             A033951    N on South-East diagonal, 4n^2 + 3n + 1

             A214250    sum N of eight surrounding cells

             A217013    permutation N of points in SquareSpiral order
                          rotated +90 degrees
             A217294    inverse

       In the two permutations the pyramid spiral is conceived as starting to the left and the
       square spiral starting upwards.  The paths here start in the same direction (both to the
       right), hence rotate 90 to adjust the orientation.

           n_start=0
             A329116    X coordinate
             A329972    Y coordinate
             A053615    abs(X)
             A339265    dX-dY increments (runs +1,-1)
             A001107    N on X axis, decagonal numbers
             A002939    N on Y axis
             A033991    N on X negative axis
             A002943    N on Y negative axis
             A007742    N on diagonal South-West
             A033954    N on diagonal South-East, decagonal second kind

           n_start=2
             A185669    N on diagonal South-East

SEE ALSO

       Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidRows,
       Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/>.