/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::PyramidSpiral

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

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/>.