Math::PlanePath::PyramidSpiral(3pm)

Usage

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

Overview

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

Home page

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

Releases

6 releases since Debian 8 (Jessie); the text changed 3 times.