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# 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
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``````

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