++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::PyramidSides -- points along the sides of pyramid

# SYNOPSIS

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

# DESCRIPTION

This path puts points in layers along the sides of a pyramid growing upwards.

``````                        21                          4
20  13  22                      3
19  12   7  14  23                  2
18  11   6   3   8  15  24              1
17  10   5   2   1   4   9  16  25    <-  y=0

^
... -4  -3  -2  -1  x=0  1   2   3   4 ...``````

The horizontal 1,4,9,16,etc at the bottom going right is the perfect squares. The vertical 2,6,12,20,etc at x=-1 is the pronic numbers s*(s+1), half way between those successive squares.

The pattern is the same as the Corner path but widened out so that the single quadrant in the Corner becomes a half-plane here.

The pattern is similar to PyramidRows, just with the columns dropped down vertically to start at the X axis. Any pattern occurring within a column is unchanged, but what was a row becomes a diagonal and vice versa.

## Lucky Numbers of Euler

An interesting sequence for this path is Euler's k^2+k+41. Low values are spread around a bit, but from N=1763 (k=41) onwards they're the vertical at x=40. There's quite a few primes in this quadratic and on a plot of the primes that vertical stands out a little denser in primes than its surrounds (at least for up to the first 2500 or so values). The line shows in other step==2 paths too, but not as clearly. In the PyramidRows the beginning is up at y=40, and in the Corner path it's a diagonal.

# FUNCTIONS

`\$path = Math::PlanePath::PyramidSides->new ()`

Create and return a new path object.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the x,y coordinates of point number `\$n` on the path.

For `\$n < 0.5` the return is an empty list, it being considered there are no negative points in the pyramid.

`\$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 points in the pyramid as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.