++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::PyramidRows -- points stacked up in a pyramid

# SYNOPSIS

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

# DESCRIPTION

This path arranges points in successively wider rows going upwards so as to form an upside-down pyramid. The default step is 2, ie. each row 2 wider than the preceding, one square each side,

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

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

The right end here 1,4,9,16,etc 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 step 2 is the same as the PyramidSides, Corner and SacksSpiral paths. For the SacksSpiral, spiral arms going to the right correspond to diagonals in the pyramid, and arms to the left correspond to verticals.

## Step Parameter

A `step` parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4

``    my \$path = Math::PlanePath::PyramidRows->new (step => 4);``

makes each row 2 wider on each side successively

``````   29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45        4
16 17 18 19 20 21 22 23 24 25 26 27 28              3
7  8  9 10 11 12 13 14 15                    2
2  3  4  5  6                          1
1                          <-  y=0

-6 -5 -4 -3 -2 -1 x=0 1  2  3  4  5  6 ...``````

If the step is an odd number then the extra is at the right, so step 3 gives

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

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

Or step 1 goes solely to the right. This is equivalent to the Diagonals path, but columns shifted up to make horizontal rows.

``````    11  12  13  14  15
7   8   9  10                    3
4   5   6                        2
2   3                            1
1                          <-  y=0

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

Step 0 means simply a vertical, each row 1 wide and not increasing. This is unlikely to be much use. The Rows path with `width` 1 does this too.

``````     5        4
4        3
3        2
2        1
1    <-y=0

x=0``````

Various number sequences fall in regular patterns positions depending on the step. Large steps are not particularly interesting and quickly become very wide. A limit might be desirable in a user interface, but there's no limit in the code as such.

## Step 3 Pentagonals

For step 3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are at the rightmost end of each row. The second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are the vertical at x=-1. Those second numbers are obtained by taking negative k, and the two together are the "generalized pentagonal numbers".

Both these are composites from 12 and 15 onwards (respectively), as in fact are the preceding values P(k)-1, P(k)-2, S(k)-1 and S(k)-2, factorizing simply as

``````    P(k)   = (3*k-1)*k/2
P(k)-1 = (3*k+2)*(k-1)/2
P(k)-2 = (3*k-4)*(k-1)/2
S(k)   = (3*k+1)*k/2
S(k)-1 = (3*k-2)*(k+1)/2
S(k)-2 = (3*k+4)*(k-1)/2``````

If you plot the primes on a step 3 PyramidRows then theses second pentagonal sequences make a 3-wide vertical gap at x=-1,-2,-3 where there's no primes. and the plain pentagonal sequences make the endmost three of each row non-prime. The vertical is much more noticeable in the plot.

``````       no primes these three columns         no primes these end three
except the low 2,7,13                     except low 5,11
|  |  |                                /  /  /
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
23 24 25 26 27 28 29 30 31 32 33 34 35
13 14 15 16 17 18 19 20 21 22
6  7  8  9 10 11 12
2  3  4  5
1
-6 -5 -4 -3 -2 -1 x=0 1  2  3  4 ...``````

In general an offset P(k)-c or S(k)-c will factorize like above using the roots of the quadratic

In general a constant offset c from S(k) is a column and from P(k) a diagonal going 2 to the rigth each time. The simple factorizations above using the roots of the quadratic for P(k)-c or S(k)-c is possible whenever 24*c+1 is a perfect square. This means the further columns S(k)-5, S(k)-7, S(k)-12, etc have no primes either. The S(k), S(k)-1, S(k)-2 columns are prominent because they're adjacent but there's no other adjacent ones of this type because the squares become too far apart for successive 24*c+1. Of course there could be lots of other reasons for other columns or diagonals to have few or many primes or above a certain point, etc.

# FUNCTIONS

`\$path = Math::PlanePath::PyramidRows->new ()`
`\$path = Math::PlanePath::PyramidRows->new (step => \$s)`

Create and return a new path object. The default step is 2.

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

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

For `\$n < 0` 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 each point in the pyramid as a square of side 1. If `\$x,\$y` is outside the pyramid the return is `undef`.