Math::PlanePath::PyramidRows -- points stacked up in a pyramid
use Math::PlanePath::PyramidRows; my $path = Math::PlanePath::PyramidRows->new; my ($x, $y) = $path->n_to_xy (123);
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.
A step parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4
step
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.
width
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.
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 P(-k), and the two together are the "generalized pentagonal numbers".
Both these sequences are composites from 12 and 15 onwards, respectively, and the preceding values P(k)-1, P(k)-2, S(k)-1 and S(k)-2 are too. They factorize 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 these second pentagonal sequences make a 3-wide vertical gap of no primes at x=-1,-2,-3. and the plain pentagonal sequences make the endmost three N of each row non-prime. The vertical is much more noticeable in a 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 a constant offset c from S(k) is a column and from P(k) a diagonal going 2 to the right each time. The simple factorizations above using the roots of the quadratic 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 columns S(k), S(k)-1, S(k)-2 are prominent because they're adjacent. There's no other adjacent ones of this type because the squares after 49 are too far apart for successive 24*c+1. Of course there could be other reasons for other columns or diagonals to have few or many primes, perhaps above a certain point, etc.
$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.
$n
For $n < 0 the return is an empty list, it being considered there are no negative points in the pyramid.
$n < 0
$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.
$x,$y
$x
$y
undef
Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::Corner, Math::PlanePath::SacksSpiral Math::PlanePath::MultipleRings
Math::PlanePath::Diagonals, Math::PlanePath::Rows
http://user42.tuxfamily.org/math-planepath/index.html
Math-PlanePath is Copyright 2010, 2011 Kevin Ryde
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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.