Math::PlanePath::PyramidSides -- points along the sides of pyramid
use Math::PlanePath::PyramidSides; my $path = Math::PlanePath::PyramidSides->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
$path = Math::PlanePath::PyramidSides->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the x,y coordinates of point number
$non the path.
$n < 0.5the 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
$yare 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.
Math-PlanePath is Copyright 2010 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.
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