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 1,4,9,16,etc along the X axis to the right are the perfect squares. The vertical 2,6,12,20,etc at X=-1 are the pronic numbers k*(k+1) half way between those successive squares.

The pattern is the same as the Corner path but turned and widened out so 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. The low values are spread around a bit, but from N=1763 (k=41) they're the vertical at x=40. There's quite a few primes in this quadratic and when plotting primes that vertical stands out a little denser 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 for instance 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.

FORMULAS

N Range

For rect_to_n_range, in each column N increases so the biggest N is in the topmost row and and smallest N in the bottom row.

In each row N increases along the sequence X=0,-1,1,-2,2,-3,3, etc. So the biggest N is at the X of biggest absolute value and preferring a positive X=k over X=-k. The smallest X conversely is at the X of smallest absolute value. When the rectangle $x1 to $x2 crosses 0, ie. $x1 and $x2 have different signs, then of course X=0 is the smallest.

SEE ALSO

Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner, Math::PlanePath::SacksSpiral

HOME PAGE

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

LICENSE

Copyright 2010, 2011 Kevin Ryde

This file is part of Math-PlanePath.

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/>.