/ 
Math-PlanePath-127
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::PyramidSides

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 ...

N=1,4,9,16,etc along the positive X axis is the perfect squares. N=2,6,12,20,etc in the X=-1 vertical is the pronic numbers k*(k+1) half way between those successive squares.

The pattern is the same as the Corner path but turned and spread so the single quadrant in the Corner becomes a half-plane here.

The pattern is similar to PyramidRows (with its default step=2), 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.

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same pyramid pattern. For example to start at 0,

n_start => 0

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

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::PyramidSides->new ()
$path = Math::PlanePath::PyramidSides->new (n_start => $n)

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.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

FORMULAS

Rectangle to 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 the positive X=k over the negative X=-k.

The smallest N conversely is at the X of smallest absolute value. If the X range crosses 0, ie. $x1 and $x2 have different signs, then X=0 is the smallest.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

   n_start=1 (the default)
     A049240    abs(dY), being 0=horizontal step at N=square
     A002522    N on X negative axis, x^2+1
     A033951    N on X=Y diagonal, 4d^2+3d+1
     A004201    N for which X>=0, ie. right hand half
     A020703    permutation N at -X,Y

  n_start=0
     A196199    X coordinate, runs -n to +n
     A053615    abs(X), runs n to 0 to n
     A000196    abs(X)+abs(Y), being floor(sqrt(N)),
                  k repeated 2k+1 times starting 0

SEE ALSO

Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner, Math::PlanePath::DiamondSpiral, Math::PlanePath::SacksSpiral, Math::PlanePath::MPeaks

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 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/>.