oracular (3) Math::PlanePath::PyramidSides.3pm.gz

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

           <http://oeis.org/A196199> (etc)

           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, 2019, 2020 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/>.