NAME
Math::PlanePath::PyramidSpiral  integer points drawn around a pyramid
SYNOPSIS
use Math::PlanePath::PyramidSpiral;
my $path = Math::PlanePath::PyramidSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path makes a pyramid shaped spiral,
31 3
/ \
32 13 30 2
/ / \ \
33 14 3 12 29 1
/ / / \ \ \
34 15 4 12 11 28 ... < Y=0
/ / / \ \ \
35 16 5678910 27 52 1
/ / \ \
36 17181920212223242526 51 2
/ \
3738394041424344454647484950 3
^
6 5 4 3 2 1 X=0 1 2 3 4 5 6 7
The perfect squares 1,4,9,16 fall one before the bottom left corner of each loop, and the pronic numbers 2,6,12,20,30,etc are the vertical upwards from X=1,Y=0.
Square Spiral
This spiral goes around at the same rate as the SquareSpiral
. It's as if two corners are cut off (like the DiamondSpiral
) and two others extended (like the OctagramSpiral
). The net effect is the same looping rate but the points pushed around a bit.
Taking points up to a perfect square shows the similarity. The two triangular cutoff corners marked by "."s are matched by the two triangular extensions.
++ 7x7 square
 . . . 31 . . .
 . . 32 13 30 . .
 . 33 14 3 12 29 .
34 15 4 1 2 11 28
3516 5 6 7 8 9 1027
36 1718 19 20 21 22 23 2425 26
37 38 3940 41 42 43 44 45 4647 48 49
++
N Start
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, with the same shape etc. For example to start at 0,
12 n_start => 0
/ \
13 2 11
/ / \ \
14 3 01 10
/ / \
15 456789
/
16171819202122...
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::PyramidSpiral>new ()
$path = Math::PlanePath::PyramidSpiral>new (n_start => $n)

Create and return a new pyramid spiral object.
$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 N in the path as centred in a square of side 1, so the entire plane is covered.
OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences as
http://oeis.org/A053615 (etc)
n_start=1 (the default)
A053615 abs(X), distance to next pronic, but starts n=0
A054552 N on X axis, 4n^2  3n + 1
A033951 N on SouthEast diagonal, 4n^2 + 3n + 1
A214250 sum N of eight surrounding cells
A217013 permutation N of points in SquareSpiral order
rotated +90 degrees
A217294 inverse
In the two permutations the pyramid spiral is conceived as starting to the left and the square spiral starting upwards. The paths here start in the same direction (both to the right), hence rotate 90 to adjust the orientation.
n_start=0
A329116 X coordinate
A329972 Y coordinate
A053615 abs(X)
A001107 N on X axis, decagonal numbers
A002939 N on Y axis
A033991 N on X negative axis
A002943 N on Y negative axis
A007742 N on diagonal SouthWest
A033954 N on diagonal SouthEast, decagonal second kind
n_start=2
A185669 N on diagonal SouthEast
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidRows, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.