Provided by: libmath-planepath-perl_122-1_all 
NAME
Math::PlanePath::PyramidSpiral -- integer points drawn around a pyramid
SYNOPSIS
use Math::PlanePath::PyramidSpiral;
my $path = Math::PlanePath::PyramidSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path makes a pyramid shaped spiral,
31 3
/ \
32 13 30 2
/ / \ \
33 14 3 12 29 1
/ / / \ \ \
34 15 4 1--2 11 28 ... <- Y=0
/ / / \ \ \
35 16 5--6--7--8--9-10 27 52 -1
/ / \ \
36 17-18-19-20-21-22-23-24-25-26 51 -2
/ \
37-38-39-40-41-42-43-44-45-46-47-48-49-50 -3
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The perfect squares 1,4,9,16 fall one before the bottom left corner of each loop, and the
pronic numbers 2,6,12,20,30,etc are the vertical upwards from X=1,Y=0.
Square Spiral
This spiral goes around at the same rate as the "SquareSpiral". It's as if two corners
are cut off (like the "DiamondSpiral") and two others extended (like the
"OctagramSpiral"). The net effect is the same looping rate but the points pushed around a
bit.
Taking points up to a perfect square shows the similarity. The two triangular cut-off
corners marked by "."s are matched by the two triangular extensions.
+--------------------+ 7x7 square
| . . . 31 . . .|
| . . 32 13 30 . .|
| . 33 14 3 12 29 .|
|34 15 4 1 2 11 28|
35|16 5 6 7 8 9 10|27
36 17|18 19 20 21 22 23 24|25 26
37 38 39|40 41 42 43 44 45 46|47 48 49
+--------------------+
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start, with the same shape etc. For example to start at 0,
12 n_start => 0
/ \
13 2 11
/ / \ \
14 3 0--1 10
/ / \
15 4--5--6--7--8--9
/
16-17-18-19-20-21-22-...
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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 South-East 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
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 South-West
A033954 N on diagonal South-East, decagonal second kind
n_start=2
A185669 N on diagonal South-East
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidRows,
Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015 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/>.