Math::PlanePath::ToothpickSpiral -- integer points in stair-step diagonal stripes
use Math::PlanePath::ToothpickSpiral; my $path = Math::PlanePath::ToothpickSpiral->new; my ($x, $y) = $path->n_to_xy (123);
This path is length=2 toothpicks placed in an anti-clockwise spiral. A single new toothpick is added at an end of the preceding. Each is as close to the origin as possible without toothpicks overlapping. Ends may touch, but no overlapping.
| 3---2--- | | | 5---4-- 1 ... | | | | ---6---7 -10--11 | | | --8---9 |
The result is a stair-step diamond spiral starting vertically. As per the other toothpick paths the vertical toothpicks are "even" points X=Ymod2 and horizontal toothpicks "odd" points X!=Ymod2.
19-18 ... 3 | | | 21-20 17-16 45-44 2 | | | 23-22 3--2 15-14 43-42 1 | | | | | 25-24 5--4 1 12-13 40-41 <- Y=0 | | | | 26-27 6--7 10-11 38-39 -1 | | | | 28-29 8--9 36-37 -2 | | 30-31 34-35 -3 | | 32-33 -4 ^ -4 -3 -2 -1 X=0 1 2 3 4
N=1,15,45,etc on the X=Y leading diagonal and N=6,28,66,etc on the X=Y-1 South-West diagonal are the hexagonal numbers k*(2k-1). The odd hexagonals are to the North-East and the even hexagonals to the South-West.
The hexagonal numbers of the "second kind" which are k*(2k-1) for k negative. They fall similarly on the X=-Y-1 North-West and X=-Y South-East diagonals.
The default is to number points starting N=1 as shown above. An optional n_start can give a different numbering of the same shape. For example to start at 0,
n_start
18-17 n_start => 0 | | 20-19 16-15 | | 22-21 2--1 14-13 | | | | 24-23 4--3 0 11-12 | | | 25-26 5--6 9-10 | | | 27-28 7--8 | ...
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ToothpickSpiral->new ()
$path = Math::PlanePath::ToothpickSpiral->new (n_start => $n)
Create and return a new staircase path object.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A059285 (etc)
n_start=1 (the default) A014634 N on diagonal X=Y, odd hexagonals A033567 N on diagonal North-West A185438 N on diagonal South-West A188135 N on diagonal South-East n_start=0 A033587 N on diagonal X=Y A014635 N on diagonal South-West, even hexagonals A033585 N on diagonal South-East
Math::PlanePath, Math::PlanePath::Staircase, Math::PlanePath::DiamondSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 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/>.
To install Math::PlanePath::HTree, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath::HTree
CPAN shell
perl -MCPAN -e shell install Math::PlanePath::HTree
For more information on module installation, please visit the detailed CPAN module installation guide.