NAME
Math::PlanePath::HexArms  six spiral arms
SYNOPSIS
use Math::PlanePath::HexArms;
my $path = Math::PlanePath::HexArms>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path follows six spiral arms, each advancing successively,
...66 5
\
6761554943 60 4
/ \ \
... 38322620 37 54 3
/ \ \ \
44 2115 9 14 31 48 ... 2
/ / \ \ \ \ \
50 27 10 4 3 8 25 42 65 1
/ / / / / / /
56 33 16 5 1 2 19 36 59 <Y=0
/ / / / \ / / /
62 39 22 11 6 713 30 53 1
\ \ \ \ \ / /
... 45 28 17 121824 47 2
\ \ \ /
51 34 23293541 ... 3
\ \ /
57 4046525864 4
\
63... 5
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
9 8 7 6 5 4 3 2 1 X=0 1 2 3 4 5 6 7 8 9
The X,Y points are integers using every second position to give a triangular lattice, per "Triangular Lattice" in Math::PlanePath.
Each arm is N=6*k+rem for a remainder rem=0,1,2,3,4,5, so sequences related to multiples of 6 or with a modulo 6 pattern may fall on particular arms.
Abundant Numbers
The "abundant" numbers are those N with sum of proper divisors > N. For example 12 is abundant because it's divisible by 1,2,3,4,6 and their sum is 16. All multiples of 6 starting from 12 are abundant. Plotting the abundant numbers on the path gives the 6*k arm and some other points in between,
* * * * * * * * * * * * * * ...
* * *
* * * * * * *
* * *
* * * *
* * * *
* * * * * * * * * *
* * * * * *
* * * * * * * * *
* * * * * * *
* * * * * * * *
* * * * * * *
* * * * * *
* * * * * * *
* * * * *
* * * * * * * *
* * * * *
* * * * *
* * * * * * *
* * * * * * * * * * *
* * * *
* * * *
* * * *
* * * * *
* *
* * * * * * * * * * * * * * *
There's blank arms either side of the 6*k because 6*k+1 and 6*k1 are not abundant until some fairly big values. The first abundant 6*k+1 might be 5,391,411,025, and the first 6*k1 might be 26,957,055,125.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::HexArms>new ()

Create and return a new square spiral object.
($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path.For
$n < 1
the return is an empty list, as the path starts at 1.Fractional
$n
gives a point on the line between$n
and$n+6
, that$n+6
being the next on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation.
Descriptive Methods
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019 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/>.