NAME¶
Math::PlanePath::HexArms -- six spiral arms
SYNOPSIS¶
use Math::PlanePath::HexArms;
my $path = Math::PlanePath::HexArms->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This path follows six spiral arms, each advancing successively,
...--66 5
\
67----61----55----49----43 60 4
/ \ \
... 38----32----26----20 37 54 3
/ \ \ \
44 21----15---- 9 14 31 48 ... 2
/ / \ \ \ \ \
50 27 10---- 4 3 8 25 42 65 1
/ / / / / / /
56 33 16 5 1 2 19 36 59 <-Y=0
/ / / / \ / / /
62 39 22 11 6 7----13 30 53 -1
\ \ \ \ \ / /
... 45 28 17 12----18----24 47 -2
\ \ \ /
51 34 23----29----35----41 ... -3
\ \ /
57 40----46----52----58----64 -4
\
63--... -5
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The X,Y points are integers using every second position to give a triangular
lattice, per "Triangular Lattice" in Math::PlanePath.
Each arm is N=6*k+rem for a remainder rem=0,1,2,3,4,5, so sequences related to
multiples of 6 or with a modulo 6 pattern may fall on particular arms.
Abundant Numbers¶
The "abundant" numbers are those N with sum of proper divisors > N.
For example 12 is abundant because it's divisible by 1,2,3,4,6 and their sum
is 16. All multiples of 6 starting from 12 are abundant. Plotting the abundant
numbers on the path gives the 6*k arm and some other points in between,
* * * * * * * * * * * * * * ...
* * *
* * * * * * *
* * *
* * * *
* * * *
* * * * * * * * * *
* * * * * *
* * * * * * * * *
* * * * * * *
* * * * * * * *
* * * * * * *
* * * * * *
* * * * * * *
* * * * *
* * * * * * * *
* * * * *
* * * * *
* * * * * * *
* * * * * * * * * * *
* * * *
* * * *
* * * *
* * * * *
* *
* * * * * * * * * * * * * * *
There's blank arms either side of the 6*k because 6*k+1 and 6*k-1 are not
abundant until some fairly big values. The first abundant 6*k+1 might be
5,391,411,025, and the first 6*k-1 might be 26,957,055,125.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$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 "$n+6" being the next on the same spiralling arm. This is
probably of limited use, but arises fairly naturally from the
calculation.
Descriptive Methods¶
- "$arms = $path->arms_count()"
- Return 6.
SEE ALSO¶
Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms,
Math::PlanePath::HexSpiral
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2011, 2012, 2013, 2014 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/>.