Math::PlanePath::HexArms -- six spiral arms
use Math::PlanePath::HexArms; my $path = Math::PlanePath::HexArms->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
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.
$arms = $path->arms_count()
Return 6.
Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This file is part of Math-PlanePath.
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