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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::HexArms \-\- six spiral arms .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HexArms; \& my $path = Math::PlanePath::HexArms\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows six spiral arms, each advancing successively, .PP .Vb 10 \& ...\-\-66 5 \& \e \& 67\-\-\-\-61\-\-\-\-55\-\-\-\-49\-\-\-\-43 60 4 \& / \e \e \& ... 38\-\-\-\-32\-\-\-\-26\-\-\-\-20 37 54 3 \& / \e \e \e \& 44 21\-\-\-\-15\-\-\-\- 9 14 31 48 ... 2 \& / / \e \e \e \e \e \& 50 27 10\-\-\-\- 4 3 8 25 42 65 1 \& / / / / / / / \& 56 33 16 5 1 2 19 36 59 <\-Y=0 \& / / / / \e / / / \& 62 39 22 11 6 7\-\-\-\-13 30 53 \-1 \& \e \e \e \e \e / / \& ... 45 28 17 12\-\-\-\-18\-\-\-\-24 47 \-2 \& \e \e \e / \& 51 34 23\-\-\-\-29\-\-\-\-35\-\-\-\-41 ... \-3 \& \e \e / \& 57 40\-\-\-\-46\-\-\-\-52\-\-\-\-58\-\-\-\-64 \-4 \& \e \& 63\-\-... \-5 \& \& ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \& \-9 \-8 \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 8 9 .Ve .PP The X,Y points are integers using every second position to give a triangular lattice, per \*(L"Triangular Lattice\*(R" in Math::PlanePath. .PP 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. .SS "Abundant Numbers" .IX Subsection "Abundant Numbers" The \*(L"abundant\*(R" 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, .PP .Vb 10 \& * * * * * * * * * * * * * * ... \& * * * \& * * * * * * * \& * * * \& * * * * \& * * * * \& * * * * * * * * * * \& * * * * * * \& * * * * * * * * * \& * * * * * * * \& * * * * * * * * \& * * * * * * * \& * * * * * * \& * * * * * * * \& * * * * * \& * * * * * * * * \& * * * * * \& * * * * * \& * * * * * * * \& * * * * * * * * * * * \& * * * * \& * * * * \& * * * * \& * * * * * \& * * \& * * * * * * * * * * * * * * * .Ve .PP 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. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::HexArms\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexArms\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HexArms->new ()" Create and return a new square spiral object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .Sp For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, as the path starts at 1. .Sp Fractional \f(CW$n\fR gives a point on the line between \f(CW$n\fR and \f(CW\*(C`$n+6\*(C'\fR, that \&\f(CW\*(C`$n+6\*(C'\fR being the next on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation. .SS "Descriptive Methods" .IX Subsection "Descriptive Methods" .ie n .IP """$arms = $path\->arms_count()""" 4 .el .IP "\f(CW$arms = $path\->arms_count()\fR" 4 .IX Item "$arms = $path->arms_count()" Return 6. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .