.\" Automatically generated by Pod::Man 2.28 (Pod::Simple 3.29) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is turned on, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{ . if \nF \{ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::PlanePath::HexArms 3pm" .TH Math::PlanePath::HexArms 3pm "2016-05-03" "perl v5.22.2" "User Contributed Perl Documentation" .\" For nroff, turn off justification. 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, 2015, 2016 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 .