.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" 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 >0, 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 "2021-01-23" "perl v5.32.0" "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, 2017, 2018, 2019, 2020 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 .