1Math::PlanePath::HexArmUss(e3r)Contributed Perl DocumentMaattiho:n:PlanePath::HexArms(3)
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NAME

6       Math::PlanePath::HexArms -- six spiral arms
7

SYNOPSIS

9        use Math::PlanePath::HexArms;
10        my $path = Math::PlanePath::HexArms->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This path follows six spiral arms, each advancing successively,
15
16                                          ...--66                      5
17                                                 \
18                    67----61----55----49----43    60                   4
19                   /                         \      \
20                ...    38----32----26----20    37    54                3
21                      /                    \     \     \
22                    44    21----15---- 9    14    31    48   ...       2
23                   /     /              \      \    \     \     \
24                 50    27    10---- 4     3     8    25    42    65    1
25                 /    /     /                 /     /     /     /
26              56    33    16     5     1     2    19    36    59    <-Y=0
27             /     /     /     /        \        /     /     /
28           62    39    22    11     6     7----13    30    53         -1
29             \     \     \     \     \              /     /
30             ...    45    28    17    12----18----24    47            -2
31                      \     \     \                    /
32                       51    34    23----29----35----41   ...         -3
33                         \     \                          /
34                          57    40----46----52----58----64            -4
35                            \
36                             63--...                                  -5
37
38            ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
39           -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9
40
41       The X,Y points are integers using every second position to give a
42       triangular lattice, per "Triangular Lattice" in Math::PlanePath.
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44       Each arm is N=6*k+rem for a remainder rem=0,1,2,3,4,5, so sequences
45       related to multiples of 6 or with a modulo 6 pattern may fall on
46       particular arms.
47
48   Abundant Numbers
49       The "abundant" numbers are those N with sum of proper divisors > N.
50       For example 12 is abundant because it's divisible by 1,2,3,4,6 and
51       their sum is 16.  All multiples of 6 starting from 12 are abundant.
52       Plotting the abundant numbers on the path gives the 6*k arm and some
53       other points in between,
54
55                       * * * * * * * * * * * *   *   *   ...
56                      *                       *           *
57                     *   *   *           *     *   *       *
58                    *                           *           *
59                   *           *                 *           *
60                  *                           *   *           *
61                 *           * * * * * *           *       *   *
62                *           *           *   *       *           *
63               *   *   *   *         *   *           *       *   *
64              *           *               *   *   *   *           *
65             *   *   *   *                 *           *   *       *
66            *           *   *             *   *       *           *
67           *       *   *                 *           *           *
68            *           *           * * *           *           *
69             *           *                 *       *           *
70              *   *       *   *   *           *   *           *
71               *           *                     *   *       *
72                *           *       *           *           *
73                 *   *       *                 *   *   *   *
74                  *           * * * * * * * * *           *
75                   *   *                         *       *
76                    *         *       *                 *
77                     *   *                         *   *
78                      *         *       *       *     *
79                       *                             *
80                        * * * * * * * * * * * * * * *
81
82       There's blank arms either side of the 6*k because 6*k+1 and 6*k-1 are
83       not abundant until some fairly big values.  The first abundant 6*k+1
84       might be 5,391,411,025, and the first 6*k-1 might be 26,957,055,125.
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FUNCTIONS

87       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
88       classes.
89
90       "$path = Math::PlanePath::HexArms->new ()"
91           Create and return a new square spiral object.
92
93       "($x,$y) = $path->n_to_xy ($n)"
94           Return the X,Y coordinates of point number $n on the path.
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96           For "$n < 1" the return is an empty list, as the path starts at 1.
97
98           Fractional $n gives a point on the line between $n and "$n+6", that
99           "$n+6" being the next on the same spiralling arm.  This is probably
100           of limited use, but arises fairly naturally from the calculation.
101
102   Descriptive Methods
103       "$arms = $path->arms_count()"
104           Return 6.
105

SEE ALSO

107       Math::PlanePath, Math::PlanePath::SquareArms,
108       Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral
109

HOME PAGE

111       <http://user42.tuxfamily.org/math-planepath/index.html>
112

LICENSE

114       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
115       Kevin Ryde
116
117       This file is part of Math-PlanePath.
118
119       Math-PlanePath is free software; you can redistribute it and/or modify
120       it under the terms of the GNU General Public License as published by
121       the Free Software Foundation; either version 3, or (at your option) any
122       later version.
123
124       Math-PlanePath is distributed in the hope that it will be useful, but
125       WITHOUT ANY WARRANTY; without even the implied warranty of
126       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
127       General Public License for more details.
128
129       You should have received a copy of the GNU General Public License along
130       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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134perl v5.34.0                      2022-01-21       Math::PlanePath::HexArms(3)
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