1Math::PlanePath::MPeaksU(s3e)r Contributed Perl DocumentaMtaitohn::PlanePath::MPeaks(3)
2
3
4

NAME

6       Math::PlanePath::MPeaks -- points in expanding M shape
7

SYNOPSIS

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

DESCRIPTION

14       This path puts points in layers of an "M" shape
15
16                41                              49         7
17                40  42                      48  50         6
18                39  22  43              47  28  51         5
19                38  21  23  44      46  27  29  52         4
20                37  20   9  24  45  26  13  30  53         3
21                36  19   8  10  25  12  14  31  54         2
22                35  18   7   2  11   4  15  32  55         1
23                34  17   6   1   3   5  16  33  56     <- Y=0
24
25                                 ^
26                -4  -3  -2  -1  X=0  1   2   3   4
27
28       N=1 to N=5 is the first "M" shape, then N=6 to N=16 on top of that,
29       etc.  The centre goes half way down.  Reckoning the N=1 to N=5 as layer
30       d=1 then
31
32           Xleft = -d
33           Xright = d
34           Ypeak = 2*d - 1
35           Ycentre = d - 1
36
37       Each "M" is 6 points longer than the preceding.  The verticals are each
38       2 longer, and the centre diagonals each 1 longer.  This step 6 is
39       similar to the "HexSpiral".
40
41       The octagonal numbers N=1,8,21,40,65,etc k*(3k-2) are a straight line
42       of slope 2 going up to the left.  The octagonal numbers of the second
43       kind N=5,16,33,56,etc k*(3k+2) are along the X axis to the right.
44
45   N Start
46       The default is to number points starting N=1 as shown above.  An
47       optional "n_start" can give a different start, in the same pattern.
48       For example to start at 0,
49
50           n_start => 0
51
52           40                              48
53           39  41                      47  49
54           38  21  42              46  27  50
55           37  20  22  43      45  26  28  51
56           36  19   8  23  44  25  12  29  52
57           35  18   7   9  24  11  13  30  53
58           34  17   6   1  10   3  14  31  54
59           33  16   5   0   2   4  15  32  55
60

FUNCTIONS

62       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
63       classes.
64
65       "$path = Math::PlanePath::MPeaks->new ()"
66           Create and return a new path object.
67
68       "($x,$y) = $path->n_to_xy ($n)"
69           Return the X,Y coordinates of point number $n on the path.
70
71           For "$n < 0.5" the return is an empty list, it being considered
72           there are no negative points.
73
74       "$n = $path->xy_to_n ($x,$y)"
75           Return the point number for coordinates "$x,$y".  $x and $y are
76           each rounded to the nearest integer which has the effect of
77           treating points as a squares of side 1, so the half-plane y>=-0.5
78           is entirely covered.
79

OEIS

81       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
82       this path include
83
84           <http://oeis.org/A045944> (etc)
85
86           n_start=1 (the default)
87             A045944    N on X axis >= 1, extra initial 0
88                          being octagonal numbers second kind
89             A056106    N on Y axis, extra initial 1
90             A056109    N on X negative axis <= -1
91
92           n_start=0
93             A049450    N on Y axis, extra initial 0, 2*pentagonal
94
95           n_start=2
96             A027599    N on Y axis, extra initial 6,2
97

SEE ALSO

99       Math::PlanePath, Math::PlanePath::PyramidSides
100

HOME PAGE

102       <http://user42.tuxfamily.org/math-planepath/index.html>
103

LICENSE

105       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
106       Kevin Ryde
107
108       This file is part of Math-PlanePath.
109
110       Math-PlanePath is free software; you can redistribute it and/or modify
111       it under the terms of the GNU General Public License as published by
112       the Free Software Foundation; either version 3, or (at your option) any
113       later version.
114
115       Math-PlanePath is distributed in the hope that it will be useful, but
116       WITHOUT ANY WARRANTY; without even the implied warranty of
117       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
118       General Public License for more details.
119
120       You should have received a copy of the GNU General Public License along
121       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
122
123
124
125perl v5.36.0                      2023-01-20        Math::PlanePath::MPeaks(3)
Impressum