Math::PlanePath::OctagramSpiral(3pm)

1Math::PlanePath::OctagrUasmeSrpiCroanlt(r3i)buted Perl DMoactuhm:e:nPtlaatnieoPnath::OctagramSpiral(3)
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NAME

6       Math::PlanePath::OctagramSpiral -- integer points drawn around an
7       octagram
8

SYNOPSIS

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

15       This path makes a spiral around an octagram (8-pointed star),
16
17                 29          25                 4
18                  | \       / |
19                 30 28    26 24    ...56-55     3
20                  |   \  /    |         /
21           33-32-31  7 27  5 23-22-21 54        2
22             \       |\  / |      /  /
23              34  9- 8  6  4- 3 20 53           1
24                \  \        /  /  /
25                 35 10  1--2 19 52          <- Y=0
26                /  /           \  \
27              36 11-12 14 16-17-18 51          -1
28             /       |/  \ |         \
29           37-38-39 13 43 15 47-48-49-50       -2
30                  |   /  \    |
31                 40 42    44 46                -3
32                  |/        \ |
33                 41          45                -4
34
35                        ^
36           -4 -3 -2 -1 X=0 1  2  3  4  5 ...
37
38       Each loop is 16 longer than the previous.  The 18-gonal numbers
39       18,51,100,etc fall on the horizontal at Y=-1.
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41       The inner corners like 23, 31, 39, 47 are similar to the "SquareSpiral"
42       path, but instead of going directly between them the octagram takes a
43       detour out to make the points of the star.  Those excursions make each
44       loops 8 longer (1 per excursion), hence a step of 16 here as compared
45       to 8 for the "SquareSpiral".
46
47   N Start
48       The default is to number points starting N=1 as shown above.  An
49       optional "n_start" can give a different start, in the same pattern.
50       For example to start at 0,
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52           n_start => 0
53
54                 28          24
55                 29 27    25 23   ... 55 54
56           32 31 30  6 26  4 22 21 20 53
57              33  8  7  5  3  2 19 52
58                 34  9  0  1 18 51
59              35 10 11 13 15 16 17 50
60           36 37 38 12 42 14 46 47 48 49
61                 39 41    43 45
62                 40          44
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FUNCTIONS

65       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
66       classes.
67
68       "$path = Math::PlanePath::OctagramSpiral->new ()"
69           Create and return a new octagram spiral object.
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71       "($x,$y) = $path->n_to_xy ($n)"
72           Return the X,Y coordinates of point number $n on the path.
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74           For "$n < 1" the return is an empty list, it being considered the
75           path starts at 1.
76
77       "$n = $path->xy_to_n ($x,$y)"
78           Return the point number for coordinates "$x,$y".  $x and $y are
79           each rounded to the nearest integer, which has the effect of
80           treating each N in the path as centred in a square of side 1, so
81           the entire plane is covered.
82

FORMULAS

84   X,Y to N
85       The symmetry of the octagram can be used by rotating a given X,Y back
86       to the first star excursion such as N=19 to N=23.  If Y is negative
87       then rotate back by 180 degrees, then if X is negative rotate back by
88       90, and if Y>=X then by a further 45 degrees.  Each such rotation, if
89       needed, is counted as a multiple of the side-length to be added to the
90       final N.  For example at N=19 the side length is 2.  Rotating by 180
91       degrees is 8 side lengths, by 90 degrees 4 sides, and by 45 degrees is
92       2 sides.
93

OEIS

95       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
96       this path include
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98           <http://oeis.org/A125201> (etc)
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100           n_start=1 (the default)
101             A125201    N on X axis, from X=1 onwards, 18-gonals + 1
102             A194268    N on diagonal South-East
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104           n_start=0
105             A051870    N on X axis, 18-gonal numbers
106             A139273    N on Y axis
107             A139275    N on X negative axis
108             A139277    N on Y negative axis
109             A139272    N on diagonal X=Y
110             A139274    N on diagonal North-West
111             A139276    N on diagonal South-West
112             A139278    N on diagonal South-East, second 18-gonals
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HOME PAGE

119       <http://user42.tuxfamily.org/math-planepath/index.html>
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LICENSE

122       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
123       2020 Kevin Ryde
124
125       This file is part of Math-PlanePath.
126
127       Math-PlanePath is free software; you can redistribute it and/or modify
128       it under the terms of the GNU General Public License as published by
129       the Free Software Foundation; either version 3, or (at your option) any
130       later version.
131
132       Math-PlanePath is distributed in the hope that it will be useful, but
133       WITHOUT ANY WARRANTY; without even the implied warranty of
134       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
135       General Public License for more details.
136
137       You should have received a copy of the GNU General Public License along
138       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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142perl v5.32.1                      2021-01-27Math::PlanePath::OctagramSpiral(3)