1Math::PlanePath::AnvilSUpsierralC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::AnvilSpiral(3)
2
6 Math::PlanePath::AnvilSpiral -- integer points around an "anvil" shape
7
9 use Math::PlanePath::AnvilSpiral;
10 my $path = Math::PlanePath::AnvilSpiral->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path makes a spiral around an anvil style shape,
15 ...-78-77-76-75-74 4
17 /
18 49-48-47-46-45-44-43-42-41-40-39-38 73 3
19 \ / /
20 50 21-20-19-18-17-16-15-14 37 72 2
21 \ \ / / /
22 51 22 5--4--3--2 13 36 71 1
23 \ \ \ / / / /
24 52 23 6 1 12 35 70 <- Y=0
25 / / / \ \ \
26 53 24 7--8--9-10-11 34 69 -1
27 / / \ \
28 54 25-26-27-28-29-30-31-32-33 68 -2
29 / \
30 55-56-57-58-59-60-61-62-63-64-65-66-67 -3
31 ^
33 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
34 The pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2 fall
36 alternately on the X axis X>0, and on the Y=1 horizontal X<0.
37 Those pentagonals are always composites, from the factorization shown,
39 and as noted in "Step 3 Pentagonals" in Math::PlanePath::PyramidRows,
40 the immediately preceding P(k)-1 and P(k)-2 are also composites. So
41 plotting the primes on the spiral has a 3-high horizontal blank line at
42 Y=0,-1,-2 for positive X, and Y=1,2,3 for negative X (after the first
43 few values).
44 Each loop around the spiral is 12 longer than the preceding. This is
46 4* more than the step=3 "PyramidRows" so straight lines on a
47 "PyramidRows" like these pentagonals are also straight lines here, but
48 split into two parts.
49 The outward diagonal excursions are similar to the "OctagramSpiral",
51 but there's just 4 of them here where the "OctagramSpiral" has 8. This
52 is reflected in the loop step. The basic "SquareSpiral" is step 8, but
53 by taking 4 excursions here increases that to 12, and in the
54 "OctagramSpiral" 8 excursions adds 8 to make step 16.
55 Wider
57 An optional "wider" parameter makes the path wider by starting with a
58 horizontal section of given width. For example
59 $path = Math::PlanePath::SquareSpiral->new (wider => 3);
61 gives
63 33-32-31-30-29-28-27-26-25-24-23 ... 2
65 \ / /
66 34 11-10--9--8--7--6--5 22 51 1
67 \ \ / / /
68 35 12 1--2--3--4 21 50 <- Y=0
69 / / \ \
70 36 13-14-15-16-17-18-19-20 49 -1
71 / \
72 37-38-39-40-41-42-43-44-45-46-47-48 -2
73 ^
75 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
76 The starting point 1 is shifted to the left by ceil(wider/2) places to
78 keep the spiral centred on the origin X=0,Y=0. This is the same
79 starting offset as the "SquareSpiral" "wider".
80 Widening doesn't change the nature of the straight lines which arise,
82 it just rotates them around. Each loop is still 12 longer than the
83 previous, since the widening is essentially a constant amount in each
84 loop.
85 N Start
87 The default is to number points starting N=1 as shown above. An
88 optional "n_start" can give a different start with the same shape. For
89 example to start at 0,
90 n_start => 0
92 20-19-18-17-16-15-14-13 ...
94 \ / /
95 21 4--3--2--1 12 35
96 \ \ / / /
97 22 5 0 11 34
98 / / \ \
99 23 6--7--8--9-10 33
100 / \
101 24-25-26-27-28-29-30-31-32
102 The only effect is to push the N values around by a constant amount.
104 It might help match coordinates with something else zero-based.
105
107 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
108 classes.
109 "$path = Math::PlanePath::AnvilSpiral->new ()"
111 "$path = Math::PlanePath::AnvilSpiral->new (wider => $integer, n_start
112 => $n)"
113 Create and return a new anvil spiral object. An optional "wider"
114 parameter widens the spiral path, it defaults to 0 which is no
115 widening.
116
118 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
119 this path include
120 <http://oeis.org/A033581> (etc)
122 default wider=0, n_start=1
124 A033570 N on X axis, alternate pentagonals (2n+1)*(3n+1)
125 A126587 N on Y axis
126 A136392 N on Y negative (n=-Y+1)
127 A033568 N on X=Y diagonal, alternate second pents (2*n-1)*(3*n-1)
128 A085473 N on south-east diagonal
129 wider=0, n_start=0
131 A211014 N on X axis, 14-gonal numbers of the second kind
132 A139267 N on Y axis, 2*octagonal
133 A049452 N on X negative, alternate pentagonals
134 A033580 N on Y negative, 4*pentagonals
135 A051866 N on X=Y diagonal, 14-gonal numbers
136 A094159 N on north-west diagonal, 3*hexagonals
137 A049453 N on south-west diagonal, alternate second pentagonal
138 A195319 N on south-east diagonal, 3*second hexagonals
139 wider=1, n_start=0
141 A051866 N on X axis, 14-gonal numbers
142 A049453 N on Y negative, alternate second pentagonal
143 A033569 N on north-west diagonal
144 A085473 N on south-west diagonal
145 A080859 N on Y negative
146 A033570 N on south-east diagonal
147 alternate pentagonals (2n+1)*(3n+1)
148 wider=2, n_start=1
150 A033581 N on Y axis (6*n^2) except for initial N=2
151
153 Math::PlanePath, Math::PlanePath::SquareSpiral,
154 Math::PlanePath::OctagramSpiral, Math::PlanePath::HexSpiral
155
157 <http://user42.tuxfamily.org/math-planepath/index.html>
158
160 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
161 Kevin Ryde
162 This file is part of Math-PlanePath.
164 Math-PlanePath is free software; you can redistribute it and/or modify
166 it under the terms of the GNU General Public License as published by
167 the Free Software Foundation; either version 3, or (at your option) any
168 later version.
169 Math-PlanePath is distributed in the hope that it will be useful, but
171 WITHOUT ANY WARRANTY; without even the implied warranty of
172 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
173 General Public License for more details.
174 You should have received a copy of the GNU General Public License along
176 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
177perl v5.36.0 2022-07-22 Math::PlanePath::AnvilSpiral(3)