1Math::PlanePath::PentSpUisrearlSCkoenwterdi(b3u)ted PerlMaDtohc:u:mPelnatnaetPiaotnh::PentSpiralSkewed(3)
2
6 Math::PlanePath::PentSpiralSkewed -- integer points in a pentagonal
7 shape
8
10 use Math::PlanePath::PentSpiralSkewed;
11 my $path = Math::PlanePath::PentSpiralSkewed->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This path makes a pentagonal (five-sided) spiral with points skewed so
16 as to fit a square grid and fully cover the plane.
17 10 ... 2
19 / \ \
20 11 3 9 20 1
21 / / \ \ \
22 12 4 1--2 8 19 <- Y=0
23 \ \ | |
24 13 5--6--7 18 -1
25 \ |
26 14-15-16-17 -2
27 ^ ^ ^ ^ ^ ^
29 -2 -1 X=0 1 2 3 ...
30 The pattern is similar to the "SquareSpiral" but cuts three corners
32 which makes each cycle is faster. Each cycle is just 5 steps longer
33 than the previous (where it's 8 for a "SquareSpiral").
34 N Start
36 The default is to number points starting N=1 as shown above. An
37 optional "n_start" can give a different start, in the same pattern.
38 For example to start at 0,
39 38 n_start => 0
41 39 21 37 ...
42 40 22 9 20 36 57
43 41 23 10 2 8 19 35 56
44 42 24 11 3 0 1 7 18 34 55
45 43 25 12 4 5 6 17 33 54
46 44 26 13 14 15 16 32 53
47 45 27 28 29 30 31 52
48 46 47 48 49 50 51
49
51 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
52 classes.
53 "$path = Math::PlanePath::PentSpiral->new ()"
55 "$path = Math::PlanePath::PentSpiral->new (n_start => $n)"
56 Create and return a new path object.
57 "$n = $path->xy_to_n ($x,$y)"
59 Return the point number for coordinates "$x,$y". $x and $y are
60 each rounded to the nearest integer, which has the effect of
61 treating each point in the path as a square of side 1.
62
64 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
65 this path include
66 <http://oeis.org/A192136> (etc)
68 n_start=1 (the default)
70 A192136 N on X axis, (5*n^2 - 3*n + 2)/2
71 A140066 N on Y axis
72 A116668 N on X negative axis, (5n^2 + n + 2)/2
73 A134238 N on Y negative axis
74 A158187 N on North-West diagonal, 10*n^2 + 1
75 A005891 N on South-East diagonal, centred pentagonals
76 n_start=0
78 A000566 N on X axis, heptagonal numbers
79 A005476 N on Y axis
80 A005475 N on X negative axis
81 A147875 N on Y negative axis, second heptagonals
82 A033583 N on North-West diagonal, 10*n^2
83 A028895 N on South-East diagonal, 5*triangular
84
86 Math::PlanePath, Math::PlanePath::SquareSpiral,
87 Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed
88
90 <http://user42.tuxfamily.org/math-planepath/index.html>
91
93 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
94 2020 Kevin Ryde
95 This file is part of Math-PlanePath.
97 Math-PlanePath is free software; you can redistribute it and/or modify
99 it under the terms of the GNU General Public License as published by
100 the Free Software Foundation; either version 3, or (at your option) any
101 later version.
102 Math-PlanePath is distributed in the hope that it will be useful, but
104 WITHOUT ANY WARRANTY; without even the implied warranty of
105 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
106 General Public License for more details.
107 You should have received a copy of the GNU General Public License along
109 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
110perl v5.38.0 2023-07-2M0ath::PlanePath::PentSpiralSkewed(3)