1Math::PlanePath::DekkinUgsCeerntCroenst(r3i)buted Perl DMoactuhm:e:nPtlaatnieoPnath::DekkingCentres(3)
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NAME

6       Math::PlanePath::DekkingCentres -- 5x5 self-similar
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SYNOPSIS

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

14       This is a variation on a 5x5 self-similar curve from
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16           F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume
17           44, 1982, pages 79-104, section 4.9 "Gosper-Type Curves"
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19       and which is a horizontal mirror image of the E-curve of McKenna 1978.
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21       The form visits the "centres" of the 5x5 self-similar unit squares of
22       the pattern.  The result is some diagonal steps, but replications
23       wholly within 5x5 areas.
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25                                     ...
26               |                     /
27             9 |  115-116 122-123-124  89--88  86--85--84
28               |    |   |    \          |    \  |       |
29             8 |  114 117-118 121-120  90  92  87  82--83
30               |    |        \   /      |/   \      |
31             7 |  113-112 106 119 102  91  94--93  81  77
32               |     /   /  |    /  |    /       /   /  |
33             6 |  111 107 105 103 101  95--96  80  78  76
34               |    |    \   \  |   |        \   \  |   |
35             5 |  110-109-108 104 100--99--98--97  79  75
36               |                                         \
37             4 |   10--11  13--14--15  35--36  38--39--40  74  70  66--65--64
38               |    |    \  |       |   |    \  |       |   |   |\   \      |
39             3 |    9   7  12  17--16  34  32  37  42--41  73  71  69  67  63
40               |    |/   \      |       |/   \      |       |/      |/   /
41             2 |    8   5-- 6  18  22  33  30--31  43  47  72  55  68  62--61
42               |      /      /   /  |    /       /   /  |    /   \          |
43             1 |    4-- 3  19  21  23  29--28  44  46  48  54--53  56--57  60
44               |         \   \  |   |        \   \  |   |        \      |   |
45           Y=0 |    0-- 1-- 2  20  24--25--26--27  45  49--50--51--52  58--59
46               +---------------------------------------------------------------
47                  X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14
48
49       The base pattern is the N=0 to N=24 section.  It repeats with rotations
50       or reversals which make the ends join.  For example N=75 to N=99 is the
51       base pattern in reverse.  Or N=50 to N=74 is reverse and also rotate by
52       -90.
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FUNCTIONS

55       See "FUNCTIONS" in Math::PlanePath the behaviour common to all path
56       classes.
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58       "$path = Math::PlanePath::DekkingCentres->new ()"
59           Create and return a new path object.
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61       "($x,$y) = $path->n_to_xy ($n)"
62           Return the X,Y coordinates of point number $n on the path.  Points
63           begin at 0 and if "$n < 0" then the return is an empty list.
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65   Level Methods
66       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
67           Return "(0, 25**$level - 1)".
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SEE ALSO

70       Math::PlanePath, Math::PlanePath::DekkingCurve,
71       Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve
72

HOME PAGE

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

77       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
78       Kevin Ryde
79
80       This file is part of Math-PlanePath.
81
82       Math-PlanePath is free software; you can redistribute it and/or modify
83       it under the terms of the GNU General Public License as published by
84       the Free Software Foundation; either version 3, or (at your option) any
85       later version.
86
87       Math-PlanePath is distributed in the hope that it will be useful, but
88       WITHOUT ANY WARRANTY; without even the implied warranty of
89       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
90       General Public License for more details.
91
92       You should have received a copy of the GNU General Public License along
93       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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97perl v5.34.0                      2022-01-21Math::PlanePath::DekkingCentres(3)
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