1Math::PlanePath::CellulUasreRrulCeo5n4t(r3i)buted Perl DMoactuhm:e:nPtlaatnieoPnath::CellularRule54(3)
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6 Math::PlanePath::CellularRule54 -- cellular automaton points
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9 use Math::PlanePath::CellularRule54;
10 my $path = Math::PlanePath::CellularRule54->new;
11 my ($x, $y) = $path->n_to_xy (123);
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14 This is the pattern of Stephen Wolfram's "rule 54" cellular automaton
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16 <http://mathworld.wolfram.com/Rule54.html>
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18 arranged as rows,
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20 29 30 31 . 32 33 34 . 35 36 37 . 38 39 40 7
21 25 . . . 26 . . . 27 . . . 28 6
22 16 17 18 . 19 20 21 . 22 23 24 5
23 13 . . . 14 . . . 15 4
24 7 8 9 . 10 11 12 3
25 5 . . . 6 2
26 2 3 4 1
27 1 <- Y=0
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29 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
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31 The initial figure N=1,2,3,4 repeats in two-row groups with 1 cell gap
32 between figures. Each two-row group has one extra figure, for a step
33 of 4 more points than the previous two-row.
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35 The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers
36 N=1,6,15,28, etc k*(2k-1). The hexagonal numbers of the "second kind"
37 1, 3, 10, 21, 36, etc j*(2j+1) are a steep sloping line upwards in the
38 middle too. Those two taken together are the triangular numbers
39 1,3,6,10,15 etc, k*(k+1)/2.
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41 The 18-gonal numbers 18,51,100,etc are the vertical line at X=-3 on
42 every fourth row Y=5,9,13,etc.
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44 Row Ranges
45 The left end of each row is
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47 Nleft = Y*(Y+2)/2 + 1 if Y even
48 Y*(Y+1)/2 + 1 if Y odd
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50 The right end is
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52 Nright = (Y+1)*(Y+2)/2 if Y even
53 (Y+1)*(Y+3)/2 if Y odd
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55 = Nleft(Y+1) - 1 ie. 1 before next Nleft
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57 The row width Xmax-Xmin is 2*Y but with the gaps the number of visited
58 points in a row is less than that, being either about 1/4 or 3/4 of the
59 width on even or odd rows.
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61 rowpoints = Y/2 + 1 if Y even
62 3*(Y+1)/2 if Y odd
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64 For any Y of course the Nleft to Nright difference is the number of
65 points in the row too
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67 rowpoints = Nright - Nleft + 1
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69 N Start
70 The default is to number points starting N=1 as shown above. An
71 optional "n_start" can give a different start, in the same pattern.
72 For example to start at 0,
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74 n_start => 0
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76 15 16 17 18 19 20 21 22 23 5
77 12 13 14 4
78 6 7 8 9 10 11 3
79 4 5 2
80 1 2 3 1
81 0 <- Y=0
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83 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
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86 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
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89 "$path = Math::PlanePath::CellularRule54->new ()"
90 "$path = Math::PlanePath::CellularRule54->new (n_start => $n)"
91 Create and return a new path object.
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93 "($x,$y) = $path->n_to_xy ($n)"
94 Return the X,Y coordinates of point number $n on the path.
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96 "$n = $path->xy_to_n ($x,$y)"
97 Return the point number for coordinates "$x,$y". $x and $y are
98 each rounded to the nearest integer, which has the effect of
99 treating each cell as a square of side 1. If "$x,$y" is outside
100 the pyramid or on a skipped cell the return is "undef".
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103 This pattern is in Sloane's Online Encyclopedia of Integer Sequences in
104 a couple of forms,
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106 <http://oeis.org/A118108> (etc)
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108 A118108 whole-row used cells as bits of a bignum
109 A118109 1/0 used and unused cells across rows
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112 Math::PlanePath, Math::PlanePath::CellularRule,
113 Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190,
114 Math::PlanePath::PyramidRows
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116 Cellular::Automata::Wolfram
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118 <http://mathworld.wolfram.com/Rule54.html>
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121 <http://user42.tuxfamily.org/math-planepath/index.html>
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124 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
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126 This file is part of Math-PlanePath.
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128 Math-PlanePath is free software; you can redistribute it and/or modify
129 it under the terms of the GNU General Public License as published by
130 the Free Software Foundation; either version 3, or (at your option) any
131 later version.
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133 Math-PlanePath is distributed in the hope that it will be useful, but
134 WITHOUT ANY WARRANTY; without even the implied warranty of
135 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
136 General Public License for more details.
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138 You should have received a copy of the GNU General Public License along
139 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
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143perl v5.30.0 2019-08-17Math::PlanePath::CellularRule54(3)