1Math::PlanePath::CellulUasreRrulCeo(n3t)ributed Perl DocMuamtehn:t:aPtliaonnePath::CellularRule(3)
2
6 Math::PlanePath::CellularRule -- cellular automaton points of binary
7 rule
8
10 use Math::PlanePath::CellularRule;
11 my $path = Math::PlanePath::CellularRule->new (rule => 30);
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This is the patterns of Stephen Wolfram's bit-rule based cellular
16 automatons
17 <http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
19 Points are numbered left to right in rows so for example
21 rule => 30
23 51 52 53 54 55 56 57 58 59 60 61 62 9
25 44 45 46 47 48 49 50 8
26 32 33 34 35 36 37 38 39 40 41 42 43 7
27 27 28 29 30 31 6
28 18 19 20 21 22 23 24 25 26 5
29 14 15 16 17 4
30 8 9 10 11 12 13 3
31 5 6 7 2
32 2 3 4 1
33 1 <- Y=0
34 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
36 The automaton starts from a single point N=1 at the origin and grows
38 into the rows above. The "rule" parameter controls how the 3 cells
39 below and diagonally below produce a new cell,
40 +-----+
42 | new | next row, Y+1
43 +-----+
44 ^ ^ ^
45 / | \
46 / | \
47 +-----+ +-----+ +-----+
48 | A | | B | | C | row Y
49 +-----+ +-----+ +-----+
50 There's 8 possible combinations of ABC being each 0 or 1. Each such
52 combination can become 0 or 1 in the "new" cell. Those 0 or 1 for
53 "new" is encoded as 8 bits to make a rule number 0 to 255,
54 ABC cells below new cell bit from rule
56 1,1,1 -> bit7
58 1,1,0 -> bit6
59 1,0,1 -> bit5
60 ...
61 0,0,1 -> bit1
62 0,0,0 -> bit0
63 When cells 0,0,0 become 1, ie. "rule" bit0 is 1 (an odd number), the
65 "off" cells either side of the initial N=1 become all "on" infinitely
66 to the sides. Or if rule bit7 for 1,1,1 is a 0 (ie. rule < 128) then
67 they turn on and off alternately in odd and even rows. In both cases
68 only the pyramid portion part -Y<=X<=Y is considered for the N points
69 but the infinite cells to the sides are included in the calculation.
70 The full set of patterns can be seen at the Math World page above, or
72 can be printed with the examples/cellular-rules.pl program. The
73 patterns range from simple to complex. For some the N=1 cell doesn't
74 grow at all such as rule 0 or rule 8. Some grow to mere straight lines
75 such as rule 2 or rule 5. Others make columns or patterns with
76 "quadratic" style stepping of 1 or 2 rows, or self-similar replications
77 such as the Sierpinski triangle of rule 18 and 60. Some rules have
78 complicated non-repeating patterns when there's feedback across from
79 one half to the other, such as rule 30.
80 For some rules there's specific PlanePath code which this class
82 dispatches to, such as "CellularRule54", "CellularRule57",
83 "CellularRule190" or "SierpinskiTriangle" (with "n_start=1").
84 For rules without specific code the current implementation is not
86 particularly efficient as it builds and holds onto the bit pattern for
87 all rows through to the highest N or X,Y used. There's no doubt better
88 ways to iterate an automaton, but this module offers the patterns in
89 PlanePath style.
90 N Start
92 The default is to number points starting N=1 as shown above. An
93 optional "n_start" can give a different start, in the same pattern.
94 For example to start at 0,
95 n_start => 0, rule => 62
97 18 19 20 21 22 23 24 25 5
99 13 14 15 16 17 4
100 7 8 9 10 11 12 3
101 4 5 6 2
102 1 2 3 1
103 0 <- Y=0
104 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
106
108 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
109 classes.
110 "$path = Math::PlanePath::CellularRule->new (rule => 123)"
112 "$path = Math::PlanePath::CellularRule->new (rule => 123, n_start =>
113 $n)"
114 Create and return a new path object. "rule" should be an integer 0
115 to 255. A "rule" should be given always. There is a default, but
116 it's secret and likely to change.
117 If there's specific PlanePath code implementing the pattern then
119 the returned object is from that class and generally is not
120 "isa('Math::PlanePath::CellularRule')".
121 "$n = $path->xy_to_n ($x,$y)"
123 Return the point number for coordinates "$x,$y". $x and $y are
124 each rounded to the nearest integer, which has the effect of
125 treating each cell as a square of side 1. If "$x,$y" is outside
126 the pyramid or on a skipped cell the return is "undef".
127
129 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
130 this path can be found in the OEIS index
131 <http://oeis.org/wiki/Index_to_OEIS:_Section_Ce#cell>
133 and in addition the following
135 <http://oeis.org/A061579> (etc)
137 rule=50,58,114,122,178,186,242,250, 179
139 (solid every second cell)
140 A061579 permutation N at -X,Y (mirror horizontal)
141
143 Math::PlanePath, Math::PlanePath::CellularRule54,
144 Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190,
145 Math::PlanePath::SierpinskiTriangle, Math::PlanePath::PyramidRows
146 Cellular::Automata::Wolfram
148 <http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
150
152 <http://user42.tuxfamily.org/math-planepath/index.html>
153
155 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
156 This file is part of Math-PlanePath.
158 Math-PlanePath is free software; you can redistribute it and/or modify
160 it under the terms of the GNU General Public License as published by
161 the Free Software Foundation; either version 3, or (at your option) any
162 later version.
163 Math-PlanePath is distributed in the hope that it will be useful, but
165 WITHOUT ANY WARRANTY; without even the implied warranty of
166 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
167 General Public License for more details.
168 You should have received a copy of the GNU General Public License along
170 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
171perl v5.32.0 2020-07-28 Math::PlanePath::CellularRule(3)