Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::CellularRule -- cellular automaton points of binary rule
SYNOPSIS
use Math::PlanePath::CellularRule;
my $path = Math::PlanePath::CellularRule->new (rule => 30);
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is the patterns of Stephen Wolfram's bit-rule cellular automatons
<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
Points are numbered left to right in rows so for example
rule => 30
51 52 53 54 55 56 57 58 59 60 61 62 9
44 45 46 47 48 49 50 8
32 33 34 35 36 37 38 39 40 41 42 43 7
27 28 29 30 31 6
18 19 20 21 22 23 24 25 26 5
14 15 16 17 4
8 9 10 11 12 13 3
5 6 7 2
2 3 4 1
1 <- Y=0
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The automaton starts from a single point N=1 at the origin and grows into the rows above.
The "rule" parameter controls how the 3 cells below and diagonally below produce a new
cell,
+-----+
| new | next row, Y+1
+-----+
^ ^ ^
/ | \
/ | \
+-----+ +-----+ +-----+
| A | | B | | C | row Y
+-----+ +-----+ +-----+
There's 8 combinations of ABC being each 0 or 1. Each such combination can become 0 or 1
in the "new" cell. Those 0 or 1 for "new" is encoded as 8 bits to make a rule number 0 to
255,
ABC cells below new cell bit from rule
1,1,1 -> bit7
1,1,0 -> bit6
1,0,1 -> bit5
...
0,0,1 -> bit1
0,0,0 -> bit0
When cells 0,0,0 become 1, ie. "rule" bit0 is 1 (an odd number), the "off" cells either
side of the initial N=1 become all "on" infinitely to the sides. Or if rule bit7 for
1,1,1 is a 0 (ie. rule < 128) then they turn on and off alternately in odd and even rows.
In both cases only the pyramid portion part -Y<=X<=Y is considered for the N points but
the infinite cells to the sides are included in the calculation.
The full set of patterns can be seen at the MathWorld page above, or can be printed with
the examples/cellular-rules.pl here. The patterns range from simple to complex. For
some, the N=1 cell doesn't grow at all such as rule 0 or rule 8. Some grow to mere
straight lines such as rule 2 or rule 5. Others make columns or patterns with "quadratic"
style stepping of 1 or 2 rows, or self-similar replications such as the Sierpinski
triangle of rule 18 and 60. Some rules have complicated non-repeating patterns when
there's feedback across from one half to the other, such as rule 30.
For some rules there's specific PlanePath code which this class dispatches to, such as
"CellularRule54", "CellularRule57", "CellularRule190" or "SierpinskiTriangle" (with
"n_start=1").
For rules without specific code, the current implementation is not particularly efficient
as it builds and holds onto the bit pattern for all rows through to the highest N or X,Y
used. There's no doubt better ways to iterate an automaton, but this module offers the
patterns in PlanePath style.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start, in the same pattern. For example to start at 0,
n_start => 0, rule => 62
18 19 20 21 22 23 24 25 5
13 14 15 16 17 4
7 8 9 10 11 12 3
4 5 6 2
1 2 3 1
0 <- Y=0
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::CellularRule->new (rule => 123)"
"$path = Math::PlanePath::CellularRule->new (rule => 123, n_start => $n)"
Create and return a new path object. "rule" should be an integer 0 to 255. A "rule"
should be given always. There is a default, but it's secret and likely to change.
If there's specific PlanePath code implementing the pattern then the returned object
is from that class and generally is not "isa('Math::PlanePath::CellularRule')".
"$n = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". $x and $y are each rounded to the
nearest integer, which has the effect of treating each cell as a square of side 1. If
"$x,$y" is outside the pyramid or on a skipped cell the return is "undef".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path can be
found in the OEIS index
<http://oeis.org/wiki/Index_to_OEIS:_Section_Ce#cell>
and in addition the following
<http://oeis.org/A061579> (etc)
rule=50,58,114,122,178,186,242,250, 179
(solid every second cell)
A061579 permutation N at -X,Y (mirror horizontal)
SEE ALSO
Math::PlanePath, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule57,
Math::PlanePath::CellularRule190, Math::PlanePath::SierpinskiTriangle,
Math::PlanePath::PyramidRows
Cellular::Automata::Wolfram
<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
of the GNU General Public License as published by the Free Software Foundation; either
version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Math-
PlanePath. If not, see <http://www.gnu.org/licenses/>.