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 based cellular automatons
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 possible 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 Math World page above, or can be
printed with the
examples/cellular-rules.pl program in the
Math-PlanePath sources. 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
and in addition the following
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 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/>.