NAME¶
Math::PlanePath::CellularRule54 -- cellular automaton points
SYNOPSIS¶
use Math::PlanePath::CellularRule54;
my $path = Math::PlanePath::CellularRule54->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This is the pattern of Stephen Wolfram's "rule 54" cellular automaton
arranged as rows,
29 30 31 . 32 33 34 . 35 36 37 . 38 39 40 7
25 . . . 26 . . . 27 . . . 28 6
16 17 18 . 19 20 21 . 22 23 24 5
13 . . . 14 . . . 15 4
7 8 9 . 10 11 12 3
5 . . . 6 2
2 3 4 1
1 <- Y=0
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The initial figure N=1,2,3,4 repeats in two-row groups with 1 cell gap between
figures. Each two-row group has one extra figure, for a step of 4 more points
than the previous two-row.
The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers
N=1,6,15,28, etc k*(2k-1). The hexagonal numbers of the "second
kind" 1, 3, 10, 21, 36, etc j*(2j+1) are a steep sloping line upwards in
the middle too. Those two taken together are the triangular numbers
1,3,6,10,15 etc, k*(k+1)/2.
The 18-gonal numbers 18,51,100,etc are the vertical line at X=-3 on every fourth
row Y=5,9,13,etc.
Row Ranges¶
The left end of each row is
Nleft = Y*(Y+2)/2 + 1 if Y even
Y*(Y+1)/2 + 1 if Y odd
The right end is
Nright = (Y+1)*(Y+2)/2 if Y even
(Y+1)*(Y+3)/2 if Y odd
= Nleft(Y+1) - 1 ie. 1 before next Nleft
The row width Xmax-Xmin is 2*Y but with the gaps the number of visited points in
a row is less than that, being either about 1/4 or 3/4 of the width on even or
odd rows.
rowpoints = Y/2 + 1 if Y even
3*(Y+1)/2 if Y odd
For any Y of course the Nleft to Nright difference is the number of points in
the row too
rowpoints = Nright - Nleft + 1
N Start¶
The default is to number points starting N=1 as shown above. An optional
"n_start" can give a different start, in the same pattern. For
example to start at 0,
n_start => 0
15 16 17 18 19 20 21 22 23 5
12 13 14 4
6 7 8 9 10 11 3
4 5 2
1 2 3 1
0 <- Y=0
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::CellularRule54->new ()"
- "$path = Math::PlanePath::CellularRule54->new (n_start =>
$n)"
- Create and return a new path object.
- "($x,$y) = $path->n_to_xy ($n)"
- Return the X,Y coordinates of point number $n on the path.
- "$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¶
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple
of forms,
A118108 whole-row used cells as bits of a bignum
A118109 1/0 used and unused cells across rows
SEE ALSO¶
Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule57,
Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows
Cellular::Automata::Wolfram
<
http://mathworld.wolfram.com/Rule54.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/>.