Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::CellularRule54 -- cellular automaton points
SYNOPSIS
use Math::PlanePath::CellularRule54;
my $path = Math::PlanePath::CellularRule54->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is the pattern of Stephen Wolfram's "rule 54" cellular automaton
<http://mathworld.wolfram.com/Rule54.html>
arranged as rows,
29 30 31 . 32 33 34 . 35 36 37 . 38 39 40 7
25 . . . 26 . . . 27 . . . 28 6
16 17 18 . 19 20 21 . 22 23 24 5
13 . . . 14 . . . 15 4
7 8 9 . 10 11 12 3
5 . . . 6 2
2 3 4 1
1 <- Y=0
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The initial figure N=1,2,3,4 repeats in two-row groups with 1 cell gap between figures.
Each two-row group has one extra figure, for a step of 4 more points than the previous
two-row.
The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers N=1,6,15,28, etc
k*(2k-1). The hexagonal numbers of the "second kind" 1, 3, 10, 21, 36, etc j*(2j+1) are a
steep sloping line upwards in the middle too. Those two taken together are the triangular
numbers 1,3,6,10,15 etc, k*(k+1)/2.
The 18-gonal numbers 18,51,100,etc are the vertical line at X=-3 on every fourth row
Y=5,9,13,etc.
Row Ranges
The left end of each row is
Nleft = Y*(Y+2)/2 + 1 if Y even
Y*(Y+1)/2 + 1 if Y odd
The right end is
Nright = (Y+1)*(Y+2)/2 if Y even
(Y+1)*(Y+3)/2 if Y odd
= Nleft(Y+1) - 1 ie. 1 before next Nleft
The row width Xmax-Xmin is 2*Y but with the gaps the number of visited points in a row is
less than that, being either about 1/4 or 3/4 of the width on even or odd rows.
rowpoints = Y/2 + 1 if Y even
3*(Y+1)/2 if Y odd
For any Y of course the Nleft to Nright difference is the number of points in the row too
rowpoints = Nright - Nleft + 1
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start, in the same pattern. For example to start at 0,
n_start => 0
15 16 17 18 19 20 21 22 23 5
12 13 14 4
6 7 8 9 10 11 3
4 5 2
1 2 3 1
0 <- Y=0
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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,
<http://oeis.org/A118108> (etc)
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, 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/>.