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 tworow groups with 1 cell gap between figures. Each tworow group has one extra figure, for a step of 4 more points than the previous tworow.
The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers N=1,6,15,28, etc k*(2k1). 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 18gonal 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 XmaxXmin 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 isundef
.
OEIS
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple of forms,
http://oeis.org/A118108 (etc)
A118108 wholerow 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
http://mathworld.wolfram.com/Rule54.html
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.