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.\" ========================================================================
.\"
.IX Title "Math::PlanePath::CellularRule54 3pm"
.TH Math::PlanePath::CellularRule54 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.SH "NAME"
Math::PlanePath::CellularRule54 \-\- cellular automaton points
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::CellularRule54;
\& my $path = Math::PlanePath::CellularRule54\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This is the pattern of Stephen Wolfram's \*(L"rule 54\*(R"
cellular automaton
.IX Xref "Wolfram, Stephen"
.Sp
.RS 4
<http://mathworld.wolfram.com/Rule54.html>
.RE
.PP
arranged as rows,
.PP
.Vb 8
\&    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
.Ve
.PP
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.
.PP
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
\&\*(L"second kind\*(R" 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.
.IX Xref "Hexagonal numbers Triangular numbers"
.PP
The 18\-gonal numbers 18,51,100,etc are the vertical line at X=\-3 on every
fourth row Y=5,9,13,etc.
.SS "Row Ranges"
.IX Subsection "Row Ranges"
The left end of each row is
.PP
.Vb 2
\&    Nleft = Y*(Y+2)/2 + 1     if Y even
\&            Y*(Y+1)/2 + 1     if Y odd
.Ve
.PP
The right end is
.PP
.Vb 2
\&    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
.Ve
.PP
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.
.PP
.Vb 2
\&    rowpoints = Y/2 + 1        if Y even
\&                3*(Y+1)/2      if Y odd
.Ve
.PP
For any Y of course the Nleft to Nright difference is the number of points
in the row too
.PP
.Vb 1
\&    rowpoints = Nright \- Nleft + 1
.Ve
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=1 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern.  For example to
start at 0,
.PP
.Vb 1
\&    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
.Ve
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::CellularRule54\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::CellularRule54\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::CellularRule54->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::CellularRule54\->new (n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::CellularRule54\->new (n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::CellularRule54->new (n_start => $n)"
.PD
Create and return a new path object.
.ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4
.el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4
.IX Item "($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number \f(CW$n\fR on the path.
.ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4
.el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4
.IX Item "$n = $path->xy_to_n ($x,$y)"
Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR.  \f(CW$x\fR and \f(CW$y\fR are each
rounded to the nearest integer, which has the effect of treating each cell
as a square of side 1.  If \f(CW\*(C`$x,$y\*(C'\fR is outside the pyramid or on a skipped
cell the return is \f(CW\*(C`undef\*(C'\fR.
.SH "OEIS"
.IX Header "OEIS"
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a
couple of forms,
.Sp
.RS 4
<http://oeis.org/A118108> (etc)
.RE
.PP
.Vb 2
\&    A118108    whole\-row used cells as bits of a bignum
\&    A118109    1/0 used and unused cells across rows
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::CellularRule,
Math::PlanePath::CellularRule57,
Math::PlanePath::CellularRule190,
Math::PlanePath::PyramidRows
.PP
Cellular::Automata::Wolfram
.PP
<http://mathworld.wolfram.com/Rule54.html>
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
.PP
This file is part of Math-PlanePath.
.PP
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the \s-1GNU\s0 General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
.PP
Math-PlanePath is distributed in the hope that it will be useful, but
\&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0
or \s-1FITNESS FOR A PARTICULAR PURPOSE.\s0  See the \s-1GNU\s0 General Public License for
more details.
.PP
You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.