Provided by: libmath-planepath-perl_129-1_all bug

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/>.