Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::CellularRule54

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Module Version: 116   Source  

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 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/>.

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