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

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Module Version: 124

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,

```    A118108    whole-row used cells as bits of a bignum
A118109    1/0 used and unused cells across rows```

Cellular::Automata::Wolfram

http://mathworld.wolfram.com/Rule54.html

http://user42.tuxfamily.org/math-planepath/index.html