Kevin Ryde > Math-PlanePath-100 > Math::PlanePath::CellularRule190

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Module Version: 100   Source   Latest Release: Math-PlanePath-103

NAME

Math::PlanePath::CellularRule190 -- cellular automaton 190 and 246 points

SYNOPSIS

``` use Math::PlanePath::CellularRule190;
my \$path = Math::PlanePath::CellularRule190->new;
my (\$x, \$y) = \$path->n_to_xy (123);```

DESCRIPTION

This is the pattern of Stephen Wolfram's "rule 190" cellular automaton

http://mathworld.wolfram.com/Rule190.html

arranged as rows,

```    66 67 68    69 70 71    72 73 74    75 76 77    78 79 80      9
53 54 55    56 57 58    59 60 61    62 63 64    65         8
41 42 43    44 45 46    47 48 49    50 51 52            7
31 32 33    34 35 36    37 38 39    40               6
22 23 24    25 26 27    28 29 30                  5
15 16 17    18 19 20    21                     4
9 10 11    12 13 14                        3
5  6  7     8                           2
2  3  4                              1
1                             <- Y=0

-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9```

Each row is 3 out of 4 cells. Even numbered rows have one point on its own at the end. Each two-row group has a step of 6 more points than the previous two-row.

On even rows Y=0,2,4,6,etc the rightmost N=1,8,21,40,65,etc is the octagonal numbers k*(3k-2). The octagonal numbers of the "second kind" 5,16,33,56,85, etc, k*(3k+2) are a straight-ish line upwards to the left.

Mirror

The `mirror => 1` option gives the mirror image pattern which is "rule 246". It differs only in the placement of the gaps on the even rows. The point on its own is at the left instead of the right. The numbering is still left to right.

```    66 67 68    69 70 71    72 73 74    75 76 77    78 79 80      9
53    54 55 56    57 58 59    60 61 62    63 64 65         8
41 42 43    44 45 46    47 48 49    50 51 52            7
31    32 33 34    35 36 37    38 39 40               6
22 23 24    25 26 27    28 29 30                  5
15    16 17 18    19 20 21                     4
9 10 11    12 13 14                        3
5     6  7  8                           2
2  3  4                              1
1                             <- Y=0

-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9```

Sometimes this small change to the pattern helps things line up better. For example plotting the Klaner-Rado sequence gives some unplotted lines up towards the right in the mirror 246 which are not visible in the plain 190.

Row Ranges

The left end of each row, both ordinary and mirrored, is

```    Nleft = ((3Y+2)*Y + 4)/4     if Y even
((3Y+2)*Y + 3)/4     if Y odd```

The right end is

```    Nright = ((3Y+8)*Y + 4)/4    if Y even
((3Y+8)*Y + 5)/4    if Y odd

= Nleft(Y+1) - 1   ie. 1 before next Nleft```

The row width Xmax-Xmin = 2*Y but with the gaps the number of visited points in a row is less than that,

```    rowpoints = 3*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`

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::CellularRule190->new ()`
`\$path = Math::PlanePath::CellularRule190->new (mirror => 1)`

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`.

`(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)`

The returned range is exact, meaning `\$n_lo` and `\$n_hi` are the smallest and biggest in the rectangle.

OEIS

This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple of forms,

```    http://oeis.org/A037576  (etc)

A037576    whole-row used cells as bits of a bignum
A071039    \ 1/0 used and unused cells across rows
A118111    /
A071041    1/0 used and unused of mirrored rule 246 ```

Cellular::Automata::Wolfram

http://mathworld.wolfram.com/Rule190.html

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