 Math::PlanePath::CellularRule190(3) 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.

The of rightmost N=1,4,8,14,21,etc are triangular plus quarter square, ie.

```    Nright = triangular(Y+1) + quartersquare(Y+1)
triangular(t)    = t*(t+1)/2
quartersquare(t) = floor(t^2/4)
```

The rightmost N=1,8,21,40,65,etc on even rows Y=0,2,4,6,etc are the octagonal numbers k*(3k-2). The octagonal numbers of the ``second kind'' N=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
```

## 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
21 22 23    24 25 26    27 28 29          5
14 15 16    17 18 19    20             4
8  9 10    11 12 13                3
4  5  6     7                   2
1  2  3                      1
0                     <- Y=0
-5 -4 -3 -2 -1 X=0 1  2  3  4  5
```

The effect is to push each N rightwards by 1, and wrapping around. So the N=0,1,4,8,14,etc on the left were on the right of the default n_start=1. This also has the effect of removing the +1 in the Nright formula given above, so

```    Nleft = triangular(Y) + quartersquare(Y)
```

## 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, 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".
"(\$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
n_start=0
A006578   N at left of each row (X=-Y),
and at right of each row when mirrored,
being triangular+quartersquare
```

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