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 the 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.

SEE ALSO

Math::PlanePath, Math::PlanePath::CellularRule54, Math::PlanePath::PyramidRows

Cellular::Automata::Wolfram

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

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath