--- name: "Math::PlanePath::CellularRule57" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::CellularRule57" --- # Math::PlanePath::CellularRule57(3pm) ## NAME Math::PlanePath::CellularRule57 -- cellular automaton 57 and 99 points ## SYNOPSIS ``` use Math::PlanePath::CellularRule57; my $path = Math::PlanePath::CellularRule57->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This is the pattern of Stephen Wolfram's "rule 57" cellular automaton \ arranged as rows ``` 51 52 53 54 55 56 10 38 39 40 41 42 43 44 45 46 47 48 49 50 9 33 34 35 36 37 8 23 24 25 26 27 28 29 30 31 32 7 19 20 21 22 6 12 13 14 15 16 17 18 5 9 10 11 4 5 6 7 8 3 3 4 2 2 1 1 <- Y=0 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 ``` The triangular numbers N=10,15,21,28,etc, k\*(k+1)/2, make a 1/2 sloping diagonal upwards. On rows with odd Y there's a solid block at either end then 1 of 3 cells to the left and 2 of 3 to the right of the centre. On even Y rows there's similar 1 of 3 and 2 of 3 middle parts, but without the solid ends. Those 1 of 3 and 2 of 3 are successively offset so as to make lines going up towards the centre as can be seen in the following plot. ``` *********** * * * * * ** ** ** ************ * * * * ** ** ** ** ********** * * * * ** ** ** *********** * * * * * ** ** ** ********* * * * ** ** ** ********** * * * * ** ** ** ******** * * * * ** ** ********* * * * ** ** ** ******* * * * ** ** ******** * * * * ** ** ****** * * ** ** ******* * * * ** ** ***** * * * ** ****** * * ** ** **** * * ** ***** * * * ** *** * ** **** * * ** ** * * *** * ** * * ** * * * * ``` ### Mirror The `"mirror => 1"` option gives the mirror image pattern which is "rule 99". The point numbering shifts but the total points on each row is the same. ``` 51 52 53 54 55 56 10 38 39 40 41 42 43 44 45 46 47 48 49 50 9 33 34 35 36 37 8 23 24 25 26 27 28 29 30 31 32 7 19 20 21 22 6 12 13 14 15 16 17 18 5 9 10 11 4 5 6 7 8 3 3 4 2 2 1 1 <- Y=0 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 ``` ### 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 22 23 24 25 26 27 28 29 30 31 18 19 20 21 11 12 13 14 15 16 17 8 9 10 4 5 6 7 2 3 1 0 ``` ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::CellularRule57-\>new ()" - "$path = Math::PlanePath::CellularRule57-\>new (mirror =\> $bool, 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"`. ## SEE ALSO Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows \ ## HOME PAGE \ ## LICENSE Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 \.