--- name: "Math::PlanePath::CellularRule54" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::CellularRule54" --- # Math::PlanePath::CellularRule54(3pm) ## NAME Math::PlanePath::CellularRule54 -- cellular automaton points ## SYNOPSIS ``` use Math::PlanePath::CellularRule54; my $path = Math::PlanePath::CellularRule54->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This is the pattern of Stephen Wolfram's "rule 54" cellular automaton \ arranged as rows, ``` 29 30 31 . 32 33 34 . 35 36 37 . 38 39 40 7 25 . . . 26 . . . 27 . . . 28 6 16 17 18 . 19 20 21 . 22 23 24 5 13 . . . 14 . . . 15 4 7 8 9 . 10 11 12 3 5 . . . 6 2 2 3 4 1 1 <- Y=0 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 ``` The initial figure N=1,2,3,4 repeats in two-row groups with 1 cell gap between figures. Each two-row group has one extra figure, for a step of 4 more points than the previous two-row. The rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers N=1,6,15,28, etc k\*(2k-1). The hexagonal numbers of the "second kind" 1, 3, 10, 21, 36, etc j\*(2j+1) are a steep sloping line upwards in the middle too. Those two taken together are the triangular numbers 1,3,6,10,15 etc, k\*(k+1)/2. The 18-gonal numbers 18,51,100,etc are the vertical line at X=-3 on every fourth row Y=5,9,13,etc. ### Row Ranges The left end of each row is ``` Nleft = Y*(Y+2)/2 + 1 if Y even Y*(Y+1)/2 + 1 if Y odd ``` The right end is ``` Nright = (Y+1)*(Y+2)/2 if Y even (Y+1)*(Y+3)/2 if Y odd = Nleft(Y+1) - 1 ie. 1 before next Nleft ``` The row width Xmax-Xmin is 2\*Y but with the gaps the number of visited points in a row is less than that, being either about 1/4 or 3/4 of the width on even or odd rows. ``` rowpoints = Y/2 + 1 if Y even 3*(Y+1)/2 if Y odd ``` For any Y of course the Nleft to Nright difference is the number of points in the row too ``` rowpoints = Nright - Nleft + 1 ``` ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start, in the same pattern. For example to start at 0, ``` n_start => 0 15 16 17 18 19 20 21 22 23 5 12 13 14 4 6 7 8 9 10 11 3 4 5 2 1 2 3 1 0 <- Y=0 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 ``` ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$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, \ (etc) ``` A118108 whole-row used cells as bits of a bignum A118109 1/0 used and unused cells across rows ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows Cellular::Automata::Wolfram \ ## 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 \.