.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::PlanePath::CellularRule57 3pm" .TH Math::PlanePath::CellularRule57 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::CellularRule57 \-\- cellular automaton 57 and 99 points .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::CellularRule57; \& my $path = Math::PlanePath::CellularRule57\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is the pattern of Stephen Wolfram's \*(L"rule 57\*(R" cellular automaton .IX Xref "Wolfram, Stephen" .Sp .RS 4 .RE .PP arranged as rows .PP .Vb 11 \& 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 .Ve .PP The triangular numbers N=10,15,21,28,etc, k*(k+1)/2, make a 1/2 sloping diagonal upwards. .IX Xref "Triangular numbers" .PP 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. .PP .Vb 10 \& *********** * * * * * ** ** ** ************ \& * * * * ** ** ** ** \& ********** * * * * ** ** ** *********** \& * * * * * ** ** ** \& ********* * * * ** ** ** ********** \& * * * * ** ** ** \& ******** * * * * ** ** ********* \& * * * ** ** ** \& ******* * * * ** ** ******** \& * * * * ** ** \& ****** * * ** ** ******* \& * * * ** ** \& ***** * * * ** ****** \& * * ** ** \& **** * * ** ***** \& * * * ** \& *** * ** **** \& * * ** \& ** * * *** \& * ** \& * * ** \& * * \& * \& * .Ve .SS "Mirror" .IX Subsection "Mirror" The \f(CW\*(C`mirror => 1\*(C'\fR option gives the mirror image pattern which is \*(L"rule 99\*(R". The point numbering shifts but the total points on each row is the same. .PP .Vb 11 \& 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 .Ve .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 1 \& 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 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::CellularRule57\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::CellularRule57\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::CellularRule57->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::CellularRule57\->new (mirror => $bool, n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::CellularRule57\->new (mirror => $bool, n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::CellularRule57->new (mirror => $bool, n_start => $n)" .PD Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each cell as a square of side 1. If \f(CW\*(C`$x,$y\*(C'\fR is outside the pyramid or on a skipped cell the return is \f(CW\*(C`undef\*(C'\fR. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows .PP .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE.\s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .