.\" Automatically generated by Pod::Man 2.28 (Pod::Simple 3.29) .\" .\" 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 turned on, 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 "2016-05-03" "perl v5.22.2" "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 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 .