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.IX Title "Math::PlanePath::FibonacciWordFractal 3pm"
.TH Math::PlanePath::FibonacciWordFractal 3pm "2014-09-03" "perl v5.20.1" "User Contributed Perl Documentation"
.\" For nroff, turn off justification. Always turn off hyphenation; it makes
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.SH "NAME"
Math::PlanePath::FibonacciWordFractal \-\- turns by Fibonacci word bits
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::FibonacciWordFractal;
\& my $path = Math::PlanePath::FibonacciWordFractal\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This is an integer version of
the Fibonacci word fractal
.IX Xref "Monnerot-Dumaine, Alexis Fibonacci Word"
.Sp
.RS 4
Alexis Monnerot-Dumaine, \*(L"The Fibonacci Word Fractal\*(R", February 2009.
.RE
.PP
It makes turns controlled by the \*(L"Fibonacci word\*(R" sequence, sometimes called
the \*(L"golden string\*(R".
.PP
.Vb 10
\& 11 | 27\-28\-29 33\-34\-35 53\-54\-55 59\-60\-61
\& | | | | | | | | |
\& 10 | 26 30\-31\-32 36 52 56\-57\-58 62
\& | | | | |
\& 9 | 25\-24 38\-37 51\-50 64\-63
\& | | | | |
\& 8 | 23 39 43\-44\-45 49 65
\& | | | | | | |
\& 7 | 21\-22 40\-41\-42 46\-47\-48 66\-67
\& | | |
\& 6 | 20 16\-15\-14 74\-73\-72 68
\& | | | | | | |
\& 5 | 19\-18\-17 13 75 71\-70\-69
\& | | |
\& 4 | 11\-12 76\-77
\& | | |
\& 3 | 10 78
\& | | |
\& 2 | 9\-\-8 80\-79
\& | | |
\& 1 | 1\-\-2\-\-3 7 81 85\-86\-87
\& | | | | | | |
\& Y=0 | 0 4\-\-5\-\-6 82\-83\-84 88\-89\-...
\& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
.Ve
.PP
A current direction up,down,left,right is maintained, starting in the up
direction. The path moves in the current direction and then may turn or go
straight according to the Fibonacci word,
.PP
.Vb 4
\& Fib word
\& \-\-\-\-\-\-\-\-
\& 0 turn left if even index, right if odd index
\& 1 straight ahead
.Ve
.PP
The Fibonacci word is reckoned as starting from index=1, so for example at
N=0 draw a line upwards to N=1 and the first Fibonacci word value is 0 and
its position index=1 is odd so turn to the right.
.PP
.Vb 9
\& N Fibonacci word
\& \-\-\- \-\-\-\-\-\-\-\-\-\-\-\-\-\-
\& 1 0 turn right
\& 2 1
\& 3 0 turn right
\& 4 0 turn left
\& 5 1
\& 6 0 turn left
\& 7 1
.Ve
.PP
The result is self-similar blocks within the first quadrant
X>=0,Y>=0. New blocks extend at N values which are Fibonacci
numbers. For example N=21 a new block begins above, then N=34 a new block
across, N=55 down, N=89 across again, etc.
.PP
The new blocks are a copy of the shape starting N=0 but rotated and/or
transposed according to the replication level mod 6,
.PP
.Vb 8
\& level mod 6 new block
\& \-\-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-
\& 0 transpose
\& 1 rotate \-90
\& 2 transpose and rotate \-90
\& 3 transpose
\& 4 rotate +90
\& 5 transpose and rotate +90
.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::FibonacciWordFractal\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::FibonacciWordFractal\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::FibonacciWordFractal->new ()"
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$n\fR on the path. Points begin
at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list.
.Sp
Fractional positions give an X,Y position along a straight line between the
integer positions.
.ie n .IP """$n = $path\->n_start()""" 4
.el .IP "\f(CW$n = $path\->n_start()\fR" 4
.IX Item "$n = $path->n_start()"
Return 0, the first N in the path.
.SH "OEIS"
.IX Header "OEIS"
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include,
.Sp
.RS 4
(etc)
.RE
.PP
.Vb 2
\& A156596 \- turn sequence, 0=straight,1=right,2=left
\& A171587 \- abs(dX), so 1=horizontal,0=vertical
\&
\& A003849 \- Fibonacci word with values 0,1
\& A005614 \- Fibonacci word with values 1,0
\& A003842 \- Fibonacci word with values 1,2
\& A014675 \- Fibonacci word with values 2,1
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::DragonCurve,
Math::PlanePath::WythoffArray
.PP
Math::NumSeq::FibonacciWord
.SH "HOME PAGE"
.IX Header "HOME PAGE"
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014 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 .