Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::FibonacciWordFractal -- turns by Fibonacci word bits
SYNOPSIS
use Math::PlanePath::FibonacciWordFractal;
my $path = Math::PlanePath::FibonacciWordFractal->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is an integer version of the Fibonacci word fractal
Alexis Monnerot-Dumaine, "The Fibonacci Word Fractal", February 2009.
<http://hal.archives-ouvertes.fr/hal-00367972_v1/>
<http://hal.archives-ouvertes.fr/docs/00/36/79/72/PDF/The_Fibonacci_word_fractal.pdf>
It makes turns controlled by the "Fibonacci word" sequence, sometimes called the "golden
string".
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
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,
Fib word
--------
0 turn left if even index, right if odd index
1 straight ahead
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.
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
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.
The new blocks are a copy of the shape starting N=0 but rotated and/or transposed
according to the replication level mod 6,
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
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::FibonacciWordFractal->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if
"$n < 0" then the return is an empty list.
Fractional positions give an X,Y position along a straight line between the integer
positions.
"$n = $path->n_start()"
Return 0, the first N in the path.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,
<http://oeis.org/A156596> (etc)
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
SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::WythoffArray
Math::NumSeq::FibonacciWord
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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 <http://www.gnu.org/licenses/>.