1Math::PlanePath::FibonaUcsceirWoCrodnFtrraicbtuatle(d3M)Paetrhl::DPolcaunmeePnattaht:i:oFnibonacciWordFractal(3)
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NAME

6       Math::PlanePath::FibonacciWordFractal -- turns by Fibonacci word bits
7

SYNOPSIS

9        use Math::PlanePath::FibonacciWordFractal;
10        my $path = Math::PlanePath::FibonacciWordFractal->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This is an integer version of the Fibonacci word fractal
15
16           Alexis Monnerot-Dumaine, "The Fibonacci Word Fractal", March 2009.
17           <https://hal.archives-ouvertes.fr/hal-00367972/>
18
19       It makes turns controlled by the "Fibonacci word" sequence, sometimes
20       called the "golden string".
21
22           11  | 27-28-29    33-34-35          53-54-55    59-60-61
23               |  |     |     |     |           |     |     |     |
24           10  | 26    30-31-32    36          52    56-57-58    62
25               |  |                 |           |                 |
26            9  | 25-24          38-37          51-50          64-63
27               |     |           |                 |           |
28            8  |    23          39    43-44-45    49          65
29               |     |           |     |     |     |           |
30            7  | 21-22          40-41-42    46-47-48          66-67
31               |  |                                               |
32            6  | 20    16-15-14                      74-73-72    68
33               |  |     |     |                       |     |     |
34            5  | 19-18-17    13                      75    71-70-69
35               |              |                       |
36            4  |          11-12                      76-77
37               |           |                             |
38            3  |          10                            78
39               |           |                             |
40            2  |           9--8                      80-79
41               |              |                       |
42            1  |  1--2--3     7                      81    85-86-87
43               |  |     |     |                       |     |     |
44           Y=0 |  0     4--5--6                      82-83-84    88-89-...
45               +-------------------------------------------------------
46                 X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
47
48       A current direction up,down,left,right is maintained, starting in the
49       up direction.  The path moves in the current direction and then may
50       turn or go straight according to the Fibonacci word,
51
52           Fib word
53           --------
54              0      turn left if even index, right if odd index
55              1      straight ahead
56
57       The Fibonacci word is reckoned as starting from index=1, so for example
58       at N=0 draw a line upwards to N=1 and the first Fibonacci word value is
59       0 and its position index=1 is odd so turn to the right.
60
61            N     Fibonacci word
62           ---    --------------
63            1       0    turn right
64            2       1
65            3       0    turn right
66            4       0    turn left
67            5       1
68            6       0    turn left
69            7       1
70
71       The result is self-similar blocks within the first quadrant X>=0,Y>=0.
72       New blocks extend at N values which are Fibonacci numbers.  For example
73       N=21 a new block begins above, then N=34 a new block across, N=55 down,
74       N=89 across again, etc.
75
76       The new blocks are a copy of the shape starting N=0 but rotated and/or
77       transposed according to the replication level mod 6,
78
79           level mod 6      new block
80           -----------      ---------
81                0           transpose
82                1                         rotate -90
83                2           transpose and rotate -90
84                3           transpose
85                4                         rotate +90
86                5           transpose and rotate +90
87

FUNCTIONS

89       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
90       classes.
91
92       "$path = Math::PlanePath::FibonacciWordFractal->new ()"
93           Create and return a new path object.
94
95       "($x,$y) = $path->n_to_xy ($n)"
96           Return the X,Y coordinates of point number $n on the path.  Points
97           begin at 0 and if "$n < 0" then the return is an empty list.
98
99           Fractional positions give an X,Y position along a straight line
100           between the integer positions.
101
102       "$n = $path->n_start()"
103           Return 0, the first N in the path.
104

OEIS

106       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
107       this path include,
108
109           <http://oeis.org/A156596> (etc)
110
111           A332298   X coordinate, starting at n=1
112           A332299   Y-1 coordinate, starting at n=1
113           A156596   turn sequence, 0=straight,1=right,2=left
114           A143668   turn sequence, 0=right,1=straight,2=left
115           A171587   abs(dX), so 1=horizontal,0=vertical
116           A265318   N at locations by diagonals
117
118           A003849   Fibonacci word with values 0,1
119           A005614   Fibonacci word with values 1,0
120           A003842   Fibonacci word with values 1,2
121           A014675   Fibonacci word with values 2,1
122

SEE ALSO

124       Math::PlanePath, Math::PlanePath::DragonCurve,
125       Math::PlanePath::WythoffArray
126
127       Math::NumSeq::FibonacciWord
128

HOME PAGE

130       <http://user42.tuxfamily.org/math-planepath/index.html>
131

LICENSE

133       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
134       Kevin Ryde
135
136       This file is part of Math-PlanePath.
137
138       Math-PlanePath is free software; you can redistribute it and/or modify
139       it under the terms of the GNU General Public License as published by
140       the Free Software Foundation; either version 3, or (at your option) any
141       later version.
142
143       Math-PlanePath is distributed in the hope that it will be useful, but
144       WITHOUT ANY WARRANTY; without even the implied warranty of
145       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
146       General Public License for more details.
147
148       You should have received a copy of the GNU General Public License along
149       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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153perl v5.36.0                      2023-M0a1t-h2:0:PlanePath::FibonacciWordFractal(3)
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