--- name: "Math::PlanePath::FibonacciWordFractal" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::FibonacciWordFractal" --- # Math::PlanePath::FibonacciWordFractal(3pm) ## 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", March 2009. \ 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, \ (etc) ``` A332298 X coordinate, starting at n=1 A332299 Y-1 coordinate, starting at n=1 A156596 turn sequence, 0=straight,1=right,2=left A143668 turn sequence, 0=right,1=straight,2=left A171587 abs(dX), so 1=horizontal,0=vertical A265318 N at locations by diagonals 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 \ ## LICENSE Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 \.