Math::...Fractal(3pm)

📦
version 129 · 3 versions
, in debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking

💖 Math::PlanePath::FibonacciWordFractal 💖

okay bestie, sit DOWN, because this is the Math::PlanePath::FibonacciWordFractal page from the User Contributed Perl Documentation (that's the Math::...Fractal(3pm) vibe you clicked on), and it's literally a fractal that turns by Fibonacci word bits and it is SO much cooler than it sounds 🦋

✨ NAME ✨

Math::PlanePath::FibonacciWordFractal — turns by Fibonacci word bits. that's it, that's the whole gig, and it slays 💅

🌸 SYNOPSIS 🌸

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

three lines. that's your whole entry point. go off 👑

🧚 DESCRIPTION 🧚

so this is an integer version of the Fibonacci word fractal, and the source of truth here is:

Alexis Monnerot-Dumaine, "The Fibonacci Word Fractal", March 2009.
<https://hal.archives-ouvertes.fr/hal-00367972/>

It makes turns controlled by the "Fibonacci word" sequence, which some people call the "golden string" (main character naming energy honestly).

here's what the path actually looks like, no cap:

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

okay so the way this works: a current direction — up, down, left, right — is maintained, and it starts in the up direction. the path moves in the current direction and then it may turn or go straight according to the Fibonacci word, like:

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 you draw a line upwards to N=1 and the first Fibonacci word value is 0 and its position index=1 is odd so you 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. iconic behavior fr fr.

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 🎀

peep "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes — that's the shared lore, iykyk.

  • "$path = Math::PlanePath::FibonacciWordFractal->new ()" — create and return a new path object. your lil baby path 🫶
  • "($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. yes, empty. nothing. void. 💀

Fractional positions give an X,Y position along a straight line between the integer positions — so you can vibe in between, it's fine.

  • "$n = $path->n_start()" — return 0, the first N in the path. zero. the beginning. the origin story ✨

🔥 OEIS 🔥

entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,

<http://oeis.org/A156596> (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

a whole reading list. nerd out bestie, no one's stopping you 📚

💜 SEE ALSO 💜

  • Math::PlanePath
  • Math::PlanePath::DragonCurve
  • Math::PlanePath::WythoffArray
  • Math::NumSeq::FibonacciWord

🌈 HOME PAGE 🌈

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

go click it, the vibes are immaculate.

👑 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 <http://www.gnu.org/licenses/>.

⚠️ okay so real talk, that warranty disclaimer is NOT a joke — no warranty, no promises, read the license fr fr.

and that's the whole page, bestie 🥰 this was perl v5.32.0, dated 2021-01-23, straight from the Math::...Fractal(3pm) corner of the manual. go draw some turns, you absolute icon 💅✨🦋

📦 where this page comes from