README.tex.md

Knacci-Utils

Introduction to -nacci Words and Curves

This is a library for k-nacci word curves in python. The k-nacci word is a string over the alphabet generated recursively via the rules:

The words are drawn via turtle-graphics (a non-branching Lindenmyer System, if you like) via a drawing rule:

(1) Initalize: declare the turning angle , set the inital drawing angle to , and place the turtle on . Set the index to 1.

(2) Move the turtle forward one step.

(3) If the th character of the word is a '0', turn the turtle's direction by .

(4) Increment the index by one.

(5) If , return to step 2.

To generate the fractal, we take each curve and rotate/scale it so that the endpoint of the curve touches . Then we take the limit of the modified curve as .

Code Snippets

Usage examples of the files included in this repo:

WordGen.py

With Knacci Utils, you can obtain all types of -nacci words easily. Get via the command f(k,n). For example,

from WordGen import f

for n in range(1,10):
    print(n, f(2,n))

You can also get via the command t(k,n).

Debugging.py

It is easy to test hypotheses about the drawing rule with random_word, which returns a random binary word of any arbitrary length. There are a few other tools.

from Debugging import random_word
w = random_word(100)       #makes a random binary word of length 100
is_alphabet_valid(w)       #checks to see if the alphabet is a subset of {'0','1'}

AngleAndVertex.py

This gives you $a(w)$, draws pictures of the word curves, and lets you have a list of vertices in the word curve.

from math import pi
from WordGen import f
from AngleAndVertex import alpha_coeff, vertices, draw_me, end_position

alpha_coeff(f(3,5))        #gives the coefficient of alpha that the drawer is pointing after drawing the word, related to net angle.
vertices(f(2,5), pi/3)     #a list of vertices in the word curve given some drawing angle
draw_me(f(2,11), pi/3)     #plt.plot(), plt.show() the vertices in the word curve.
end_position(f(2,17), pi/3)      #gives both the coefficient of alpha after drawing and also the last vertex in the word curve

WordSplit.py

For discovering word generation patterns. There also exist some word comparison generators, not sure what I was doing with those.

from WordGen import f
from WordSplit import split_to

# outputs all possible word decompositions given the subwords in the dict.
split_to(f(2,7), {"f(2,6)":f(2,6), "f(2,5)":f(2,5), "f(2,4)":f(2,4), "f(2,3)":f(2,3)}) 

HighLevelEndpoints.py

Determine the endpoints of a composition of words. This file uses numpy.

from math import pi
from WordGen import f, t
from HighLevelEndpoints import wordmat, ep_for_words

# z(w) + wordmat(w).z(v) = z(wv)
wordmat(f(2,7), pi/3) #gives the translation/rotation matrix for drawing another word curve after drawing a word.

ep_for_words(pi/2, f(2,7),f(2,8),t(2,5)) #gives the endpoint and drawing angle coefficient after drawing all words given.