Inspired by the KnitYak Kickstarter, I thought I would code up a simple Python program that could generate the same sort of patterns that are used in the scarves in the video. If you want to learn more about the mathematical theory of these cellular automata, google for keywords like “1D” “cellular automata” and “steve wolfram”. Without much further explanation, here is the code:
[sourcecode lang=”python”]
#!/usr/bin/env python
# __
# ___ __ __ _ _ _ / _|
# (_-</ _/ _` | ‘_| _|
# /__/\__\__,_|_| |_|
#
# Inspired by the KnitYak Kickstarter, which offers
# mathematical scarves whose patterns are algorithmically
# generated, I thought it would be good to dust off my old
# knowledge and generate the same sort of patterns using
# Python. The patterns are simply printed using ASCII,
# but the code could be adapted to output in other forms
# easily.
#
# Written by Mark VandeWettering
#
import sys
import random
WIDTH=79 # How wide is the pattern?
w = WIDTH * [0] # create the current generation
nw = WIDTH * [0] # and the next generation
w[WIDTH/2] = 1 # populate with a single one
# or alternatively, you can populate it with a random
# initial configuration. If you want to start with
# just a single one, comment the next two lines out.
#for i in range(WIDTH):
# w[i] = random.randint(0, 1)
# How wide is the neighborhood of cells that are
# examined? The traditional Wolfram 1D cellular
# automata uses a neighborhood of 3…
NEIGHBORHOOD=3
# rtab is space for the rule table. It maps all
# numbers from [0, 2**NEIGHBORHOOD) to either a 0 or 1.
rtab = (2**NEIGHBORHOOD) * [0]
# The "rule" is a number which is used to populate
# rtab. The number is in the range [0, 2**(2**NEIGHBORHOOD))
# Many rules generate uninteresting patterns, but some
# like 30 generate interesting, aperiodic patterns.
rule = 30
# This fills in the table…
for i in range(2**NEIGHBORHOOD):
if ((2**i) & rule) != 0:
rtab[i] = 1
def dump(r):
for x in r:
if x == 1:
sys.stdout.write(‘X’)
else:
sys.stdout.write(‘ ‘)
sys.stdout.write(‘\n’)
# and generates 100 lines…
for y in range(100):
dump(w)
for x in range(WIDTH):
sum = 0
for d in range(NEIGHBORHOOD):
sum = sum + (2**d) * w[(x+d+WIDTH – NEIGHBORHOOD/2) % WIDTH]
nw[x] = rtab[sum]
w, nw = nw, w
[/sourcecode]
The pattern this generates with these settings follows. Try adjusting rule to other numbers up to 255 to generate other patterns. Lots will be boring, but some will be pretty interesting.
X
XXX
X XX
XXXX XX
X X XX
XXX XXXX XX
X X X XX
XXXXXX XXXX XX
X XXX X XX
XXX X XX XXXX XX
X XX XXXX X X XX
XXXX X X XX XXXX XX
X X XX XX XXX X XX
XXX XX XXX XXX XX XXXX XX
X X XXX X XXX X X XX
XXXXXXX XXXXX X X XX XXXX XX
X XXX X XXXX XXX X XX
XXX X XX XX XXX XX XXXX XX
X XX XXXX XXX XX X XXX X X XX
XXXX XXX X X XXXXXX X XX XXXX XX
X X XX XXX XXXX XXXX XXX X XX
XXX XXX X X X XX X XXX XX XXXX XX
X X X X XX XXX X XX XXX X XXX X X XX
XXXXXX XX X X XXXX X X XXX X XX XXXX XX
X X X XXXXX X XXXXX XX XXXX XXX X XX
XXX XXXXX X XX X X XX X XXX XX XXXX XX
X XX X XX XX X XX XXXX X XX X XXX X X XX
XXXX X XX X XX X XXXX XXX X X X XXX X XX XXXX XX
X X X XXXX X X XXX XX XX XXXX XXXX XXX X XX
XXX XX XXX XXXX XX X XXX X X XX X XXX XX XXXX XX
X X X XX X X XXXXXX X XXXXX X X XX X XXX X X XX
XXXXXXXXX X X XX XXXX XXXX XXXX X X XXX X XX XXXX XX
X X X X X XX X XX X X XXXX XXXX XXX X XX
XXX XX X XXXXX X XX XXX X XX XXX XX XX X XXX XX XXXX XX
X XX X X X XXXX X X X X X XX X X XX X XXX X X XX
XXXX XX XX X XX X XXXXX XX XXXXX XXXX XXXX X X XXX X XX XXXX XX
X X XXX X X XX XXX X X X X X X XXXX XXXX XXX X XX
XXX XXXX X X XXX X X XX XXXXXX XXX XXX XX XX X XXX XX XXXX XX
X X XXXX X X XX XX XXX XX X XXX XXX XX X X XX X XXX X X XX
XXXXXX X X XX XX X XXX XX X X XXX XXX X XXXX X X XXX X XX XXXX XX
XXXXX XX X XXXXXX XXX XX XX X XXX XXXXXX X XXXX XXXX XXX X
X X XXXXXX XXX X X X XX X XXX XX XX XX X XXX XX XXX
XXX XXXX XX X XXXXXXXXXX X XXX XX X X XX X X XX X XXX X XX
X XXX XX X XX XXXX XXXX XXX XX XX XXX XXXX X X XXX X XX X XX
XXXXX XX X XX XX X XX X XX X X X X X X XXXX XXXX X X X
XXX X X X XXXXX X XX XXX X X XXXXXXXXXXX XXX XX XX X XXXX XXX
X X X XXXXXX XXXX XX X X X X X XXX XX X X XX X X XX
X XXXXX X XX X XXX XX XXXXX X XX XXX X X XXXX X X XX XXX X X
X X XX X XXXXX X X X X X XX X X XXXXXX X XXXX X X X
XX XX XX XX X XXXXXXXX XX XXX XX XXXXX XX XX XXXXX XX XXX
XX X XXX XXX XX XX XX X X X XX X XX X X XX X X X
X X X X X XX X XX X X X XX XXXX XX XXX X XX XX XXX XXXXX XXXXXX
X X XX XXX XXXX X X XX XX X X X X X XXXX XXX X X XXX XX X
X X X X X X XXX XX X X X XXXXX XXXXXX X XXX X XX XXX X XX X X
X XXXXX XXX XX X X XXXX X X XXX X XX X XXXX X XX XXXX XX XX XXX
X X XXX X XX XXXX X X XX X XX XX X XXX XXXXX X X X X X
XX XXX X X X X XX XX X XX XXXX XX X XXXX XX X X X XX XXXXXXXXX X
XX X X XXXX XXXXX X X X XXX X X X X XX X X XX XX X XXX XXX
X X XXXX X XXXX XXXX X XX XXXXX XX X X X X XXX X XXX XX X XX
X X XX XXX X X XX XX XXX X XXXX X X XXX X X XXX XX XXXX
X XX X XXX XX XXX XXX X X XXX XX XXXX X X X XXXX XX X X XX X XXX
X XXXX XXX X X XXXX XXX XXX XXX XX XX X XX X X X XXXXXX XXXXX X
XXX XX X X XX XXX X X XXX X XX X X X X X XX XXXXX X X XX X
XX X X XXXX X X XX XXX X XXXXX X X X XXXX XXXX X XX XXX XX X
X X X X XXXXX XX X X XXX X X X X X XXXXX X XX X XX X XXXXX
XX X X XX X X XXXXX XX XX XX X X XX XXX X XX XX XXXXXX X X XX
X X X XXXXX XXXX X XX X XXX X X X XXX XXXXX X X XXX X X XX X
XX X XXX XXX XX XXXX X X X X X XXX XXX XXXX XXX XX XX X XX XXX
X X XX X XX X XXX X X XX XX X X XXX XX X X XXX XX X X XXX X X
XX XX X XXXXXX X X XX XX X X X X XX X XXX XXXXX XXX X X X X X X XX XXX
X X X X X XX X X X XXXXXXX X X XXX X X X XXXXXXX X XX XX X
XXXXX XX XX X X X XXXX X XXXX XXXXX XXX XX X X X XXXXX
X XX X X XXXX X XX XX XX X XX X X XX XX XXXXXX XX X
X XXXX XXXX X X XX X XX X XX X X XX X X XXXXXX XX X X XX X X X
XXX X X XX XX XXXX XX XX XXXX X XXXX X X XX XX XX X XXXX X
XX XXX XX XXX XXX XXX X XXX X XXX X XX XXXX X X XX XX X XXX
XX X XXX XXX X XX X X XXX XX XX XX XX XX X X XXXXXX XXX XX XX X
X XX X X XXXXX X X XXXX XXX X XX X X XXX XXXXX XX X X XXX XX X
XX X XXXXX X X X X XX X X XXX X X XXX X X XX XXX XXXX X X XX
XXXX X XX XX X XX X X XXXX X X X XXXXX XXXX XX X X XXXXXXX
X XX XX XXX X X XXXX X XX XX X XX X XX X X XXXXXXXX X XX
XX X XX X XXX X X XXX X XX X X X X X XX X X XX XXXX XXXXX X XXX
X X X X XXXX X XX XX XXXX XXXX XXXX XXXX X X XX X XX XX
XX XXXX XX X X XX X X XXX X X XXX X XXXXX X XX XXX X XXX XX
X X X XX XX X X XXX XX XXX XXX X XX XX XXXX XX X XXXX X XX X
XX XXXXX X XXXX X XXX X XXX XXXXXX X XX X XXX X XXX XX XXXX X X
XXX XXXXXX X XX X X XX X XXX X XXX XX XXX X X X XX X X X X X
XX XX X XX XX X XXXX X XXX XX XX X X X XXXX XX X X X XX XX X X
XXX XXXXX X X XXXX XXXX XXX XX X XX XXXXXX XX X X X X X XXX X X
X X XX XX XXX XX X XX X X X X X X XX XXXXX X X XXX X X XX
XXXXX X X X XX X XXXXX X X XXXXXXX XXXXX XXXX XXX X X X XXXX X X
XX XX XXXXX X X X X X X X XX X X XX XX X XX X X XXXX
X X X X X X XX XX X XX XXX X XXXXX XXX X XXX X X X XX XX XX
X XX XXXXX XX X X XX X X X XX X XX XX X X X X X XXXX X XX X X
X X XX X X X XXX X X X XXX XX XXXX X XX XXX XX XX XX X XXXXXX XXXX
XXXXX X X X X X X X X X X X X XXX XX X X X X X XX X X XXX
XX XX X X X XX XX X X XX XXXXXX XX X X XXXXXXXXXXXXX XXXXX XXX X
X X X X X X X X X X X XXX XX XXXXX XXX XX X XXXXX
XX XXXX X X XXXXXXX X XXXXX X X X XX X XX X X XXX XX
I will experiment with larger rules: the space of possible behaviors with a neighborhood of 5 is quite a bit larger (there are 2**32 possible rules instead of just 256).
1 line of APL, 6 if you document:
fBW?{i w n?? ? ‘ X'{?[1+?]}?(n?{({(,?)/,???}?(8/2)??)?{(2??)??}¨3,/0{?,?,?}?}){i x?? ? i=0:? ? x{(??),?}?(i-1)(?? x)}i(w?(-1+2×w)?1)}
fBW←{i w n←⍵ ⋄ ‘ X'{⍺[1+⍵]}↑(n∘{({(,⍵)/,⍳⍴⍵}⌽(8/2)⊤⍺)∘{(2⊥⍵)∊⍺}¨3,/0{⍺,⍵,⍺}⍵}){i x←⍵ ⋄ i=0:⍬ ⋄ x{(⊂⍺),⍵}∇(i-1)(⍺⍺ x)}i(w⌽(-1+2×w)↑1)}