Note
Go to the end to download the full example code.
Langton’s loop
Chris Langton created the “loop” cellular automata to illustrate how a cellular automata can self-reproduce.
Here is a video of Langton’s loop.
The code below is adapted from an implentation by Romain Fontaine (Copyright 2018)
import numpy as np
import matplotlib.pyplot as plt
from pylab import rcParams
def show_grid(ax,grid,make_animation):
N=np.size(grid,1)
if make_animation:
frame = ax.imshow(grid, cmap=plt.get_cmap('Blues'), interpolation='nearest',animated=True)
else:
frame = ax.imshow(grid, cmap=plt.get_cmap('Blues'), interpolation='nearest')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['left'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.set_ylim(0,N)
ax.set_xlim(0,N)
ax.set_xticks([])
ax.set_yticks([])
ax.axis('equal')
return frame
def iterate(grid,new_value):
N=np.size(grid,1)
new_grid = np.zeros((N, N), dtype="int")
for i in range(N):
for j in range(N):
new_grid[i,j]=new_value(i, j,grid)
return new_grid, grid
Now we set up the cellular automata
N = 120
STEPS = 800
def initialize():
init_pattern = ["02222222200000",
"21701401420000",
"20222222020000",
"27200002120000",
"21200002120000",
"20200002120000",
"27200002120000",
"21222222122222",
"20710710711111",
"02222222222222"]
for i, line in enumerate(init_pattern):
for j, char in enumerate(line):
grid[i+int(N/2)-5, j+int(N/2)-15] = int(char)
And set up the rules and a function which applies them.
rules = { # format: "CNESW":"C"
"00000":"0","00001":"2","00002":"0","00003":"0","00005":"0","00006":"3","00007":"1","00011":"2","00012":"2","00013":"2","00021":"2",
"00022":"0","00023":"0","00026":"2","00027":"2","00032":"0","00052":"5","00062":"2","00072":"2","00102":"2","00112":"0",
"00202":"0","00203":"0","00205":"0","00212":"5","00222":"0","00232":"2","00522":"2","01232":"1","01242":"1","01252":"5",
"01262":"1","01272":"1","01275":"1","01422":"1","01432":"1","01442":"1","01472":"1","01625":"1","01722":"1",
"01725":"5","01752":"1","01762":"1","01772":"1","02527":"1","10001":"1","10006":"1","10007":"7","10011":"1","10012":"1",
"10021":"1","10024":"4","10027":"7","10051":"1","10101":"1","10111":"1","10124":"4","10127":"7","10202":"6",
"10212":"1","10221":"1","10224":"4","10226":"3","10227":"7","10232":"7","10242":"4","10262":"6","10264":"4",
"10267":"7","10271":"0","10272":"7","10542":"7","11112":"1","11122":"1","11124":"4","11125":"1","11126":"1",
"11127":"7","11152":"2","11212":"1","11222":"1","11224":"4","11225":"1","11227":"7","11232":"1","11242":"4",
"11262":"1","11272":"7","11322":"1","12224":"4","12227":"7","12243":"4","12254":"7","12324":"4","12327":"7",
"12425":"5","12426":"7","12527":"5","20001":"2","20002":"2","20004":"2","20007":"1","20012":"2","20015":"2",
"20021":"2","20022":"2","20023":"2","20024":"2","20025":"0","20026":"2","20027":"2","20032":"6","20042":"3",
"20051":"7","20052":"2","20057":"5","20072":"2","20102":"2","20112":"2","20122":"2","20142":"2","20172":"2",
"20202":"2","20203":"2","20205":"2","20207":"3","20212":"2","20215":"2","20221":"2","20222":"2","20227":"2",
"20232":"1","20242":"2","20245":"2","20252":"0","20255":"2","20262":"2","20272":"2","20312":"2","20321":"6",
"20322":"6","20342":"2","20422":"2","20512":"2","20521":"2","20522":"2","20552":"1","20572":"5","20622":"2",
"20672":"2","20712":"2","20722":"2","20742":"2","20772":"2","21122":"2","21126":"1",
"21222":"2","21224":"2","21226":"2","21227":"2","21422":"2","21522":"2","21622":"2","21722":"2","22227":"2","22244":"2",
"22246":"2","22276":"2","22277":"2","30001":"3","30002":"2","30004":"1","30007":"6","30012":"3","30042":"1",
"30062":"2","30102":"1","30122":"0","30251":"1","40112":"0","40122":"0","40125":"0","40212":"0","40222":"1",
"40232":"6","40252":"0","40322":"1","50002":"2","50021":"5","50022":"5","50023":"2","50027":"2","50052":"0",
"50202":"2","50212":"2","50215":"2","50222":"0","50224":"4","50272":"2","51212":"2","51222":"0","51242":"2","51272":"2",
"60001":"1","60002":"1","60212":"0","61212":"5","61213":"1","61222":"5","70007":"7",
"70112":"0","70122":"0","70125":"0","70212":"0","70222":"1","70225":"1","70232":"1","70252":"5","70272":"0"
}
def new_value(i, j,grid):
c = str(grid[i, j])
a = [str(grid[(i-1)%N, j]), str(grid[i, (j+1)%N]), str(grid[(i+1)%N, j]), str(grid[i, (j-1)%N])]
for i in range(4): # Try every rotation of this array: [TOP, RIGHT, BOTTOM, LEFT]
try:
return rules["".join([c, a[i%4], a[(i+1)%4], a[(i+2)%4], a[(i+3)%4]])]
except KeyError:
pass
return grid[i, j] # if no rule is found, return the previous value
Now we simulate the model printing out every 50th step
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import matplotlib.animation as animation
grid = np.zeros((N,N), dtype="int")
rcParams['figure.figsize'] = 20/2.54, 20/2.54
initialize()
fig,axs=plt.subplots(4,4)
count=0
count2=0
for i in range(STEPS):
grid, old_grid = iterate(grid,new_value) # Iterate & swap the two grids
if (np.mod(i,50)==0):
show_grid(axs[count][count2],old_grid,0)
count=count+1
if count>=4:
count2=count2+1
count=0
plt.show()
Make a video
make_animation=0
if make_animation:
grid = np.zeros((N,N), dtype="int")
initialize()
img = [] # some array of images
frames = [] # for storing the generated imagesfig, ax = plt.subplots()
fig,ax=plt.subplots(1)
for step in range(STEPS):
grid, old_grid = iterate(grid,new_value) # Iterate & swap the two grids
im = show_grid(ax,old_grid,1)
frames.append([im])
ani = animation.ArtistAnimation(fig, frames, interval=50, blit=True,
repeat_delay=1000)
# set output file
writer = animation.FFMpegWriter(fps=15, metadata=dict(artist='Me'), bitrate=1800)
ani.save("Langtons_Loop_movie.mp4", writer=writer)
Total running time of the script: (0 minutes 22.675 seconds)