Python | Langton’s Ant

Langton’s Ant is a 4-state two-dimensional universal Turing machine. It was invented by Chris Langton in 1986. It is basically an ant, sitting on a square lattice of cells, which are initially white. The ant moves on the plane and changes the color of cells creating patterns on it. But the movement of the ant is not random; it follows the following set of rules :

  • If the ant is on a black square, it turns right 90 degrees and moves forward one unit.
  • If the ant is on a white square, it turns left 90 degrees and moves forward one unit.
  • When the ant leaves a square, it inverts the color.

As the ant starts, it creates a black and white pattern while moving. Initially, the changes are not distinctive but as we iterate it over and over again, a beautiful pattern emerges. But if we further increase the number of iterations (say ~ 10000), the ant starts repeating its path with a gradual shift, instead of making new patterns. Thus, we obtain a highway like pattern that is infinite. The ant keeps moving on that highway and gives the following pattern.

Refer the visual explanation of Langton’s Ant from here. It helps to visualize how exactly the ant works.

The Python-3 code for Langton’s Ant is given below :

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# importing turtle module
import turtle
  
def langton():
  
    # Initializing the Window
    window = turtle.Screen()
    window.bgcolor('white')
    window.screensize(1000,1000)
  
    # Contains the coordinate and colour
    maps = {}
  
    # Initializing the Ant
    ant = turtle.Turtle()
      
    # shape of the ant
    ant.shape('square')    
      
    # size of the ant
    ant.shapesize(0.5)
      
    # speed of the ant
    ant.speed(10000)                                 
      
    # gives the coordinate of the ant                
    pos = coordinate(ant)                             
      
    while True:
          
        # distance the ant will move
        step = 10                                     
        if pos not in maps or maps[pos] == "white":
              
            #inverts the colour
            ant.fillcolor("black")        
              
            #stamps a copy of the ant on the canvas
            ant.stamp()                                 
            invert(maps, ant, "black")
            ant.right(90)
              
            #moves the ant forward
            ant.forward(step)                         
            pos = coordinate(ant)
              
        elif maps[pos] == "black":
            ant.fillcolor("white")
            invert(maps, ant, "white")
              
            ant.stamp()
            ant.left(90)
            ant.forward(step)
            pos = coordinate(ant)
  
def invert(graph, ant, color):
    graph[coordinate(ant)] = color
  
def coordinate(ant):
    return (round(ant.xcor()), round(ant.ycor()))
  
langton()

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Output :

Here, the white cells are denoted by ‘ ‘ (space) and the black cells are denoted by ‘•’ (dot).



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