Conway's Game of Life
Author scheinerman
1 Star
Updated Last
1 Year Ago
Started In
April 2020


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If A is a zero-one (integer) matrix, then life_step(A) gives the next generation from A in Conway's cellular automata Game of Life. The life_run function shows the progression of the cellular automata on the screen. Use life_movie to create an animated GIF.



Cellular automata are represented by an integer matrix with 1s representing live cells and 0s dead cells. Given such a matrix A, the function life_step(A) returns a new matrix that is the one-step progression of the cellular automata. Optionally, life_step(A,true) signals that the matrix wraps around (left and right edges are identified, as are top and bottom edges) and so the domain is a torus.

We also provide life_step!(A,wrap=false) which overwrites A with the next generation. life_step! returns true if the new A is different from the old A and false if there was no change.


The function A = random_life(n,m) creates a random n-by-m zero-one matrix as a starting point for the cellular automata.

random_life(n,m,p) creates an n-by-m random matrix in which the the probability a cell is a one is p.

  • random_life(n) is equivalent to random_life(n,n).
  • random_life(n,p) is equivalent to random_life(n,n,p).


The life_run(A) function is used to run and visualize the cellular automata. The run goes on endlessly, but will stop if there are no live cells, if two consecutive generations are identical, or if it enters a period-2 oscillation. If life_run halts, it returns the number of iterations.

This function takes the following optional named arguments:

  • pause=0.0 is the number of extra seconds between iterations. Note: The first image may take a while to appear as the plotting software initializes.
  • wrap=false determines if the cellular automata field wraps. If true then the domain is a torus.
  • counter=false shows the iteration number under the image.
  • max_steps places a limit on the number of steps to run.


life_movie(A) creates an animated GIF file. Use the following named arguments to control the result:

  • file_name="life.gif": name of the GIF file in which to save
  • wrap::Bool=false: edge wrapping (same as life_run)
  • max_steps: limit on the number of steps (frames)
  • rate=5: animation frames per second



The life_step function is reasonably fast, but life_run is slow for large boards because of the visualization.

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