Cellular Automata: 1D, 2D, Game of Life

Cellular Automata: 1D, 2D, Game of Life


A Cellular automaton is a collection of colored cells on a grid. At each iteration (generation) the cells may change state (color) depending on the state of the other neighboring cells. We will look at automata in which the cells may have 2 states only (dead or alive). Such simple automata may produce many kinds of interesting behavior from dull extinction to life-like growth patterns and chaos. In the Cellular automata module of Mathgrapher you may try 3 types of automata. The first 2 ( 1D and 2D) are taken from the book by S. Wolfram entitled “A new kind of Science”. The third one, The Game of Life, is a more general type of automata, sometimes called Life. Dead (uncolored) cells may come to life and living cells may survive or die depending on the status of their neighboring cells. The initial state may be random, or some predefined configuration. Such a configuration may be drawn and stored under some name (Glider, Gosper_gun, block, puffer, etc. see the list). You may start with one of these stored configurations by choosing it from the list. When you start the Game of Life the system evolves according to the Rule given by 2 numbers. Rule 23 / 3 (Conway’s Game of Life) means that (alive) cells with 2 or 3 neighbors will survive and dead cells with 3 neighbors become alive.

More information on Cellular automata: Wikipedia | Variations on the game of life | Life lexicon

Cellular Automata – 2D rule


The second type of automata is a simple two-dimensional one also from Stephen Wolfram’s book “A new kind of Science”. The initial state is a single black cell on the middle of a grid of 300×300 cells. Now the color is determined by the number of black neighbors in the vertical and horizontal direction and by its present color as follow. For example for the number 942, which is 1110101110 in binary notation, we have nr of neighbors:

4 3 2 1 0

color 11 10 10 11 10

meaning that if a cell has 3 neighbors its color will become 10, i.e. black (1) if it was black and white (0) if it was white. The total number of recipes will be 1024.