## 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

## Cellular Automata – 1D rule

The first one is a simple one-dimensional Automaton from Stephen Wolfram’s “A new kind of Science”. The colour of a cell is given by 0 (white) or 1 (black). The initial state is either a single black cell on the middle of the first row or a randomly choice of black and white cells on the first row. At each iteration the pixels status of the next row is determined by the status of the 3 nearest neighbours in the row above it according to the rule number written in binary format. In the example given below the Rule number is 30: = 00011110. There are eight possible combinations for the colours of the 3 nearest neighbours:

111 110 101 100 011 010 001 000.

For each combination the colour of the cell is given by the number 30 in binary format:

0 0 0 0 1 1 1 0 = 30. So the first 4 combinations (111, 110, 101 and 100) yield 0 (white cell), and the fifth (011) yields 1 (black).

The total number of combinations is 256 (2^8).