[Game of Life animated gif.]
[Game of Life animated gif.]
Cellular automata were first investigated by John von Neumann and Stanislaw Ulam while studying self replicating systems. A cellular automaton is a spatial lattice which can have any one of a finite number of states and which are updated synchronously in discrete time steps according to a local (nearby neighbor) rule. One of the most popular two-dimensional examples of cellular automaton is the Game of Life devised by the British mathematician John Horton Conway in 1970.
Description from Wikipedia.
The Game of Life is a zero-player game, meaning
that its evolution is determined by its initial state, needing no input from human players. One interacts with
the Game of Life by creating an initial configuration and observing how it evolves. A variant exists where two
players compete. Rules: The universe of the Game of Life is an infinite two-dimensional orthogonal grid
of square cells, each of which is in one of two possible states, live or dead. Every cell interacts with its eight
neighbors, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step
in time, the following transitions occur:
The initial pattern constitutes the 'seed' of the system. The next generation is created by applying the above rules simultaneously to every cell in the seed - births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick. (In other words, each generation is a pure function of the one before.) The rules continue to be applied repeatedly to create further generations.)
The following EJS models show examples of cellular automata. These models are listed in order of EJS programming complexity.
The core idea that very simple rules often generate great complexity is now an established idea in science. For example, many millions of organic compounds having an immense range of physical properties can be constructed from carbon, oxygen, and hydrogen and can be understood using the Schrödinger equation. Although the stronger conjecture that simple automata rules may underlie physical laws predates Steven Wolfram's book A New Kind of Science, this book provides a comprehensive overview of automata and many examples of the applicability of this idea.
Cellular automata models are also import in the social sciences. See, for example, the following books.
Cellular automata are often discussed in books on chaos and nonlinear dynamics. The following books connect the mathematics with physical systems.
This implementation of the Game of Life was created by Wolfgang Christian and Francisco Esquembre using the Easy Java Simulations (EJS) authoring and modeling tool. You can examine and modify a compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open Ejs Model" from the pop-up menu. You must, of course, have EJS installed on your computer.
Information about Ejs is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.