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Cellular Automata

Our page on grids discusses why one might want to describe an image as a discrete grid of color values. If such a description is taken as "realistic" (rather than as an arbitrary representation), it implies an image generation strategy: to specify classes of images by means of local processes on color grids. The mathematical articulation of this idea is the cellular automaton.

A cellular automaton is a grid of cells whose "color" changes in time. Time moves in discrete steps, and at every step, every cell assumes a new color. An update rule specifies how the new state of a cell depends on its old state and on the old states of its neighboring cells.

Different cellular automata can be specified, varying in the geometry of the grid, the number of states of the cells, the definition of "neighborhood", and the nature of the update rule. For instance, the grid can be one-dimensional, two-dimensional or three-dimensional. The cells in a two-dimensional grid are usually square, but they may also be triangular or hexagonal. The state of every cell may be a Boolean (black and white), a scalar (grey-tones), or a three-tuple of scalars (colors). The neighborhood is often the area within one cell distance, but it can also be larger; diagonal neighbors may be included or excluded. The update rule can be deterministic or probabilistic. Often it is a "voting rule", which only counts the number of occurrences of the states of the neighbors, without taking into account their positions.

Because of the local nature of the update rules, the behaviour of cellular automata may be reminiscent of physical, biological or social processes. The structures observed in visualizations of the resulting configurations are typical examples of "emergent phenomena": they are not pre-programmed, but arise out of an unforeseen interaction between the update rule and the initial conditions. Often, the emerging shapes have a natural, organic look. Cellular Automata thus demonstrate that constructivism may be process art, and vice versa.

 1D  1-Dimensional Cellular Automata

  2-Dimensional Cellular Automata

  3-Dimensional Cellular Automata

General overviews:

A.W. Burks (ed.): Essays on Cellular Automata. Chicago: University of Illinois Press, 1970.

Gary W. Flake: "Cellular Automata." In: The Computational Beauty of Nature. Cambridge, MA: MIT Press, 1998.

Herbert W. Franke: Zellulare Automaten - Modelle für die physikalische Welt. In: Naturwissenschaftliche Rundschau nr. 9, 1996.

Tommaso Toffoli and Norman Margolus: Cellular Automata Machines. A New Environment for Modeling. Cambridge, MA: MIT Press, 1987.

Stephen Wolfram's Articles on Cellular Automata (1983-1988).

Stephen Wolfram: Cellular Automata and Complexity. Collected Papers, 1994.

Stephen Wolfram: "Computer Software in Science and Mathematics." Scientific American 251, 3 (September 1984), pp. 188 - 203.

Stephen Wolfram: A New Kind of Science, 2002.

Applications (Physics):

Dale P. Bentz, Peter V. Coveney, Edward J. Garboczi, Michael F. Kleyn and Paul E. Stutzman: "Cellular Automaton Simulations of Cement Hydration and Microstructure Development,"    Modelling and Simulation in Materials Science and Engineering, 2, (1994), pp. 783-808.

Michael Creutz: "Deterministic Ising Dynamics," Annals of Physics 167 (1986), pp. 62-76.

Michael Creutz, "Xtoys: cellular automata on xwindows," Nuclear Physics B (Proc. Suppl.) 47 (1996), pp. 846-849.

Yoshiaki Takai: Graphics Applications of Cellular Automata, 1994-2000

Applications (Social Sciences):

Edwin Dethlefsen and Carlisle Moody: "Simulating Neighborhood Segregation. A BASIC program gives surprising insights into some of the forces that hamper integration." BYTE, July 1982, pp. 178–206.

Garret Wilson: The Complex World: Nonlinear Dynamical Systems as a Paradigm for International Relations Theory. University of London, School of Oriental and African Studies. MA Thesis International Studies and Diplomacy, 1999.


Mirek's Java Cellebration


Tim Tyler's Cellular Automata Links

Mirek Wójtowicz's Cellular Automata Links