(Detail; click for
Julesz: Computer Graphic, 1965.
Julesz used random black & white grids in his psychological research
on stereopsis. The piece illustrated here looks as if it may have been
derived from such a random pattern by one or two applications of a simple
voting rule. Julesz displayed this piece in an early computer art show
(Howard Wise Gallery, New York, 1965), though he denies any artistic intention.
[Art History students]: Reconstruct the 1965 show by Kenneth
Knowlton and Béla Julesz at the Howard Wise Gallery. Find
descriptions of how this piece (and other, similar pieces) came
Assignment [Computer Science students]: Experiment with
Cellular Automata which yield this kind of pattern from a random
initialization. Devise tests which characterize this kind of pattern
and apply them to the results of the Cellular Automata.
John Conway: The Game of Life, 1970.
A simple but cleverly designed voting rule generates
complex dynamic patterns in a black & white square grid.
Gardner: The fantastic combinations of John Conway's new solitaire game "life", Scientific American 223 (October 1970), pp. 120-123.
W. Flake: The
Computational Beauty of Nature (Cambridge, Mass.: MIT Press,
1998). [Choose "Conway's game of life" in the menu
on the Java-Applets
Meertens and Leo Geurts: Kristalstructuren, 1970.
Kristalstructuren ("crystal structures") is a family of algorithms which
all use majority voting. Some of them use larger neighborhoods than
the immediately adjacent cells. The update regime is incremental:
cells are updated one by one.
Geurts: Kristalstructuren, een experiment in computer-kunst.
Amsterdam: Stichting Mathematisch Centrum, 1973. (Vacantiecursus
Peter Struycken: FIELDS, 1979/1980.
FIELDS uses a square grid with 8-bit color
cells. Update-rule: the color of a cell adapts in the direction
of the average color of the surrounding cells. Repeated application
of the rule thus ends with a uniformly colored surface. But before
this stable endpoint is reached, the automaton goes through a
sequence of configurations of irregular shapes.
An automaton generating snowflake-shapes on a grid with hexagonal
A New Kind of Science,
Paul Coddington and Enzo Marinari: Visualizations
of spin models, 1993
Antiferromagnetic triangular lattice
Ising model (detail)
Antiferromagnetic triangular lattice
Potts model (detail)
O(3) vector spin model
Michael Creutz: XToys,
16 State Potts Model
The interactions between the spin
directions in neighboring atoms of ferromagnetic and antiferromagnetic
materials can be nicely modelled by certain versions of cellular
automata though they can not be captured by the basic
one-layer automaton where all cells are updated simultaneously by
the same update rule at each time step. Simultaneous update of neighboring
cells can be avoided by doing all updates sequentially (as in Meertens & Geurts (1970), mentioned above), or by imposing an update
regime which alternates between "even" and "odd"
positions in the grid. Toffoli & Margolus (1987) use the latter
approach and discuss spin modelling in detail.
The visualizations by Coddington & Marinari were created by
efficient algorithms which maintain descriptions of regions of cells
with identical-looking neighborhoods.
J. Apostolakis, P. Coddington & E.
Marinari: "New SIMD Algorithms for Cluster Labeling on Parallel
Computers" International Journal of Modern Physics C,
Michael Creutz: "Deterministic
Ising Dynamics," Annals of Physics 167 (1986), pp.
Michael Creutz, "Xtoys:
cellular automata on xwindows," Nuclear Physics B (Proc. Suppl.) 47 (1996), pp. 846-849.
Tommaso Toffoli and Norman Margolus: Cellular
Automata Machines. A New Environment for Modeling. Cambridge,
MA.: The MIT Press, 1987.
Turbulence in a Cyclic Cellular Automaton, 1994
Soap Bubble Clustering of a Plurality Vote Rule, 1994
The 'stepping stone' rule:
First, choose a number between 0 and 1; this will be the update
probability for all cells. At every time step, generate a random
number between 0 and 1 for every cell. If the number is lower
than the update probability, the color of the cell changes to
that of one of its neighbors (selected uniformly at random).
(The neighbors are defined here as the four orthogonally adjacent
cells: north, east, south, west.)
Left: the initial condition: 256 colors distributed randomly.
Right: the result of applying the rule repeatedly.
recursive cellular automaton. At every time-step, every cell is
split into 4 cells. The colors of the newly generated cells are
probabilistic variations on the color of the "mother-cell" and the colors of the neighboring cells.