Local Structure Theory: Calculation on hexagonal arrays, Interaction of rule and lattice.

Howard A. Gutowitz and Jonathan D. Victor

1:Laboratory of Mathematical Physics
The Rockefeller University
1230 York Avenue
New York, New York 10021-6399

2:Department of Neurology
Cornell University Medical College
1300 York Avenue
New York, New York 10021

3:Laboratory of Biophysics
The Rockefeller University
1230 York Avenue
New York, New York 10021-6399

J. Stat.Phys. 1987

Abstract:

We apply local structure theory calculations to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway's Life. The local structure theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study of Life itself.

As in Life, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on different lattices. The differences between the behavior of these rules on different lattices are dramatic. The mean field theory cannot reflect these differences. However, a generalization of the mean field theory, the local structure theory, does account for the rule/lattice interaction.

• Introduction
• Empirical Results
  • Outer-totalistic rules
  • Outer-totalistic rules in one-dimension
• Analytic Approximation of Statistical Properties
  • The mean field theory
  • Local structure theory of one-dimensional Life analogs
  • local structure theory on the hexagonal lattice
  • Extension of a block probability function on the hexagonal lattice
  • The triangle approximation,
  • 
  • The hexagon approximation,
  • Numerical studies of rule (3422)
• Conclusions
  • Acknowledgements
• References
• Figure Captions