Complex Systems

Geometry and Arithmetic of a Simple Cellular Automaton Download PDF

Burton Voorhees
Faculty of Science, Athabasca University,
Box 10,000 Athabasca (AB), Canada T0G 2R0


This paper presents the results of a study of the geometric and arithmetic properties of the graph of a simple cellular automaton, considered as a mapping of the unit interval to itself. The graph provides an example of a strictly self-similar figure and exhibits some numeric properties relating to Fermat numbers. In addition an interesting density result is proved: the predecessor set of any number in [0,1] is dense in the interval, and the set of kth-order predecessors of any number is uniformly distributed over a partition of the interval into uniform segments.