## Geometry and Arithmetic of a Simple Cellular Automaton

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

#### Abstract

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 *k*th-order predecessors of any number is uniformly distributed over a partition of the interval into uniform segments.