The 3x+1 Problem: A Quasi Cellular Automaton
Thomas Cloney
Laboratory for Computer Science,
Massachussets Institutes of Technology,
Cambridge, MA 02139, USA
Eric Goles
Departamento Matemáticas Escuela de Ingeniería,
Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
Gérard Y. Vichniac
Plasma Fusion Center,
Massachussets Institutes of Technology,
Cambridge, MA 02139, USA
and
BBN Communications Corporation,
Cambridge, MA 02238, USA
Abstract
Computer graphic techniques are applied to a celebrated problem. The iterates of the "
" function are studied in base two as a quasi cellular automaton. This geometric approach brings to a problem, previously studied mainly from the viewpoint of number theory, questions and methods associated with automata theory and the study of discrete dynamical systems. Generic behaviors associated with classes of seed values are demonstrated. Periodic and chaotic structures within iterate patterns are investigated. Fluctuations from a steady decay are found to be independent of iterate size.
