Time-space pattern and periodic property of elementary cellular automata — Sierpinski gasket and partially Sierpinski Gasket —
Cellular automata have interested many researchers because of the unexpected complex time-space patterns that are generated by very simple rules of cellular automata. Especially Wolfram   classified the cellular automata by computer simulations, and his work has inspired the research on cellular automata since then.
Braga et al.  gave an effective classification of cellular automata according to the behavior on the set of all the 0-finite configurations and showed an algorithmic method to classify 0-quiescent elementary cellular automata. They presented in the thesis a conjecture on the cellular automaton with rule number 180 by graphically observing the time-space patterns of the cellular automaton. The conjecture insisted that the dynamics determined by rule 180 was shift like, which they called generalized shift dynamics, and was proved to be true by Cattaneo et al. .
It is graphically understood that the time-space patterns of rule 180 are basically same to those cut off from the time-space patterns of rule 90. Braga et al.  and Cattaneo et al.  did not necessarily explain such the feature. They mainly concerned the topological behavior of the dynamics of cellular automata, and not on the time-space patterns.
In this article we logically prove that the time-space patterns of rule 180 are those cut off from the time-space patterns of rule 90, and then prove the conjecture of Braga et al.  in an extended case.
Key wordscellular automata rule number 180 rule number 90 periodic trajectory period
- C.G. Langton, Life at the edge of chaos. Artificial Life II, Proceedings of the Workshop on Artificial Life (eds. C.G. Langton, C. Taylor, J.D. Farmer and S. Rasmussen), February 1990, Santafe, New Mexico, Addison-Wesley Publishing Company, 1990, 41–91.Google Scholar