Chapter 26 - On the congruence of binary patterns generated by modular arithmetic on a parent array

Abstract

Because material microstructure plays a vital role at nanoscopic scales, studies on gaskets have become of considerable importance in materials science. Inevitably, these and similar developments in other fields have spawned interest in cellular automata and patterns for their own sake, and several reports have come forth. This chapter communicates the results of the investigations on congruence. A paper by Sakamoto and Takagi is also focused––on patterns with residue arithmetic that was brought to attention recently. Presented is the idea of the congruence of binary gaskets generated using modular arithmetic operations on a parent array. The binary gaskets of prime orders are nontrivial, while the binary gaskets of nonprime orders are congruences.