A note on pattern reproduction in tessellation structures

摘要

Tessellation structures that reproduce arbitrary patterns are special cases of tessellation structures having local transformations that are linear operators. We introduce a novel formulation of tessellation structures which emphasizes the connection between these structures and concepts of functional analysis. Using this formulation a behavioral analysis technique is developed which implies the earlier results on pattern reproduction and generalizes them to tessellation structures whose state alphabets are arbitrary fields of non-zero characteristic and whose tessellation arrays are arbitrary countable abelian groups. It is also shown that a local transformation can be chosen to produce at a specified time any desired set of “copies” of an initial pattern each multiplied by a specified scalar. We then indicate that connections exist between linear tessellation structures and linear partial differential equations which describe wave propagation by giving an example of a classical form of pattern reproduction.