Metric Pattern Theory

Given a set of laws R for the regular structure, they induce natural probability measures over the configuration space T(R) and associated image algebras. This topic — metric pattern theory — was introduced in Section 2.10 of Volume I and we shall pursue it further in this chapter, extend the results to great generality and deepen some of them. When doing this we shall concentrate our attention on the configurations and neglect the corresponding questions for images; see Notes A. A reader can therefore in the present chapter think of the identification rule R as EQUAL, treating images as identical to configuration. Important advances have been made in metric pattern theory after the appearance of Volume I, some of which are contained in two reports by Hwang and Thrift, see Bibliography; much of this chapter is devoted to presenting their results.