The focus of this paper is on some antinomies that arise in the analysis of urban-rural relations. First of all: who wins along the centuries? Second: the fight between res extensa, territory, and res cogitans, nets and nodes. Third: the contrast between the theory of Central Places and the Reticular theory. And then the study of the form of boundaries between urban and rural: are they minimizing paths or tend to become fractals so long as possible? In order to give non emotive answers a scientific framework is required, that has the capability of classifying and distinguishing. Mathematics supplies some consistent definitions that can be tailored in order to correspond to real experience. The main concepts of measure theory are presented in an easy-to-grasp form. Some critical and appealing cases are shown, involving the “popular” fractals. In particular the most surprising property of fractional dimension is defined and shown by examples. The section that joins theory with actual world is important since it shows that some approximating mathematical tools can be used as powerful instruments for dealing with real world. Examples of spatial boundaries are given, namely hard boundaries versus distributed boundaries. At the end the long lasting confrontation between countryside and urban structures appears again in three tables dedicated especially to the form of their boundaries
