Central Place Theory after Christaller and Losch && : Some Further Explorations

This paper deals with the critical reevaluation of the methodology of classical Christaller Losch && Central Place Theory. In the beginning of the paper the reconstruction of Central Place Geometry on the basis of the Mobius && Barycentric Calculus was considered. On this basis a superposition model of the actual Central Place System is constructed. Building blocks of this model are the Beckman-McPherson models representing the main tendencies of optimal organizations of space acting simultaneously in the actual Central Place system. The weights of these building blocks represent the level of realization of the specific extreme tendencies in the real system. The algorithm of decomposition of an actual Central place system into the weighted sum of the Beckman-McPherson building blocks is elaborated and presented in detail. This algorithm generates also the description of the interconnections of hexagon coverings on the sequential hierarchical levels with the help of convex combinations of hexagon coverings and homothetic transformation of the coverings. Next, the (jumping) catastrophic dynamics of the Central Place hierarchies presented with the help of geometrical scheme of the movement of the point representing actual Central Place system in the polyhedron of all admissible Central Place systems. Two main applications of this conceptual framework are elaborated: • The enumeration of all structurally stable optimal (minimal cost) transportation flows in the hierarchical Central Place system and • The merger of two major theories in the Regional Science: the classical Input-Output theory of Leontief and the classical Christaller Losch && Central Place theory. We hope that this critical reevaluation of the geometrical and conceptual basis of Central Place theory will contribute to narrowing the existing gap between the formal theory and empirical studies.