Fractal dimension complexity of gravitational fractals based on hexagonal and square lattices: Christaller and Lösch revisited

Abstract

The central place theory (CPT) formulated by Walter Christaller in 1933 and in the same spirit by Lösch was widely introduced into the economic geography literature in the 1950s. CPT has become one of the leading theoretical frameworks describing the spatial order of settlements in the space-economy, and consequently, the geographical differentiation and specialization of economies in various regions and countries. In a geometric sense, CPT is essentially based on a comparative study of three types of regular network configurations that completely cover a plane, i.e., hexagonal, square, and equilateral triangle networks. This paper examines spatial interaction patterns between central places using a dynamic gravitational model applied to square and hexagonal lattices. The study focuses on the formation of attraction basins and their geometric properties, which are analyzed through numerical simulations and visualized as color maps. The model represents spatial interactions as movements of agents in a potential field generated by central places, with transport costs expressed through a spatial resistance parameter. The results show that, for low spatial resistance, the system exhibits irregular and fractal-like structures, reflecting complex and highly sensitive spatial interactions. As spatial resistance increases, these patterns gradually converge toward regular geometric configurations consistent with central place theory. A comparative analysis demonstrates that neither square nor hexagonal lattices exhibit a systematic advantage in terms of attraction basin size. Instead, the spatial structure is primarily determined by the level of transport costs and the relative importance of individual centers. The findings are further illustrated using empirical examples from real-world city networks in the United States. These examples confirm that actual spatial interaction patterns deviate from ideal geometric forms and are shaped by heterogeneous spatial conditions. The study contributes to the understanding of spatial structure by providing a reproducible modeling framework that links geometric theory, dynamic processes, and empirical observations.