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

The central place theory (CPT) formulated by Walter Christaller in 1933 and widely introduced into the economic geography literature in the 1950s 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. The original formulation by Christaller - and in the same spirit by Lösch - elaborated in many subsequent applications, is dominated by the hexagonal network, while square and triangular networks are used only sporadically. From an algebraic point of view, CPT is related to number theory and allows for the determination of new properties of numbers called Löschian numbers. However, the basic physical feature of CPT is its stationarity. This means that the hierarchy of central centers results from the “rotation” of market areas, rather than from the social, economic, or cultural properties of these centers. It can be shown that urban centers in a hexagonal network may generate irregular areas of attraction (attractors) whose complexity and shape depend primarily on transportation costs. Furthermore, it can be shown that, if transport costs – i.e., the “resistance” of space – are sufficiently high, the areas of attraction of cities located in a hexagon exactly with the shapes and ranges determined by central place theory. In the present paper we undertake a novel numerical experiment compared to the original hexagonal form, but now with reference to a square grid. The main objective of this work is to compare the relationship between the costs of space resistance occurring in a hexagonal network and the cost implications of space resistance characterizing a square network. This will allow us to determine at which critical level of transport costs the gravitational fractals transform into regular networks predicted by central place paths. The findings of our research will be illustrated by several visualized computer color simulations.