Hexagonal Distributions on Hexagonal Lattice

Abstract

An infinite hexagonal lattice is introduced as a two-dimensional discretized uniform space for economic agglomeration. This chapter focuses on an analysis of geometrical characteristics of the lattice, as a vital prerequisite for the group-theoretic bifurcation analysis of this lattice that will be conducted in Chaps. 6–9. Hexagonal distributions on this lattice, corresponding to those envisaged by Christaller and Losch in central place theory (Sect. 1.2), are explained, parameterized, and classified, and their two-dimensional periodicities are presented. A finite n × n hexagonal lattice with periodic boundaries is introduced as a spatial platform for economic activities of core–periphery models. The symmetry of the n × n hexagonal lattice is described by the group composed of the dihedral group D6 expressing local regular-hexagonal symmetry and the group \(\mathbb{Z}_{n} \times \mathbb{Z}_{n}\) (direct product of two cyclic groups of order n) expressing translational symmetry in two directions. Subgroups relevant to hexagonal distributions of this group are obtained by geometrical consideration and classified in accordance with the study of central place theory.