Some fundamental differences between Christaller K = 3, 4, and / systems: spatial autocorrelation at work

Abstract

This essay highlights mathematically derived and geographically grounded differences in the properties between Christaller’s three canonical central place schemes (K = 3, 4, and 7), with special emphasis on spatial autocorrelation (SA). It reveals the following groundbreaking insights: (1) implications from judiciously augmented increases in the minimum geographic landscape size that accompany displaying a six-tier hierarchy across the three K values; (2) contiguous SA dominates hierarchical SA; (3) negative SA reigns; (4) SA variance inflation in population counts is considerable, masking the Poisson random variable nature of a system of cities’ populations; and, (5) much can be gained by studying geographic landscapes that are substantially larger than those portrayed in textbooks and almost all research publications. Local indicators of spatial association (LISA) and Griffith-Moran eigenvector spatial filtering (GMESF) analyses confirm systematic differences in both local and global SA within these separate ideal geographic landscapes. Number-theoretically, K = 3, 4, and 7 serve as canonical prime generators of many higher-order Löschian numbers. These distinctions provide a robust, multidimensional SA-anchored framework for characterizing and discriminating among central place hierarchies, suggesting broader potential for integrating number theory and other mathematical subdisciplines with spatial analysis in settlement research.