Christaller understood the power of this model of an economic landscape (now referred to as an isotropic plane):

This model is “correct” in itself, even if it is never to be found in the reality of settlement landscape in pure form: mountain ranges, variable ground; but also variable density of population, variable income ratios and sociological structure of the population, historical developments and political realities bring about deviations from the pure model. (Christaller, 1972, p. 608)

In effect, the model allows Christaller to separate the signal (pattern) from the noise (deviations) and, at the same time, to offer explanations for both signal and noise. He also recognized the link between space and time, pattern and change:

I thus did not satisfy myself with setting up a model for an invariable and constant economic landscape (thus, for a static condition)—but instead I also tried to show how the number, size and distribution of the central places change, when the economic factors change…. (Christaller, 1972, p. 608)

The explanations were built on the concept of a central place. Christaller saw settlements of all sizes as serving functions (commercial, governmental, educational, recreational, medical, etc.) for themselves and for a surrounding population. To use this conceptual building block, Christaller had to find a way of measuring the quality of centrality, and to do so, he used the number of telephone connections as an indicator, scaled to the population size of the place. Here we can see echoes of his childhood preoccupation with “statistics.”

According to Christaller (1972, p. 609), given this connection indicator: “I was able to assign all central places in South Germany value indication numbers from 1 to 2825 (Munich).” After arraying the values into size classes: “I was able to mark in the central places in South Germany, according to their significance as central places, on a map, to measure the distances among them, and to determine their rank in the hierarchy of their ‘complementary areas’ which belonged to them (i.e., of their surrounding territory).”

At this point, the mapped patterns and the theoretical patterns intersect in a brilliant exercise in spatial thinking. Settlements are scaled by size—and therefore distance—classes, and these classes are organized into a hierarchy that captures size, number, spacing, and spatial arrangements. Different types of hierarchy arranged settlements on the basis of different logics of grouping and dependency: K-3 for efficiency or markets, K-4 for optimal traffic flows, and K-7 for efficiency of administration:

… the lowest ranked central places have a surrounding territory with a radius of 4–5 kilometers, and an average population of 3,000 (in the central place and surrounding territory) and their distances from the nearest central places, which according to the model lie within hexagonal (six-sided) connections, amount to seven kilometers. Within the completely regular hexagonal system of central places, the distances of all higher ranked central places can be derived, inasmuch as one multiplies the distance of the next lowest ranking central place by √3; the sequence of distances is thus: 7–12–21–36–62–108–185 … km. (Christaller, 1972, p. 609)

Christaller had explained the size, number, and distribution of settlements in one theoretical structure that has both an economic and a geographic rationale to it. It has both an algebraic and a geometrical interpretation. It is both static and dynamic. It is both abstract and concrete: “Thus, I was able to find surprising concurrences between geographical reality and the abstract schema of the central places (the theoretical model) especially in the predominantly agrarian areas of North and South Bavaria” (Christaller, 1972, p. 609).

The process of development of this intellectual structure also showed echoes of Christaller’s youth:

On Sundays, or also Saturday and Sunday, I used to hike around in the beautiful landscapes of

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