This attempt at explaining the spatial economy as a whole is considered the highlight of all location theories. The most important theory is the central place theory which was created Walter Christaller (1893-1969) in 1933. Christaller´s aim was to explain the location and size of settlements in an urban system while assuming rational consumer and seller behavior and homogeneous surface (i.e. geographical factors). His theory was based on the analysis of retail and services network in the south of Germany - this was where his assumption of homogeneous surface came from. Christaller made an unambiguous generalization about space - the larger the settlements are in size, the fewer in number they will be (Blazek and Uhlíř, 2002). The result of Christaller´s work is the conclusion that with the neoclassical assumptions of perfect mobility, etc. the optimal solution is hexagonal lattice where the hexagons neighbor one another and localized centers are in their middles.

Hexagons were selected as they ideally divide space - in experiments with circles there were problematic areas created by areas not covered by circles or by overlapping circles (see fig below). In the former case there were areas which were not serviced, in the latter there were overlaps on the serviced areas.

Geometric models of central place lattice (Source, adapted from Pacione, 2009)

Hexagonal lattice reflects two basic concepts of the theory: the minimum market - threshold and maximum distance - range. The minimum market is the lower limit of spatial expansion of a particular service. The upper limit is then the maximum distance from the center (from the service provided) which consumers are prepared to travel to acquire service. The space determined by the upper limit is a sphere of influence (see fig). The size of the centers affects the type of service which the center provides (Krejčí and coll., 2010).

Sphere of influence of the center and the limits of its size (Source: own)

Focusing services in the centers generates a hierarchy of central places - the higher the order of the services provided the higher order the center occupies in the hierarchy. Christaller was working with the assumption (too simplified to be used for reality) that higher order center provide all functions (services) of lower order centers (Blazek and Uhlíř, 2002). The hierarchy of centers is illustrated by the following picture.

Basic spatial center hierarchy (Source: Ježek, 2002)

Spatial hierarchy of centers have, according to Christaller, three possible layouts: marketing, transport and administrative. Every layout is defined by a K-value. These show how much the sphere of influence of the central places takes in as opposed to the centers of lower order (or how many centers are subject to a center of higher order).

Marketing principle is based on minimal number of centers needed to service an area and is defined as K-3 principle. To minimize the number of centers the centers of lower order are located at the corners of a hexagons and each high-order settlement gets 1/3 of each satellite settlement. A high-order centers is surrounded by six satellite centers - for every high-order center there are 3 lower-order centers on average - see figure below for an illustration. Consequently, a center of higher-order serves three times the area than a center one order lower.

Marketing principles of central place theory (Source: Christaller, 1933; modified)

Transport principle (K-4) involves the minimization of the length of roads connecting central places. In hexagon lattice the shortest connection is using the edges of hexagons - lower order centers are all located along the edges and every one of them is for two higher-order center. The market area of a higher-order place includes a half of the market area of each of the six neighboring lower-order places. For each higher order center, there are now four centers of immediate lower order (see fig).

Marketing principles of central place theory (Source: Christaller, 1933; modified)

Administrative principle (K-7) - the market areas of the smaller settlements are completely enclosed within the market area of the larger settlement and all lower-order center are located inside the higher-order center - i.e. tributary ares are allocated exclusively to a single higher-order place. Therefore, in a hexagonal lattice the area serviced by a higher-order center is seven times larger than the area serviced by one order lower.

Administrative principles of central place theory (Source: Christaller, 1933; modified)

Besides these three basic approaches Christaller proposed other modifications which, together with a more detailed description of central place theory, may be found in the work of Ježek (2002). Christaller´s theory was used, besides Germany, in the Polish areas annexed by Germany in WWII (see Gregory et al., 2009). It has also been applied in the Dutch polder near Emmeloordu, in Israel during the settlement of new immigrants and providing services for remote farms (Yiftachel a kol., 2001). In the Czech Republic, as pointed out by Musil (2001), the central place theory was used when creating centers of settlements in former Czechoslovakia.

Christaller´s theory inspired many followers to further modify it. Some of these include August Lösch (1906-1945) who assumed the motivation of actors will be sparked by maximizing profit via monopoly standing which will fail to lead to equilibrium in the market. Lösch abandoned some of the simplifying assumptions of Christaller – he did not, for instance, add the functions of lower-order centers automatically for higher-order centers(Blazek a Uhlíř, 2002), experimented with industrial and agricultural sectors, used other geometric shapes besides hexagons and he built his model starting with the lowest-order centers. Due to this, the centers could specialize and differentiate their supply from other centers. His efforts are reflected in a model in which there is a dominant center divided into sectors with a different number of centers of lower-order.

Lösch central place system (Source: Ježek, 2002, own)