Chrystaller found that the number of settlements at any level in either of these hierarchies is directly related to which hierarchy it is.
In a k=3 system for each of the largest settlements in the hierarchy there are on average 3 proximal settlemnents of the next size down in the hierarchy, for each of these again there will be on average 3 proximal settlemnents of the next size down in the hierarchy and so on down the hierarchy to the smallest sized settlement. Christaller noted that this type of hierarchy prevailed where it was most important for society to ensure equal provision of goods and services. Examining the pattern using hexagons Christaller found that central places emerge at the center of a hexagon, containing six lower-order settlements. If transportation costs are to be minimized Chrystaller found that by rotating and enlarging the hexagon that central places emerged where there are 4 proximal settlement of a given size. Finally, for administration purposes Christaller believed the need for 7 proximal settlements of the highest order to drive the creation of the central one into the next order in the hierarchy.
There are various interpretations of Christaller's CPT. Often too much is read into the 3 main systems he studied. The key is the fractal type nature of settlement hierarchies. In the interpretation I've outlined above using proximity K could attain any integer value. It is quite possible that the value of K could be calculated for subareas of Europe and would be helpful in creating EU population surfaces.
Many other geographers and economists have developed
theories to explain central places and they have been applied in many areas
of the world. The following is a list of links to CPT related web sites: