A geographical solution to the geometrical approach to Christaller's centrality problem
In 1933 in « Die zentralen Orte in Süddeutschland » Walter Christaller formulated the problem of the spatial form of the distribution of the central good around a central place in an inconsistent manner. On the empirical level, he asserts that the (geographical) form of the limit of the central good is irregular. On the theoretical level, he assumes that the geometrical form of the central limit is circular (ring).
In 1984, it has been demonstrated that the geometrical solution to the problem of the distribution of the central good in a ring around a central place proposed by Walter Christaller is geometrically false. It has been demonstrated moreover that the exact geometrical solution to the centrality problem stated by Walter Christaller allows the use of any geometrical figure whatever (regular or irregular) of 3, 4, 5 or 6 sides to represent the arrangement of central places around an initial central place.
It has also been demonstrated that the equilateral triangle and the regular hexagon are special cases or extreme cases of the general solution and that the probability of an empirical observation of them is practically nil.