International Encyclopedia of Systems and Cybernetics

The subject was tackled by W. CHRISTALLER (1933 & 1937) and again by A. LÖSCH (1944).

L.F. RICHARDSON, in turn, studied it in an appendix to his "Statistics of deadly Quarrels" (1961, p.139-87), in relation to the influence on wars (foreign or civil) of population densities and locations.

He defined the notion of "homoplaty", i.e. the condition of a closed surface about as broad and as long as possible: "The geometric figure which satisfies completely the property of homoplaty is the circle. It is known, however, that a two-dimensional region cannot be divided into contiguous circles".

Thus, the search for homoplaty becomes the search for the best possible contiguity. As already discovered by CHRISTALLER, this is obtained by clustering space hexagonally. RICHARDSON found that 2-dimensional space can be thus partitioned in hexagonal cells in any extension and at any level by using the formula:

3K – 3K + 1

… in which K must be an integer.

Moreover, any hexagonal grid can be fractalized with cells of 1/7th, or 1/7 or 1/7, etc. of the area.

The general rationale behind this is that hexagonal clusters express iso-pressure between human populations in territory occupation.

CHRISTALLER had also discovered that growing density of occupation tends to lead to the appearance of population (and services) concentration at the center of every hexagon.

This kind of implosion has been explained later by P. ALLEN et al, through dissipative structuration due to massive inputs of energy, thus adding a dynamic dimension to these models of space occupation.

RICHARDSON had however discovered that hexagonal grids could become deformed under some strain, as for example centrifugal or centripetal and gave formula for what he respectively called "central desert" and "central town".

This effect he called "conformal representation" which "allows the honeycomb pattern to be strained in all those manners which leave the cells almost regular hexagons but no longer all equal" (1961, p.154).

More recently, M. MARUYAMA observed that there is an inhibiting effect exerted by any element on elements of the same class, which leads to what he called "spacing". He adds: "Spacing contributes to heterogeneization" (1994, p.81).

This rule also seems to be valid for biological organisms and in interpersonal contacts (see for instance E. HALL's proxemic research, specially "intrusion distance" – 1959, 1966, 1977).

→ Hexagonal space filling