International Encyclopedia of Systems and Cybernetics

2nd Edition, as published by Charles François 2004 Presented by the Bertalanffy Center for the Study of Systems Science Vienna for public access.


The International Encyclopedia of Systems and Cybernetics was first edited and published by the system scientist Charles François in 1997. The online version that is provided here was based on the 2nd edition in 2004. It was uploaded and gifted to the center by ASC president Michael Lissack in 2019; the BCSSS purchased the rights for the re-publication of this volume in 200?. In 2018, the original editor expressed his wish to pass on the stewardship over the maintenance and further development of the encyclopedia to the Bertalanffy Center. In the future, the BCSSS seeks to further develop the encyclopedia by open collaboration within the systems sciences. Until the center has found and been able to implement an adequate technical solution for this, the static website is made accessible for the benefit of public scholarship and education.


SPACE STRUCTURATION (Two dimensional) 2)4)

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


  • 1) General information
  • 2) Methodology or model
  • 3) Epistemology, ontology and semantics
  • 4) Human sciences
  • 5) Discipline oriented


Bertalanffy Center for the Study of Systems Science(2020).

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Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]

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