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I. Historical aspects

An Investigation of the Relationships between Empirical Trends in Both Entropy and Maximum Urban Area Size Over the Last 5,000 Years  (Download pdf)


Antony Harper, Independent Scholar


The previous paper (Harper 2017a) revealed a relationship between modeled system entropy and phases of punctuation and stasis of modeled system population sizes over time. This paper investigates the same relationship between system entropy and population, but does so using empirical data for both entropy, based on total world-system population size, maximum urban area population size, and gamma, γ, a fitted constant and natural log-transformed population data for maximum urban area, all over the last 5,000 years. First, basic entropy trends over human history are investigated, and then by selectively removing outlier points, a process called skeletonization, more detailed patterns of entropy are revealed and analysed. It is shown that a pattern of alternating increasing and decreasing system entropy is associated with alternating periods of stasis and punctuation of maximum urban area size. Also, it is shown that embedded within the pattern of alternating increasing and decreasing entropy are linear and non-linear entropy patterns. Some speculation on the adaptive function of such geometries is shared. Further, the potential of the entropy pattern as a standard of comparison for other historical trends is discussed.

Keywords: entropy, maximum urban area size, stasis, punctuation, skeletoni-zation.


1. The macroscopic view of both maximum urban area population and the entropy associated with the world-system supporting those maximum urban areas exhibit the same hyperbolic pattern of growth.

2. The fine scale of the entropy pattern over the last 5,000 years is revealed by a process of skeletonization in which outlier points are removed.

3. Initial skeletonization reveals an alternating pattern of periods, groups of centuries, of entropy decrease followed by entropy increase. The periods of entropy decrease are approximately 1.7 times longer than the periods of entropy increase.

4. At one level of skeletonization there are four periods of decreasing and increasing entropy. Further skeletonization reduces this number to three.

5. Further skeletonization again reveals two parabolic point distributions, which are separated by two shorter periods of linear change.

6. The parabolas by themselves are sequentially concave up followed by concave down, and the periods of linear change alternate between an initial period having a negative slope and a second period, the one we are currently in, having a positive slope.

7. A composite of these four entropy point distributions provides a model against which the types of world-system historical change may be compared.

8. The future of the world-system may in broad terms only be predicted, given the verity of the combined pattern of entropy change.


The author wishes to extend his heartfelt gratitude to Evgeniya Stolyarova for very professional editing of this paper.


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