A universal rule for the distribution of sizes

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Abstract
Human artifacts, ranging from small objects all the way up to large buildings and cities, display a variety and range of subdivisions. Repeating structural and design elements of the same size will define a particular scale. Most pleasing designs obey an inverse power-law distribution: the product of the relative multiplicity p of a substructure with an algebraic power of its size x is a constant, pxμ = constant, which is derived here from basic principles . This means that the logarithmic plot of p versus x has a slope of -μ , where typically 1 ≤ μ ≤ 2, and this is a widely observed relationship in both the natural and social sciences. Departures from this rule apparently result in incoherent, alien structures. Three applications to urban systems are proposed. It is argued that in living cities: (i) the distribution of path lengths; ( ii) the allocation of project funding; and (iii) the distribution of built elements all follow the derived multiplicity rule. By violating all three, modernist cities create inhuman urban environments.

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