Gli Alberi non bastano (e neppure i semireticoli): Una lettura probabilistica dell’inadeguatezza di alberi e semireticoli alla descrizione delle strutture urbane. (L'architettura delle città - The Journal of the Scientific Society Ludovico Quaroni, 2021)
The famous essay A city is not a tree by Christopher Alexander (1965) proposes the use of two mathematical structures, trees and semi-lattices, to describe the urban environment. In particular, Alexander recognizes in the tree structure the key to the failure of many artificial cities, at the same time indicating in the semi-lattice structure the main of spontaneous or natural settlements. In this work we try to show from a quantitative point of view how in reality both structures involve too stringent constraints on urban planning, making it impossible to satisfy both axioms identified by Alexander as the population increases (hence the title). Although these limits have already been highlighted in the literature, we have tried to use an approach as close as possible to the set theory language used by Alexander himself, using probabilistic and combinatorial calculus tools. The results of the work are anticipated by an exposition of how it is possible to describe with the theory of sets the elements of the urban environment that are of primary importance in modeling the problem. Quantitative are followed by a discussion of possible alternative approaches to the urban planning problem presented in the literature.