Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

The making of geometry

Geometry has been a source of inspiration in the design of the manmade world for millennia; it also provides representational means enabling development of a concept into a built object. In the past three decades computing methodologies have provided the designer with unprecedented tools to explore highly complex forms, create digital models and fabricate them. This paper describes a computational methodology for the transition of forms from abstract geometric configurations to physical objects: a parametric design process assists from the initial ideation to the final prototyping with 3D printing technologies. The five regular polyhedra are used as a case study; this paper explores how parametric based procedures develop these geometric shapes into digital models of structures to be fabricated in different sizes and materials

Non-omega-overlapping TRSs are UN

This paper solves problem #79 of RTA’s list of open problems [14] — in the positive. If the rules of a TRS do not overlap w.r.t. substitutions of infinite terms then the TRS has unique normal forms. We solve the problem by reducing the problem to one of consistency for “similar” constructor term rewriting systems. For this we introduce a new proof technique. We define a relation ⇓ that is consistent by construction, and which — if transitive — would coincide with the rewrite system’s equivalence relation =R. We then prove the transitivity of ⇓ by coalgebraic reasoning. Any concrete proof for instances of this relation only refers to terms of some finite coalgebra, and we then construct an equivalence relation on that coalgebra which coincides with ⇓

Explicating structural realism in the framework of the structuralist metatheory

A form of structural realism affirms that, when our theories change, what is always retained is their structural content and that there is structural continuity between our theories, even through radical theory change. I first introduce and discuss structural realism, with a focus on structural realism and change theory. Then, I will consider some critiques on structural realism. In order to address them, I introduce the framework of the so-called structuralist metatheory and allude to the notion of reduction, arguing that this notion provides the formal elucidation of the notion structural continuity. This aims to get a precise notion of continuity of structure, which is central to structural realism and to the understanding of theory change. In this sense, I propose a new way of formulating structural realism in an appropriate formal framework, namely, the framework of structuralist metatheory.Uma forma de realismo estrutural afirma que, quando nossas teorias mudam, o que sempre é retido é seu conteúdo estrutural e que há continuidade estrutural entre nossas teorias, mesmo com a mudança radical da teoria. Em primeiro lugar, apresento e discuto o realismo estrutural, com foco no realismo estrutural e na teoria da mudança. Depois, vou considerar algumas críticas ao realismo estrutural. Para abordá-las, introduzo o quadro da chamada metateoria estruturalista e faço alusão à noção de redução, argumentando que esta noção fornece a elucidação formal da noção de continuidade estrutural. Isto visa obter uma noção precisa de continuidade da estrutura, que é central para o realismo estrutural e para a compreensão da mudança teórica. Neste sentido, proponho uma nova forma de formular o realismo estrutural num quadro formal apropriado, nomeadamente, o quadro da metateoria estruturalista

Relational Complexes

In this proposal the aim is to analyse the fortified city of Arezzo from unpublished archival documents. The Johannite Commandery of S. Jacopo, today no longer existing, was part of the urban setting of Arezzo and was located near the Porta Santo Spirito. This ancient fortification survives today. It stands as a very important example of military constructions for its massive polygonal town walls which were built between 1538 and 1560 by Antonio da Sangallo il Giovane on the site of the old Medieval citadel. The Church of S. Jacopo was destroyed to make way for new urban plans in the post-war period. Still in the urban area traces of the Order of Malta’s architecture survive. Our explanation attempts to explore the connection of this commandery with the fortified city. From such perspective it is interesting to analyse the setting up and functioning of the commandery within a fortified-urban framework. In this analysis studying the drawings produced by the land surveyors from the cabrei is of utmost importance. These unpublished documents, part of the ancient archive of the Priory of Pisa, offer in fact an unusual representation of a fortified city, which is now preserved in the Archivio di Stato in Florence