Introduction

This special issue of the Nexus Network Journal presents the second set of papers chosen for development and expansion from presentations given at the 13th international, interdisciplinary conference “Nexus 20/21: Relationships Between Architecture and Mathematics” in July 2021 in Kaiserslautern, Germany (Fig. 1). Starting with over 50 presentations at the conference, a short list was prepared by the scientific committee for further development and peer review. At the end of this process a set of 22 papers were selected for publication in two special issues of the Nexus Network Journal. Vol. 24(2) presented the first of these papers and this issue contains the second set.

Fig. 1
figure 1

Screenshot of the Nexus 20/21 conference website https://nexus2021.architektur.uni-kl.de

Papers in this Special Issue

The research papers presented in this special issue of the Nexus Network Journal are organised into four themes: (i) spatial distortion and perception, (ii) computational design methods, (iii) design concepts in historical examples and (iv) stereotomy, vaults and domes. The Didactics section of the journal features research which examines teaching and learning themes which are pertinent to the education of architects to use mathematical theories, processes, or methods.

Spatial Distortion and Perception

The first two papers of the issue have a common interest in spatial distortion and perception. The first paper, “Surface Ornamentation Techniques for Spatial Distortion” by Javier Martín and Daniel Martín Fuentes describes different types of surface-related ornamentations that can distort the perception of space. The paper draws on both historical and recent examples to develop a classification of three different ornamental techniques: quadrature, varinism, and sciography. The suggested taxonomy is exemplified by works of contemporary artists.

The next paper in this section, “The Masterly Perspective Design of Bramante’s ‘Mirabile Artificio’ in Milan” is by Giorgio Buratti, Giampiero Mele and Michela Rossi. It analyses an early example of a perspectival distortion in architecture in Bramante’s church of Santa Maria Presso San Satiro in Milan. The research method employed in this paper uses a laser scanner survey to find the station point on the digital model and to identify the “Mirabile Artificio” as an anamorphosis.

Computational Design Methods

In this section two different computational design methods are presented, the first related to urban design and the second to architectural form. In “Cellular Automata for Infill Designs in Historic Urban Quarters” Pınar Çalişir Adem and Gülen Çağdaş present a computational approach to generating infill designs in historic urban quarters. To do this they utilize the urban knowledge embedded in architectural morphology and a Cellular Automata-based model which generates conceptual building massing proposals.

The second paper in this section, “Form Follows Function in Hyperboloidical Cooling Tower”, is by Rachele A. Bernardello and Paolo Borin. This paper presents a computational method to study and test the relationship between shape and performance in the built environment. The method is demonstrated by means of a case study of a 1938 cooling tower in Marghera, Venice. The method enables shape optimization by varying parameters.

Design Concepts in Historical Examples

In this section two mathematical design concepts in historical examples are studied. The paper, “The Modulor in Manuel Tainha’s Oliveira do Hospital Pousada” by Teresa Belo Rodeia and João Miguel Couto Duarte, assesses the application of the Modulor, the system of proportions by Le Corbusier, to the Oliveira do Hospital Pousada designed by Manuel Mendes Tainha. The analysis reveals the importance of geometry in the design theory and practice of Tainha.

The use of polyhedral forms in the unusual ceiling design of the Picasso Museum in Malaga, Spain is the subject of the next paper in the issue, “Polyhedra and Honeycombs from the Octagonal Wooden Ceiling of Picasso’s Museum in Malaga” by Antonia Redondo Buitrago. The honeycombs forming the wooden ceiling can be considered as a polyhedral generalization of the planar pattern of regular octagons and four-pointed stars, constructed by means of special truncations of a cube. Such design concepts demonstrate a way for generating and understanding new designs.

Stereotomy, Vaults and Domes

The study of a rectangular plan sail vault built by brick slices in the Roman villa of Carranque in Spain is the subject of the first paper in this section, “Geometry and Actual Construction in Brick Vaults by Slices. The Case of Carranque in Spain” by Ana López-Mozo, Enrique Rabasa-Díaz, José Calvo-López, Miguel Ángel Alonso-Rodríguez, and Alberto Sanjurjo-Álvarez. The research is part of a broader project that analyses the constructive configuration of this kind of vaults in the Mediterranean region. For this paper automated photogrammetry was used to obtain a three-dimensional model, allowing a hypothesis to be tested about the general form and brick arrangement of the partially ruined state of the vault.

The next paper in this section, “Stonecutting and Early Stereotomy in Fatimid Walls of Cairo” by Macarena Salcedo Galera and Ricardo García Baño, is dedicated to the analysis of the Bab el Nasr gate, one of the three remaining gates in the Cairo Fatimid walls built around 1090. A great variety of stonework constructions were found inside the walls and gate revealing the stereotomic knowledge of the constructors. A photogrammetric survey supported the research along with a close reading of Renaissance stonecutting texts.

In the last paper in this section, some research results about defining parametric tools for modelling complex decorative apparatuses of domes are presented. In “Parametric Tools for Majolica Domes Modelling” by Mara Capone and Emanuela Lanzara, an adaptive tool, “Majolica Tiles on Domes”, was developed to reproduce the look of existing domes and virtually reconstruct modified or destroyed domes. The geometric problem which is addressed in this paper, is how to evenly distribute flat tiles of the same size on doubly curved surfaces.

Didactics

The paper “Learning by Doing: Integrating Shape Grammar as a Visual Coding Tool in Architectural Curricula” by Deena El-Mahdy, assesses the use of shape grammars as a visual teaching method to initiate the manual (that is, without digital software) exploration of computational thinking in early-stage architectural curricula. First-year architecture students at the British university in Egypt fabricated self-structured pavilions in a visual design course. A comparison of design parameters was then conducted including material qualities. The study concludes that by applying shape grammars in early design studios, students were better prepared for the following computational design approaches. The final paper in this issue, “A Mathematical Course Model for Architectural Education: Geometry of Design” by Ayçe Döşemeciler and Andrée Sonad Karaveli Kartal, describes the rationale for, and content of, a specialized course for architectural students to support the integration of mathematics into design.

Conclusion

The papers in these two special issues of the Nexus Network Journal – Vol. 24(2) and Vol. 24(3) – demonstrate the wide range of themes researchers explored about the interrelations between architecture and mathematics. The researchers also undertook this work under challenging circumstances, gathering data, conducting surveys and seeking peer review during a global pandemic. A team of more than 40 referees, and members of the scientific committee assessed the papers in these two issues, offering detailed feedback and advice to improve the quality of every paper accepted. The next Nexus conference is planned for June 2023 in Torino, Italy, hopefully as a face-to-face meeting to deepen the personal exchange of the network on architecture and mathematics.