Introduction

The research in Vol. 25(3) of the Nexus Network Journal: Architecture and Mathematics addresses three types of mathematical constructions, vision, surface, and form. All three are well-developed in past research and continue to provide fertile grounds for new exploration as scholars seek to provide insights into the past or develop novel ideas and applications for the future.

A significant early application of mathematics in architecture is found in the use of projective geometry to construct rigorous visual representations of the world. Historically, projective geometry has been employed to create perspective views and envision the extent of shadows (Sbacchi 2009). These methods reached a level of refinement during the Renaissance, although debate continues to the present day about their precise development and application (Xavier 2015). This issue features two examples of research about the use of projective geometry for the construction of vision and shadows.

In two and three dimensions, tessellated, tiled, and woven surfaces are another example of the symbiotic relationship between architecture and mathematics. Whether the tiling is of a surface (Wichmann and Wade 2018) or developed through stereotomy (Calvo-López 2020), there are practical, decorative, and symbolic dimensions to the construction of plane- and form-filling polygons (Ostwald 2021). Two research papers about applications of complex tiling in Islamic architecture are included in this issue.

The use of mathematics in the construction of architectural forms is a topic that has continued to evolve throughout history. In the ancient world, mathematics provided a practical tool for architects to construct large-scale figures—triangles, circles, ovals, and polygons—for laying out building plans (Duvernoy 2021). More recently, architects and scholars have used computers to parametrically evolve or generate designs (Lastro 2021), either for analytical purposes or to create new works replicating a particular style (Gu et al. 2021). In much the same way that computational Shape Grammar research demonstrated the dual potential of rule-based analysis and generation, parametric scripting can foster deep insights into a historic building and create new instances of that style. Five papers in this issue concern the development of geometric forms as frames for architectural thinking and production and three are about using computational methods to extract or evolve new forms.

Vision, Surface, and Form

The first pair of papers in this issue address the construction of visual representations. ‘Desargues’s Perspective Theory: A Critical Interpretation of the Fundamental Theorem’ by Leonardo Baglioni and Riccardo Migliari is about the debate surrounding one aspect of Abraham Bosse’s and Girard Desargues’s seventeenth-century Manière Universelle. The focus of this research is Desargues’s perspectival elucidation, which Baglioni and Migliari argue, offers an innovative foundation for a general theory of perspective. Andrés Martín‑Pastor’s and Francisco González‑Quintial’s ‘Approaching Developable Surfaces Through Shadow and Penumbra’ examines the graphic development of shadow and penumbra forms in architectural representations. Their research presents a reconstruction of the ways artisans and artists used mechanical and physical models to explore these features before proposing two new methods for generating them in digital models.

The next two papers are about the construction of surface tiling. Occasionally, with the support of our reviewers and the editorial board, the Nexus Network Journal will publish pairs of papers on important topics from leading scholars. Peter R. Cromwell is responsible for one of these sequences, focussed on constellation patterns, an Islamic geometric tiling combining star forms into an intricate matrix. Cromwell’s research in ‘Creating Constellation Patterns I: Composition’ and ‘Creating Constellation Patterns II: Enumeration’ starts by demonstrating a systematic method using circle packing to compose patterns of this type and then identifies 30 complex periodic constellation patterns.

The following five papers have a common interest in the construction of shapes and forms. In ‘Observations on the Geometry of the Monument at the Kasta Tumulus at Amphipolis’ by Demetrius Savvides, an underlying geometric structure, based on the ad-quadratum and octature principles, is proposed. Savvides’ research uses this construction to position the Kasta tumulus in a lineage of Macedonian tombs, illuminating the funerary tradition of the Hellenistic era. ‘Octagonal Layouts: Project Genesis of the Cathedral of Valencia’ by Concepción López González, Jorge Luis García-Valldecabres and Luis Cortés Meseguer reports the results of a 3D laser scan survey of the famous cathedral and explores arguments about the construction and use of the octagon in its design. In ‘The Circular Helical Staircase at Palazzo Spada’ by Matteo Flavio Mancini and Laura Farroni, a detailed investigation is recorded of the properties of this circa 1660 stair located in the Palazzo Spada’s wing on Via del Polverone in Rome. Using a three-dimensional digital model, Mancini and Farroni analyse the geometry of the helical lines and slope angle, identifying a connection to Palladian proportions. The fourth paper on this theme, ‘A Method for Recognising Oval Forms: The Case Study of a Copperplate Engraving of the Bibliotheca Wolfenbüttel 1705/23’ by Angelo Alessandro Mazzotti and Achim Ilchmann, starts with the observation that interpretations of historic geometric constructs often don’t take account of the tools, or capacity of the architect, to produce them. The lack of a plausible explanation for one of these constructs, oval forms, is examined in this paper through an analysis of an eighteenth-century engraving representing the Bibliotheca Wolfenbüttel. In ‘Invisible Cube: Aldo Rossi’s Early Works with Gianugo Polesello’ by Yuji Katagiri, three-dimensional models are employed to examine examples of cubic architectural forms designed by Polesello and Rossi. Katagiri identifies three variations of the cube—‘literal’, ‘phenomenal’ and ‘fluctuating’—and classifies those most critical in Polesello’s and Rossi’s design vocabulary.

The final three papers in this issue are about generative methods in architecture, the first proposing an algorithm for expanding the corpus of an architectural style, the second a parametric definition of part of a famous style, and the last visualising an unfinished design. In ‘A Recursive Algorithm for the Generative Study of Seljuk Muqarnas in Kayseri and Sivas’ by Sabri Gökmen, Yusuf Aykin, Altan Basik, and Sema Alacam, the use of generative algorithms to support a comparative study of structures belonging to a common period, style, or area is described. The authors use four different octagonal muqarnas to present a series of generative hierarchical rules to evolve muqarnas structures. Elisabetta Caterina Giovannini’s ‘Making Palladio Digitally Explicit: Geometrical Parameters in Door’s Ornaments’ expands her research about patterns in Palladian architecture and their capacity to be parametricised, both for analytical and generative purposes. The focus of this particular research is on moulding elements in Palladio’s villas. Finally, ‘A Virtual Reconstruction of Gaudi’s Skyscraper Hotel Attraction Using Physics‑Based Simulation’ by Arnau Luque‑Sala and Federico Luis del Blanco García uses an algorithmic model based on a simulation of weighted hanging chains (Gaudi’s method) to visualise this famous unbuilt design. This final paper in the issue closes the loop opened by the first, which focussed on visualisation. Luque-Sala’s and del Blanco García’s research is as much about visualising and celebrating form, perspective, and shadow as it is about the methods used to generate the building. Where once different mathematical techniques were used for constructing vision, surface or form in architecture, contemporary computing has begun to combine the three in new ways.