Introduction

In 2015 Kim Williams and I published ‘Mathematics in, of and for Architecture: A Framework of Types’ which offered a classification system for defining connections between architecture and mathematics (Ostwald and Williams 2015). That framework was development over three-stages. First, the foundation myths of architecture were examined using an aetiological logic (the investigation of the cause of something from earliest principles or beliefs). Second, contemporary architectural connections to mathematics were examined using teleological reasoning (the investigation of phenomena in terms of the purposes it serves). Finally, the framework that had evolved through the first two stages was tested using the contents of the Nexus Network Journal and the Nexus Conference books, to confirm if it could classify all the research published by the network at the time. The resulting framework classified connections between architecture and mathematics into three overarching categories and thirteen types, several of which were interconnected. Two of these types are concerned with regularity and rules, which are the themes that define the content of Vol. 25(2) of the Nexus Network Journal.

One example of regularity in historic architecture is the proportional system. The adjective ‘regularity’ describes the arrangement of individual instances, that collectively constitute a larger structure or theme. Such patterns of elements often serve practical or functional purposes, and in historic buildings proportions were also an important part of the symbolic language of the architecture. In a proportional system, however, the process that combines or extrapolates an individual instance into a whole (or conversely, which divides a whole into sub-components) is typically a rule. The word ‘rule’ is a noun that describes the regulating principles which govern actions or practices. Rules in architecture are often algorithmic or generative, being scripted (e.g., software codes for parametric design) or expressed in terms of mathematical logic (e.g., shape grammars). But at a simpler level, rules define the patterns, proportions and modules that shape architecture. Therefore, the two themes in this issue – the regular and the rule – are closely connected.

The Regular in Architecture

The first paper in this section is ‘The Fibonacci Rectangles in Lalibela and Their Significance for the Reconstruction of the Vanished Cathedral of Aksum’ by Matiyas Bekele Fantaye and Tibebu Assefa. The origins of the rock-hewn churches of Lalibela in Ethiopia are typically traced to the thirteenth century, and this paper reports on the results of a photogrammetric analysis of orthogonal views of three of these structures. The next paper is ‘A Statistical Assessment of Sinan’s Central Domed Mosques’ by Can Uzun. This research examines proportions in 47 mosques designed by the sixteenth century Ottoman architect Mimar Sinan. This is followed by ‘Geometry in Veli Pasha Mosque, Crete’ by Antonis Katsarakis. The Veli Pasha Mosque was constructed in the mid seventeenth century, as one of a series of structures built by the Ottomans in Greece. In this paper Katsarakis reports on a survey and analysis of the planning of the mosque, focussing on proportional and typological features of the building.

The next three papers in this section are about proportions in twentieth century architectural design. The first of these is Francesco Maggio’s, ‘Proportions in Salvatore Caronia Roberti Between Theory and Practice.’ Maggio’s research is about Caronia Roberti’s 1949 work Introduzione allo studio della composizione architettonica, wherein he sought to reposition geometry and proportion at the core of post Art Nouveau architectural aesthetic practice. With a focus on a similar era and geographic location, the following paper is ‘An Exemplary Case Study of Post‑WWII Reconstruction in Milan’ by Giulio Barazzetta, Camilla Guerritore and Marco Simoncelli. Their research examines proportional systems in a fifteenth century cloister and a twentieth century rationalist reconstruction by architects Ignazio Gardella and Giovanni Romano. The last paper in this section is about the ‘Marburg building system’, which was developed in Germany in the post-war years, and realised in the early 1960s. It is a famous modular construction system that employed standardized components. In ‘Measure and Module of Helmut Spieker’s Marburg Building System 1960–1970’, Robin Rehm and Silke Langenberg observe that although this system is famous for its rational logic, it has multiple connections to famous architectural proportional systems. Rehm and Langenberg examine connections to ‘Vitruvian man’, Le Corbusier’s Modulor and Neufert’s ‘measure of all things’.

The Rule in Architecture

The first paper in this section is ‘Finding Generative Rules in Settlements and Houses by Means of an Ideographic Language’ by Kyung Wook Seo. This research employs an early space syntax technique, ideographic representation, to examine patterns in Korean vernacular planning from the fifteenth to the twentieth century. The goal of this research is to develop generative rules for design. The next paper is ‘The Mathematics of Spatial Structure Evolution: Using Syntactical Data to Compare the Humble Administrator’s Garden in the Sixteenth and Nineteenth Centuries’ by Tiantian Zhang, Gaoxing Tang and Zefeng Lian. Traditional Chinese gardens are a particular type of spatial type that have fascinated researchers for many decades, but recently computational analysis has opened new avenues for examining their properties. This paper uses axial maps, a space syntax approach, to understand the rules governing movement paths and spatial depth in the famous Humble Administrator’s Garden in Suzhou, China.

A 1968 cocktail bar is the focus of Heather Ligler’s paper, ‘The Subsymmetry Analysis and Rule‑Based Synthesis of John Portman’s Midnight Sun’. In this research Ligler decomposes the patterns in the Midnight Sun, developing a rule-based interpretation or Portman’s architecture, that can also be implemented in a shape grammar. The final paper in this section is ‘Smooth Poly‑hypar Surface Structures: Freeform Shells Based on Combinations of Hyperbolic Paraboloids’ by Ting Cao, Toni Kotnik and Joseph Schwartz. A smooth poly-hypar surface could be considered a special type of parabolic net possessing structural efficiencies. This paper describes a novel approach to the design of these structures and defines two key geometrical constraints or rules governing their generation.

Geometer’s Angle

The last paper in this issue is ‘Geometric Construction of Rumî’ by Nadide Ebru Yazar. In the decorative arts a Rumî is a type of motif from the Seljuk and Ottoman eras that is often found in architecture. In this paper a set of design and construction rules for Rumî are examined. Appropriately, this paper refers to both rules and patterns, bringing together the dual themes developed in this issue. This is also the third paper is this issue about Ottoman architecture.

Conclusion

Issue 25(2) of the Nexus Network Journal marks my first practical issue as sole Editor-in-Chief. In issue 24(4) founding Editor-in-Chief, Kim Williams, announced that after 24 years editing the journal, the last eight sharing the role with me, she would be retiring. I first encountered Kim in the months after the 1996 Nexus Conference in Fucecchio (Florence), Italy. Because email was not yet available, we began our correspondence the old-fashioned way, by posting letters and parcels around the world. At the 1998 Nexus Conference in Mantova, Italy, I was part of the discussion about the future of the network. Like Kim, I advocated for the creation of the world’s first scholarly journal dedicated to research about architecture and mathematics. At the time I was motivated by the twin desire to ensure our research would remain accessible over time and to celebrate its quality. Creating a journal for the society seemed the right way to achieve these goals, and 25 years later I think most people would agree we made the right decision.

Since the formation of the Nexus Network Journal, I have held a range of roles and responsibilities for its production and content. In 2015, after Kim and I edited the two volume Architecture and Mathematics from Antiquity to the Future, she asked me to join her as co-Editor-in-Chief of the Nexus Network Journal. Despite no longer sharing this role with me, Kim is still involved in the transition process for the journal, and we are in regular contact about its operations. She also continues to be director of the conference series ‘Nexus: Relationships Between Architecture and Mathematics’.

Finally, in late 2022 the Nexus Network Journal team expanded to include Dr. Nick Mols, architectural scholar and academic from Cardiff University, Wales. Nick has taken over the initial copy-editing and text reviewing process from myself and Kim, and his efforts are already shaping the quality of our journal. Thanks to the entire Nexus Network Journal team, and especially to all our reviewers, for their efforts.