This paper introduces persistent homology as a complementary framework for the topological analysis of architectural floor plans. While Space Syntax relies on graph-based metrics to quantify spatial accessibility, it remains insensitive to higher-order topological features including circulation loops. Applying cubical persistent homology with distance-transform filtration to five synthetic typologies (Palladian, Corbusian, Wrightian, corridor, and courtyard) and validating against Villa Rotonda and Villa Savoye, this study demonstrates that plans with nearly identical Space Syntax integration values exhibit fundamentally different first-homology signatures. The Wasserstein-2 distance between persistence diagrams provides a stable metric for typological classification, capturing loop structures invisible to graph methods. A correlation analysis over an augmented sample confirms that the strength of the dominant loop (H1 maximum persistence) is statistically independent of integration, while resolution tests, morphological-robustness tests, and comparison with simpler topological baselines establish the stability and specific advantage of the proposed framework. Persistent homology and Space Syntax together provide a more complete characterization of architectural spatial configurations.
Beneath the intricate surfaces of muqarnas—key elements of Islamic architecture—lie complex two- and three-dimensional geometric systems. Despite their widespread historical use, these geometric characteristics remain insufficiently examined. This research offers a comparative analysis of thirteen muqarnas from the main gates of Istanbul mosques, all designed by the architect Sinan. Plans and elevations were generated with their constituent cell types, after which each muqarnas was algorithmically encoded and analyzed in three dimensions. The study identifies notable compositional parallels among several main gate muqarnas, evident in both their plans and three-dimensional configurations, despite minor internal variations. Additional similarities appear in star patterns, stratification, and cell typologies, while rare forms such as oyster and nail cells are also documented.
This study investigates the geometric and topological structure of Armenian Khachkar motifs to assess their computational reproducibility. Characterized by complex curvilinear networks and interlaced patterns, these motifs are analyzed through the decomposition of their fundamental geometric components within a grid-based system and represented using node–edge relationships. The proposed method employs a parametric modeling approach that controls curve generation and over–under crossing relations at vertex points. In this framework, the motif is treated not as a static composition but as a rule-based system defined by variable parameters. The performance of the model is evaluated based on rule compliance, curve continuity, and interlacing readability, demonstrating its capability to generate accurate reconstructions and controlled variations.
La Pedrera, an emblematic building designed by Antoni Gaudí, is distinguished by the undulating and continuous profile of its façade, thus establishing it as an iconic landmark of Catalan Art Nouveau. The present study proposes a harmonic, descriptive analysis of these undulations, with the aim of exploring their regularity, structure and formal coherence from a mathematical perspective. In order to achieve this objective, high-precision 3D laser scanning techniques and Fourier transform analysis are utilised to disaggregate the geometry of the façade into its constituent frequency components. The results provide a quantitative reading of the undulating rhythm of the surface, revealing harmonic patterns that contribute to understanding the design of the façade from a geometric and systematic approach.
This study presents a geometric analysis of the macsura vaults of the Mosque–Cathedral of Córdoba, focusing on the reconstruction of their underlying design principles. Based on high-resolution three-dimensional data, the methodology combines geometric reconstruction with metric validation to investigate the relationship between design and built form. The results reveal a coherent geometric framework defined through elementary operations executable with simple drafting instruments, such as the straightedge, square, and compass. Within this system, variations between central and lateral vaults are interpreted as controlled adaptations of a unified geometric logic. The high correspondence between reconstructed and measured geometry indicates that the vaults can be explained through geometric definition rather than empirical adjustment. Dimensional relationships further support the plausibility of the cubit as part of the design process.
This issue of the Nexus Network Journal presents thirteen expanded articles selected from the papers originally presented at the Nexus Conference 2025, held in Vicenza, Italy. A city deeply associated with the mathematical culture of architecture through the legacy of Andrea Palladio, Vicenza provided an appropriate setting for renewed discussions on the relationships between architecture and mathematics. Following the conference, selected contributions underwent further development and peer review. The resulting articles explore themes ranging from historical geometry, vault construction, and architectural representation to algorithmic design, artificial intelligence, digital fabrication, and architectural education. Together, they demonstrate the continuity between historical mathematical knowledge and contemporary computational approaches, reaffirming the central role of mathematics in architectural research and practice.
This article examines the digital reconstruction of kundekari, a traditional woodworking craft characterised by geometric patterning and interlocking wooden elements. The study reinterprets the craft’s design and fabrication processes using contemporary tools such as AutoCAD and Rhino–Grasshopper, supported by workshop observations and geometric analyses carried out with practising artisans. Through this combined approach, the research proposes a systematic method for reproducing kundekari in a digital environment. The study reframes an embodied, practice-based tradition—transmitted through intergenerational learning—as a form of local knowledge adaptable to contemporary design workflows. By translating tacit craft knowledge into computational processes, the research highlights how digital tools can help preserve endangered crafts. The developed algorithm generates geometrically accurate digital models and supports the creation of new variations derived from traditional logic.
This article explores the pattern-based design of traditional Iranian bazaars, emphasizing visual congruence of the components through quasi-fractal patterns. According to a context-oriented order, traditional bazaars, as integral parts of Iranian cities, follow an organized, mathematical approach. Using the box-counting method in ImageJ software, 13 samples from bazaars in Tabriz, Tehran, Kashan, Kerman, and Yazd were analyzed at macro, meso, and micro scales. The fractal dimensions, ranging from 1.48 to 1.66, align with traditional rather than modern architecture. The fractal dimension of Timchehs is notably higher than that of Sarās. Further studies at finer scales are recommended.
This paper presents a computational approach based on formal rules and essential geometric parameters for generating simplified 3D models of historic churches, designed to support post-earthquake inspection activities. The methodological framework assumes the decomposition of the architectural organism into structurally significant units as the foundation for a digital shape grammar, capable of translating metric and typological information into spatial configurations consistent with reality. The system allows the rapid creation of visual representations adaptable to different existing morphological structures and aims to provide a digital tool that enables even users without advanced modelling skills to configure the model. Finally, through application tests on real cases, the study highlights the potential of parametric abstraction as a cognitive device for documenting built heritage in emergency conditions, opening prospects for more accessible and interoperable digital tools.
The contemporary evolution of the concept of the model in architecture and engineering has shifted the model from a representational artifact to a cognitive infrastructure that supports knowledge organization and decision-making across the life cycle of the built environment. This paper combines a theoretical analysis of the epistemic role of the model with a methodological framework that integrates parametric modeling and Building Information Modeling (BIM) as a semantic and relational environment. The modeling and inspection of existing bridges are used as a reference domain due to the scale of infrastructure inventories, typological recurrence, and the need for continuous updating of inspection data. A geometric–relational scripting approach is presented for the large-scale generation of parametric bridge models, based on the integration of recurring inspection datasets within an openBIM workflow. The method produces interoperable and queryable models in which geometry and information are structured into a coherent system of entities, relationships, and properties, enabling reuse across heterogeneous platforms. The results show that the integration of parametric modeling and openBIM is not only an optimization of modeling and update processes, but also a mechanism for the formalization, management, and transmission of infrastructural knowledge. The digital model therefore becomes a shared environment in which technical, past, and operational information can be continuously constructed, verified, and updated.
This essay explores 18th-century Sino-European cultural exchange through the lens of linear perspective, focusing on Jesuit missionaries as intermediaries of science and art. It examines the influence of Andrea Pozzo's Perspectiva pictorum et architectorum on Nian Xiyao's Shixue. Developed with Giuseppe Castiglione, this work became the first Chinese treatise on perspective, adapting European geometry to Chinese visual culture. The paper analyses the structure of the Shixue, identifying three different relationships with European sources: by borrowing, interpreting and adding to them. It also discusses the application of perspective in xianfahua paintings at the Qing court, drawing an analogy with European quadrature and demonstrating the application of linear perspective rules in the artwork Qianlong Emperor Viewing Peacocks. In conclusion, the essay considers perspective to be a tool for intercultural dialogue, giving rise to artworks characterised by both European and Chinese elements.
Scholarship on English church architecture in the seventeenth and eighteenth centuries rarely discusses the dimensions and design proportions of the church structures themselves. This overlooks valuable evidence about the process of church design. This study investigates the influence of Sir Christopher Wren’s Church of St James Piccadilly on the most significant architects who followed after; James Gibbs, Nicholas Hawksmoor, and John James. Data from recent laser-assisted surveys was used to calculate the proportional relationships that define the interior space of a representative church by each architect; St Martin-in-the-Fields, Christ Church Spitalfields, and St George Hanover Square. The study findings suggest that Gibbs used the simple whole-number ratios advocated by Alberti, Serlio, and Palladio, as did James (for the most part). Hawksmoor followed Wren in combining these with square-root ratios. Wren’s mixed economy of whole-number and square-root ratios echoes the hybrid Gothic and Classical design at St Paul’s.
This study measures architectural originality in AI-generated facades by looking at measurable structural differences. It uses a controlled prompting method with four levels of semantic conditions: traditional, minimal, radical, and experimental. Two multimodal systems, ChatGPT and Gemini, created facade outputs under the same design limits. Each image was examined in a shared embedding space and evaluated using three formal metrics: symmetry difference, Shannon grid entropy, and a convex-hull-based mass articulation index. The results indicate that increasing semantic intensity does not lead to consistent structural changes. Instead, facade configurations shift into defined areas of transformation. Changes at the embedding level mostly relate to symmetry relaxation and, to a lesser extent, to volumetric articulation, while grid entropy shows a weaker link. Throughout all conditions, typological coherence remains statistically stable. These findings suggest that originality in AI-generated facades can be seen as a controlled structural adjustment within defined formal systems. This contributes to the mathematical study of architectural form in computational design contexts.
This article explores the relationship between architecture and the kaleidoscope, tracing its history from its origins to its modern-day applications using advanced digital tools. The kaleidoscope, an object that functions through the application of optical principles, is primarily known today for its recreational use. Originally designed to aid in the creation of decorative patterns for architectural applications, it soon became a paradigm of an era and its technological and formal characteristics. Used as a conceptual and formal reference in architecture since the nineteenth century, the kaleidoscope has continued to inspire artistic and architectural experiences to the present day, where new parametric digital design technologies can be used, as will be demonstrated in this article, to shape new built objects and spatial experiences.
The research, entitled “Numbers, Shapes, Geometries, Models. From the Measure the contemporary project of re-use”, started in 2016 with a Survey-and-Project workshop of an eighteenth-century building in Castri, a small town in Salento (Puglia). We experimented with a method applicable to various scales, which takes at its basis the analysis of the geometries and the logical-compositional structures of the pre-existing architectures, to be used as a memory for the project of reuse, and for the regeneration of historical sites that have lost their character over time. The research consists in the study of the cultural and geographical context, in the description of the historical buildings and of the soundscape aspect of the places, in the following geometric analysis from which to derive the historical measure to layout the new projects. We therefore studied what the relationship is between knowledge of space and its practical use, between rational control of the problematic nature of measurement and the geometric representation of simple or complex shapes and the processing of the project, between mathematics and architecture, two disciplines that express contemporary thought, knowledge, and the adoption of new paradigms.
The study focuses on the construction techniques and form control of Chinese squinch vaults from 3rd to fifth century, through a case analysis of eight vaults from six conserved tombs in the middle and lower Yangtze region. Three-dimensional models generated by photogrammetry and analytical drawings based on the survey were produced through site visits. Additionally, hypotheses on construction procedures and possible control methods are proposed to recover the construction technique used in that remote era. Comparison with other cases along the Silk Road in Central Asia and Persia reveals both similarities and distinctive characteristics of Chinese squinch vaults, thereby placing the studied cases within a global context of the technique by slices.
This paper examines the intersection of structural innovation and procedural geometry in Guarino Guarini’s Chapel of the Holy Shroud. Analyzing a unique stereotomic device, this research investigates how Guarini bypassed traditional domes to create a specific discontinuous geometry, with symbolic layers evoking wonder. The study moves beyond the traditional debate of prospective acceleration to propose a generative logic rooted in proportional reduction and complex surfaces. Results are discussed by coordinating Guarini’s treatises and drawings with VPL-implemented parametric models. Arches were modeled as elliptical elements based on a hexagonal layout, enabling precise representation and digitally simulating Guarini’s plan-to-elevation workflow. The research confirms that Guarini’s geometry serves as the “constant” of his logic, bridging the gap between speculative mathematics, symbolic intent, and the rigorous demands of stereotomic fabrication.
This paper presents a procedural, AI-assisted workflow for translating the proportional and com-positional logic of classical architecture into a parametric modeling system based on shape grammar theory. Grounded in Scamozzi's L'Idea dell'architettura universale, the research formalizes a corpus of architectural elements by encoding their geometric rules into modular, reproducible data structures. Leveraging Blender's open-source Python API and large language models for natural-language-to-code generation, the method has shifted from primitive-based approximations to analytically defined Bezier and NURBS profiles, enabling accurate reconstruction of moldings, entasis, and proportional dependencies. The resulting Blender add-on, Ars Modulandi 2.0, provides an accessible tool for generating classical components while enhancing transparency and scientific traceability. Overall, the study demonstrates how AI-driven scripting can act as a mediator between textual description, mathematical reasoning, and architectural form, opening new avenues for procedural heritage reconstruction and computational design.
This study explores the reinterpretation of the traditional kündekârî interlocking woodcraft system through contemporary digital design and fabrication tools. The minbar of the Eşrefoğlu Mosque in Beyşehir was selected as a case study, and the original kündekârî pattern’s geometric structure was digitally reconstructed using Rhinoceros 3D and Grasshopper. Initially, the structural logic of the pattern was analyzed through a set of shape grammar rules, followed by the development of a three-dimensional algorithmic model to achieve the computational reconstruction of the system. By modifying geometric parameters such as angles, thicknesses, and modular relationships within the parametric model, a series of design variations were generated and evaluated in terms of their applicability to contemporary design workflows. The findings reveal that the kündekârî technique, beyond being a craft worthy of preservation, can be redefined in the digital age as a generative design language.
