Using 3D Modeling to Decode the Mathematics of Roman Architecture

Roman architecture, celebrated for its grandeur, precision, and technical innovations, has fascinated historians and enthusiasts for centuries. By blending functionality and aesthetics, it transformed the urban landscapes of antiquity and left a legacy that continues to influence contemporary architecture. Iconic structures such as the Colosseum, the Pantheon, and Roman aqueducts exemplify Roman ingenuity in using materials like concrete and implementing advanced techniques, such as the arch and the vault, which ensured durability and efficiency in their constructions. However, much of the theory and knowledge that underpinned these remarkable feats has been lost over time, leaving intriguing gaps in our understanding of their methods and practices.

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Courtesy of Autodesk

Nestled in the rolling hills of Tuscany, Italy, the charming city of Volterra has played a crucial role in advancing our knowledge of antiquity. With over 3,000 years of history, Volterra is a historical treasure trove, home to sites dating back to the 4th century BCE. Known as one of the oldest continuously inhabited cities in the world, it boasts remarkable landmarks such as the world's oldest Etruscan arch. The city also features one of the finest examples of a Roman theater, discovered just 50 years ago. These discoveries have been driven by the efforts of a research team led by Wladek Fuchs, Professor at the School of Architecture and Community Development at the University of Detroit Mercy and Executive Director of the Volterra-Detroit Foundation, alongside groundbreaking technological innovations. Modern tools, developed by companies like Autodesk, have enabled new and profound analyses of the mathematical sophistication behind these ancient architectural projects.

The project took on the ambitious task of scanning Roman structures, leveraging cutting-edge technology and academic insights to uncover the secrets of these millennia-old constructions. Using drones, photogrammetry, and terrestrial laser scanning, the teams captured high-resolution 3D models of the city's ancient structures and artifacts. With support from Autodesk, these models have not only aided urban planning and restoration efforts but also allowed global audiences to explore Volterra's heritage through virtual platforms and 3D-printed replicas. As Tristan Randall, Senior Business Development Executive at Autodesk, explains, the data processed through tools like ReCap Pro and Revit facilitated the creation of detailed Building Information Models (BIM), providing deeper insights into these structures and enabling the construction of models and virtual reality tools.

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Courtesy of Autodesk

A New Lens on Vitruvian Ideals

When Professor Wladek Fuchs began his work with the Volterra-Detroit Foundation, he could not have anticipated the groundbreaking revelations the project would uncover.

My original area of expertise was architectural visualization – from hand drawings to computer renderings and animations. Part of my fascination lay in demonstrating the original forms of ancient structures, now in ruins, through virtual 3D reconstructions. In Volterra, Italy, remnants of the 1st-century BCE Roman theater remain, but little of the structure is intact today. I wanted to envision its original state. To create its CAD 3D model, I needed to understand how it was initially designed. However, existing documentation from past hand measurements was confusing, with significant discrepancies between drawings. - Professor Wladek Fuchs

In August 2016, the Volterra-Detroit Foundation organized the city's first Digital Reservation Workshop, scanning the theater ruins in 3D with laser technology. This effort produced a precise digital replica, offering reliable material for further study. Like any researcher of Roman architecture, Fuchs relied on Vitruvius's Ten Books on Architecture as a primary reference, including its sections dedicated to theater design. Yet, it soon became apparent that Volterra's theater deviated significantly from the ancient author's descriptions. Vitruvius described an auditorium based on the geometry of a regular 12-sided polygon, but the layout in Volterra was based on a 7-sided polygon, with each side divided into three equal parts, resulting in a 21-sided figure. Consequently, stage doors for actors could not align with specific corners of a dodecagon as described, as no dodecagon was used. Fuchs had to identify alternative geometric principles to determine their placement. He explains: "Vitruvius wrote that if a colonnade has two levels, the upper level's height should be ¾ of the lower level; in Volterra, the ratio is unmistakably greater—8:9. Almost everything I examined differed from Vitruvius's prescriptions, leading to a dilemma: should I trust my observations or assume the ancient author's descriptions were correct?"

Archaeological evidence shows that many Roman structures deviate from Vitruvius's descriptions. While scholars often attribute these differences to later design modifications or construction errors, the precise execution of structures like the Volterra theater suggests intentional planning rather than on-site adjustments. This prompted Fuchs to expand his research to 50 Roman theaters, revealing that only seven followed Vitruvius's 12-sided polygon layout, while others used polygons with 7 to 13 sides. Roman stages (scaenae) were often based on square grids unmentioned by Vitruvius, further indicating alternative planning methods. Similarly, investigations of Roman temples revealed modular grids in their layouts, while circular structures like the Pantheon demonstrated that Romans used a more precise value of π (22/7) than Vitruvius's 3.125, enabling practical and proportionate designs.

According to Fuchs, Vitruvius focused on general design principles (utilitas, firmitas, venustas) and aimed to make architectural language accessible to a broad audience, rather than detailing practical methods. This likely explains the discrepancies between his recommendations and actual Roman practices.

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Courtesy of Autodesk

The Intersection of Technology and Tradition

Bringing these architectural insights to light required technological innovation, with Autodesk playing a pivotal role in the digital preservation of Volterra's ruins. Using terrestrial laser scanners and drone photogrammetry, Tristan Randall was part of a team led by the Volterra-Detroit Foundation and supported by Case Technologies that captured detailed 3D models of these ancient sites. "The process of creating these scans," Randall shares, "involved over 100 individual scans just for the Roman theater, ensuring millimeter-level precision. The resulting point cloud provided a flawless digital replica of the existing conditions." These data sets were then processed using Autodesk ReCap Pro, allowing seamless integration into 3D models and even scaled replicas through 3D printing.

Although the workflow was groundbreaking, it was not without its challenges. From precisely maneuvering drones at the sites to managing vast data sets, the project demanded innovation at every stage. Nevertheless, as Randall emphasizes, the effort paved the way for a scalable and replicable approach to heritage preservation worldwide. "These tools are more accessible and affordable than ever, making it possible to meet the needs of digital preservation across the globe."

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Courtesy of Autodesk

Lessons for the Present and Future

The Volterra Project is more than an academic exercise; it serves as a bridge between the ancient and the modern. Fuchs emphasizes the potential of these discoveries to influence contemporary architectural education. "We often teach the history of architecture as a catalog of dramatic forms," he observes. "But understanding Roman design methods—how they optimized geometry and proportions—can make history more relevant and transformative for today's designers."

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Courtesy of Autodesk

I would risk a statement that the geometry and mathematics of Roman structures are like a mix between the fingerprints of the architects and the mitochondrial DNA of all Roman architecture. Of course, it will take a long time to fully understand them—if that is even possible—because much of the material must be re-studied, and it is often in ruins. Therefore, I feel that I am merely laying the groundwork for future knowledge. But this already offers a completely different perspective on ancient architecture: that it was not homogenous, as Vitruvius's writings might suggest, but rather highly diverse, with much to be learned about its variations. -  Professor Wladek Fuchs

Randall echoes this sentiment, emphasizing the role of technology in preserving historical memory. "What we've accomplished in Volterra can serve as a model, bringing together professionals, technologists, and public groups to sustainably breathe life into these incredible places." 

We have an almost insurmountable global need to protect and preserve these important sites. What we've achieved collaboratively in Volterra can be used as a template to organize resources worldwide and unite experts, technologists, and the public sector to safeguard these extraordinary locations. -  Professor Wladek Fuchs

Fuchs also notes that the principles of Roman design can inspire new approaches to solving architectural challenges today. "Their proportional systems weren't just about aesthetics; they simplified logistics and construction, offering a level of precision that minimized errors. If today's architects can integrate that logic with modern materials and techniques, the possibilities are endless."

The Volterra Project has illuminated a richer and more intricate tapestry of Roman architecture than Vitruvius's writings alone suggest. As Fuchs concludes, "The geometry and mathematics in Roman structures are like the architects' fingerprints and the DNA of the architectural tradition. They reveal a diversity and ingenuity that challenge our assumptions and invite deeper exploration." By blending advanced technologies with rigorous historical investigation, the Volterra Project is not only preserving the past—it is redefining how we understand and utilize it. In doing so, it ensures that the wisdom of ancient architects continues to inspire future generations.

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Cite: Eduardo Souza. "Using 3D Modeling to Decode the Mathematics of Roman Architecture" 10 Dec 2024. ArchDaily. Accessed . <https://www.archdaily.com/1024418/using-3d-modeling-to-decode-the-mathematics-of-roman-architecture> ISSN 0719-8884

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