Nexus Network Journal: Architecture and Mathematics Online
Nexus 2008, 23-25 June 2008, San Diego, USA
John Poros
College of Architecture, Art, and Design, Mississippi State University
P.O. Box AQ, Mississippi State, MS 39762 USA
Bell Banner geometry, Priory of the Annunciation, digital model by author
The use of ruled surfaces proliferated in post-WWII architecture due to the influence of engineer/ architects such as Nervi and Candela, who saw the ruled surface as the geometrical underpinning for highly efficient shell surfaces. Marcel Breuer, while using ruled surfaces in structurally advantageous ways, also looked to these geometries for architectural expression. This paper will explore the use of these ruled geometries in structural and non-structural roles in Breuer's work. The use of rules surfaces both structurally and non-structurally indicates that Breuer thought primarily of these geometries for their qualities of space and form rather than their structural role. Breuer also used ruled geometries in ways that contradicted their structural role to create unexpected contradictions and tension in his work.
About the author
John Poros is an Associate Professor in the School of Architecture at Mississippi State University. He is currently the director of the Carl Small Town Center, a community design and outreach component of the school. He is also the former director of the Educational Design Institute, a research center for improving school design. Before joining the faculty at Mississippi State ten years ago, Prof. Poros had worked with the architecture firm of Kieran Timberlake Associates in Philadelphia for seven years. He has taught as an adjunct professor at Philadelphia College as well. Prof. Poros received his Masters of Architecture Degree from the Harvard Graduate School of Design and his Bachelor of Arts from Columbia University.
The correct citation for this paper is:
John Poros, "The Ruled Geometries of Marcel Breur", pp.233-242 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Michael Ostwald, Josephine Vaughan, Chris Tucker
University of Newcastle, School of Architecture and Built Environment, Faculty of Engineering and Built Environment
New South Wales, 2308 AUSTRALIA
Third stage grid placed over the east elevation of the Tomek house showing box-counting
In the late 1970s Mandelbrot argued that natural systems frequently possess characteristic geometric or visual complexity over multiple scales of observation, suggesting that systems which have evolved over time may exhibit certain local visual qualities that also possess deep structural resonance. In mathematics this realization, founded on seemingly irrational or "monstrous" numbers, lead to the formulation of fractal geometry and was central to the rise of the sciences of non-linearity and complexity. During the last decade this concept was developed in relation to architectural design and urban planning, and more recently architectural scholars have suggested that such approaches might be used in the analysis of historic buildings. At the heart of this approach, in both its theoretical and computational forms, is a set of rules for analyzing buildings. However, the assumptions implicit in this method have never been adequately questioned. The present paper returns to the origins of the conventional "box counting" method of fractal analysis for historic buildings to reconsider the initial interpretations of the architecture of Le Corbusier and Frank Lloyd Wright. This new analysis uses "Archimage" software, developed by the authors, to undertake a multi-dimensional review of the fractal dimension of the houses of Wright and Le Corbusier and, in doing so, to develop a more consistent method for the application of such mathematical tools to the analysis of historic buildings.
About the authors
Dr This email address is being protected from spambots. You need JavaScript enabled to view it. is Professor and Dean of Architecture at the University of Newcastle, Australia. He is a Visiting Professor at RMIT University in Melbourne and a Professorial Research Fellow at Victoria University Wellington. He has a Ph.D. in architectural philosophy and a D.Sc. in the mathematics of design. He is co-editor of the journal Architectural Design Research and on the editorial boards of Architectural Theory Review and the Nexus Network Journal. He has authored more than 200 scholarly publications and his recent books include The Architecture of the New Baroque (2006), Homo Faber: Modelling Design (2007) and Residue: Architecture as a Condition of Loss (2007).
This email address is being protected from spambots. You need JavaScript enabled to view it. is a research higher degree candidate at the University of Newcastle, where she is also a member of the architectural computing research group. Her postgraduate studies are focused on the fractal dimensions of buildings. Josephine directs the design firm, One Thousand Years, which specialises in sustainable houses and community facilities and she also tutors in design at the University. Her designs have been exhibited and installed regionally and nationally.
This email address is being protected from spambots. You need JavaScript enabled to view it. is a lecturer in the School of Architecture at the University of Newcastle and is a director of the architectural practice Herd. Chris has been awarded regional, state and international prizes for architecture and his buildings and designs have been widely exhibited and published. His research interests revolve around the development of software tools to analyse the geometric and mathematical qualities of buildings. He is currently completing a research higher degree on the algorithmic analysis of streetscapes.
The correct citation for this paper is:
Michael Ostwald, Josephine Vaughan, Chris Tucker, "Characteristic Visual Complexity: Fractal dimensions in the architecture of Frank Lloyd Wright and Le Corbusier", pp. 217-231 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Michael Ytterberg
BLT Architects, 1216 Arch Street, Philadelphia, PA 19107 USA
Sant'Andrea in Mantua by Leon Battista Alberti
Sant'Andrea in Mantua is the last of Alberti's churches yet it is the most complete, and the one in which his intentions seem to be clearest. It takes the form of a Latin cross, but evidence suggests that Alberti had intended a basilican plan. Alberti specified that his proposal was for a church of the type "known among the ancients as the Etruscan," but it is not planned like an Etruscan temple. The description in Alberti's treatise adhered precisely to the account of Vitruvius only in the presence of the unusual proportion of 5: 6. In spite of numerous attempts to discover the proportional system in Sant'Andrea, the present study is the first to have found the presence of the proportion 5:6 in the completed building. This paper demonstrates the systematic strategy that Alberti employed to bring every detail of the building into a coherent spatial framework related to the perceiving body, not as an abstract exercise, but as an enveloping web of meaning.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. received undergraduate and graduate degrees in architecture from Rice University and a Ph.D. in the history, theory, and criticism of architecture from the University of Pennsylvania. He teaches urban design and the history of architectural theory at Drexel University in Philadelphia. He is a registered architect in a number of US states and a design principal and member of the executive committee of BLT Architects, a 130-person firm headquartered in Philadelphia. Currently under design are high rise residential towers in Philadelphia and Newark, NJ, and a new casino resort on the strip in Las Vegas, with five hotels, shopping mall and convention center. His research interests include Hadrian's Villa, the subject of his Ph.D. dissertation, architectural theory before the eighteenth century, and the changing relationship material culture - and architecture in particular - to the society it serves.
The correct citation for this paper is:
Michael Ytterberg, "Alberti's Sant'Andrea and the Etruscan Proportion", pp. 201-216 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Steven Fleming
University of Newcastle, Mark A. Reynolds - School of Architecture and Built Environment
Callaghan, NSW, 2308, AUSTRALIA
Mark A. Reynolds
667 Miller Avenue
Mill Valley, California, 94941, USA
Salk Institute for Biological Studies, La Jolla, California, Louis I. Kahn, architect (photograph by Steven Fleming)
This paper presents a geometrical analysis of the Salk Institute for Biological Studies by Louis Kahn, 1959-65. With recourse to construction drawings, a digital plan, photographic evidence, and various forms of textural evidence, we emphasize the role played by musical ratios in Kahn's thinking about architecture. The Salk is widely regarded as a coauthored work, the product of a unique collaboration between scientist Jonas Salk and architect Louis Kahn. Not dissimilarly, the paper is coauthored by a geometer and an historian, the former having the role of identifying proportions, the later charged with reconciling prima facie findings with other available evidence: job correspondence, oral histories, known influences, etc. The research is occasioned, naturally, by the Nexus 2008 conference venue and the Salk's commanding presence north of San Diego, but it also highlights a way in which artifacts, texts and narrative can be given equal consideration when historical inquiry and geometrical analysis are coupled in a joint enterprise.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. lectures in the history and theory of architecture at the University of Newcastle, Australia. He received his Ph.D. in 2003 from the Department of Architecture at The University of Newcastle, with a thesis on Classical Platonism with respect to Louis I. Kahn's concept of "form". He has worked as a practicing architect in Australia and in Singapore.
Mark Reynolds is a visual artist who works in oils, drawing/mixed media, and printmaking. He received his Bachelor's and Master's Degrees in Art and Art Education from Towson University in Maryland. He was also awarded the Andelot Fellowship to do post-graduate work in drawing and printmaking at the University of Delaware. Mr. Reynolds is also an educator who teaches geometry for art and design students, and sacred geometry and geometric analysis for graduate students at the Academy of Art University in San Francisco, California. Additionally, Reynolds is a geometer, with expertise in geometric analysis. Some of his studies can be found in the Nexus Network Journal. Mark also lectures on his work in geometric analysis at international conferences on architecture and mathematics. For more than thirty years, Mr. Reynolds has been at work on an extensive body of drawings, paintings, and prints that incorporate and explore the techniques of contemplative geometry. His work has been widely exhibited in both group competitions and one-person shows, especially in California. His work is in corporate, public, and private collections throughout the United States and Europe. In 2004, Mark had over thirty drawings accepted into the permanent collection of the Leonardo da Vinci Museum and Library, the Biblioteca Communale Leonardiana, in Vinci, Italy. Mr. Reynolds is also a member of the California Society of Printmakers and the Los Angeles Printmaking Society. Examples of his art can be found online http://www.markareynolds.com
The correct citation for this paper is:
Steven Fleming, Mark A. Reynolds, "The Salk: A Geometrical Analysis Supported by Historical Evidence", pp. 185-200 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Avril Behan
Department of Spatial Information Sciences, School of Spatial Planning
Dublin Institute of Technology, Bolton Street
Dublin 1, IRELAND
Rachel Moss
Department of the History of Art and Architecture
School of Histories and Humanities, University of Dublin, Trinity College
Dublin 2, IRELAND
St. Nicholas' Collegiate Church, North Window E
The aim of this paper is to examine the extent to which detailed empirical analysis of the metrology and proportional systems used in the design of Irish ecclesiastical architecture can be analysed to provide historical information not otherwise available. Focussing on a relatively limited sample of window tracery designs as a case study, it will first set out to establish what, if any, systems were in use, and then what light these might shed on the background, training and work practices of the masons, and, by association, the patrons responsible for employing them.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. is a lecturer in Geomatics (specifically remote sensing, photogrammetry, CAD, and land surveying) at the Department of Spatial Information Sciences, Dublin Institute of Technology. She holds a Master of Science Degree from Dundee University, Scotland, in Remote Sensing, Digital Image Processing and Applications and is completing Ph.D. studies at the Department of History of Art and Architecture, University of Dublin, Trinity College on the application of geomatics techniques to the analytical study of medieval window tracery in Connaught and Ormond, Ireland. Her other research interests include heritage applications of terrestrial laser scanning, CAD and visualisation, satellite remote sensing, airborne laser scanning, and the usage of Web 2.0 applications for higher education. She has presented at conferences such as CIPA/VAST 2006 on "Close-Range Photogrammetric Measurement and 3D Modelling for Irish Medieval Architectural Studies" and ISPRS Congress 2000 "On the Matching Accuracy of Raterised Scanning Laser Altimeter Data".
Dr This email address is being protected from spambots. You need JavaScript enabled to view it. is a lecturer at the Irish Art Research Centre in Trinity College, Dublin. A specialist in the field of medieval architecture and sculpture, research projects with which she has been involved include the electronic capture and archiving of medieval architectural details with the department of Computer Science at Trinity College Dublin, and she is currently working on a project that looks at medieval buildings as documents of social and economic change. She was also involved, in a consultative role, with the establishment of a state-sponsored national database of movable Irish field antiquities. She has published numerous articles on medieval art and architecture and edited/co-edited two books, Art and Devotion in Late Medieval Ireland (2006) and Making and Meaning in Insular Art (2007).
The correct citation for this paper is:
Avril Behan, Rachel Moss, "Metrology and Proportion in the Ecclesiastical Architecture of Medieval Ireland", pp. 171-183 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Peter Saltzman
P.O. Box 9003, Berkeley, CA 94709 USA
A portion of the right half spandrel at the Darb-I Imam, Isfahan
For nearly 150 years, scholars have analyzed the symmetries of Islamic ornamental designs, constituting the most highly developed chapter in cultural symmetry studies. Yet these studies hardly exhaust the mathematically significant properties of Islamic designs. Over the past 30 years, mathematicians have given increasing attention to "quasi-periodic" geometric structures (such as the Penrose tilings) which exhibit infinite repetition of their bounded subparts and crystallographically forbidden symmetries, occupying an important niche between random and highly ordered, periodic structures. This paper provides a critical review of the literature and argues that the dualization of quasi-periodic tiling fragments from two designs from twelfth- and fifteenth-century Iran helps to inform their aesthetic complexity.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. is a mathematician and lawyer living in Berkeley, California.
The correct citation for this paper is:
Peter Saltzman, "Quasi-Periodicity in Islamic Geometric Design", pp. 153-168 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Alfonso Ramírez Ponce
E 21, M XII Educación, Coyoacán 04400 México, D. F.
Administrative Center, Jalapa, Veracruz. Arch. Alfonso Ramírez Ponce, designer and builder. Main atrium: a circle inscribed in a square
Architecture and music and other disciplines, have their objectives and goals, and the respective means or ways to reach them. Mathematics and geometry are some of the essential means that permits architecture goes beyond reason to the land of emotion.
Wherever a thought can go back,
there is hope for tradition
Derek Walcott
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. has a Master's in Architecture, and is a designer, builder, professor, writer and lecturer. He lives in Mexico City and teaches in the National Autonomous University of Mexico (UNAM). His specialization is the use of the "leaning brick" technique to build economical vaults and domes, following the traditional technique with innovations in the shapes coverings, an argument he presented at Nexus 2004 in Mexico City.
The correct citation for this paper is:
Alfonso Ramírez Ponce, "Poetic, Musical and Architectural Regionalism", pp. 129-138 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Maria Zack
Department of Mathematical, Information and Computer Sciences
Point Loma Nazarene University
3900 Lomaland Drive
San Diego, CA 92106 USA
Robert Hooke's Fire Monument (photo by Maria Zack)
After the Great London Fire of 1666, Robert Hooke was appointed along with Christopher Wren to lead the massive effort to rebuild the City of London. Hooke was involved extensively in all aspects of the rebuilding of London, both the mundane (widening streets and establishing property boundaries) and the creative (designing churches and civic buildings). Although very little of Hooke's architectural work has survived the passage of time, the Monument to the Great Fire is a shining example of his creativity. As a monument, it is fairly conventional - a column resting on a prism -, but as a scientific instrument it is ingenious. At the time of the monument's design, Hooke was conducting experiments on both the motion of the earth and the measurement of gravity. To further this research, the monument was constructed to contain a zenith telescope as well as a gravitational "lab." This paper will discuss how the scientific uses of the Monument were integrated into its design.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. received her BA (1984) and Ph.D. (1989) in Mathematics from the University of California at San Diego. She has held posts at Texas Tech University, The Center for Communications Research and Point Loma Nazarene University, where she is currently a Professor as well as the Chair of the Department of Mathematical, Information and Computer Sciences. Her research interests include the history of mathematics in seventeenth- and eighteenth-century England.
The correct citation for this paper is:
Maria Zack, "Robert Hooke's Fire Monument:Architecture as a Scientific Instrument", pp. 117-126 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Niels Bandholm
Klostervej 30, Kloster - DK6950 Ringkoebing, DENMARK
The Geometric Riddle
The stereographic projection of the Heavenly Sphere used in an astrolabe is a remarkable metaphor for the connection between Heaven and Earth, and as the medieval churches are treasure shrines for this holy union, their placements and proportions are chosen accordingly. Stereographic projection is proposed as the key to the locations of nearly all of the fifteen medieval churches built from 1150 to 1250 on the island of Bornholm in the Baltic Sea. The stereographic projection generates the specific ratio of the square root of 7 divided by the square root of 3 and angles of 33.2° and 56.8°, and leads to a unique astrolabe construction with angles of 40.9° and 10.9°. This ratio and these angles (to the north) are found for nine pairs of distances and half-distances between churches. The high degree of accuracy makes it highly improbable that the arrangement was by chance. Furthermore, some vectors point to pre-Christian holy places: Ertholmene and Ales Stenar in Sweden. The same key has been found in the measure and proportions of pre-Gothic buildings, and in Christian and Celtic iconography.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. earned his Master of Science in Chemistry and Physics from Aarhus University, Denmark in 1971, with a M.Sc. thesis in the History of Science in 1972. He is the author of books on chemistry, computer science and energy storage for the Danish upper secondary schools, where he was employed as a teacher and a senior master in physics, chemistry and astronomy from 1966 until his retirement in 2005. In addition, he has organised many conferences on interdisciplinary didactics and teaching. The present study is an example of an interdisciplinary project and was kindly hosted by the History of Science Department (now Department of Science Studies) at Aarhus University 2006-07. He has been greatly interested in environmental issues since 1987, initiating organisations, newsletters and eco-villages. He is now going to build his own dome house in a sustainable society with 250 persons 15 km north of Aarhus.
The correct citation for this paper is:
Niels Bandholm, "The Celestial Key: Heaven Projected on Earth", pp. 95-116 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
Tessa Morrison
The School of Architecture and Built Environment
The University of Newcastle, Callaghan
NSW, 2308, AUSTRALIA
Villalpando's Floor Plan of Solomon's Temple. From Villalpando and Prado, Ezechielem Explanationes
The second volume of Ezechielem Explanationes by Juan Battista Villalpando, published in 1604, contains a re-creation of the Temple of Solomon illustrated by a portfolio of exceptionally detailed architectural drawings. His designs were built on the principles of Platonic musical harmonies and his interpretation of ancient measurements. Villalpando envisaged the temple as a building encapsulating the entire formal grammar of classical architecture. Villalpando's architecture, harmonic proportions and measurements appear to be a flawless system and his design exerted an extraordinary influence on the architects and historians of architecture in Europe for at least the next two centuries. His reconstruction inspired not only other commentaries and other reconstructions of Solomon's Temple, but it also stimulated discussion on the very origins of architecture. However, his reconstruction was not without its critics. In the seventeenth and eighteen centuries critics included Louis Cappel, Samuel Lee, Louis Compiègne de Veil, Nicolaus Goldmann and others who produced alternative reconstructions of Solomon's Temple. In the twentieth century criticism from what appears to be an unusual source was uncovered. In Sir Isaac Newton's unpublished manuscripts he claimed that although Villalpando had created the best of the reconstructions of the Temple of Solomon, the reconstruction had many problems. This paper examines Villalpando's reconstruction of the Temple in the light of Newton's unpublished commentary.
About the author
Tessa Morrison is an Australian Research Council post-doctoral fellow in the School of Architecture and Built Environment at the University of Newcastle, Australia. Her academic background is in art history, mathematics and philosophy. Her current research project focuses on sixteenth- and seventeenth-century sacred architecture, particularly that of Juan Battista Villalpando and Isaac Newton's reconstructions of the Temple of Solomon. She has also published extensively on geometric, spatial symbolism and has an interest in examining and reconstructing the plans and structures of architecture in medieval poetry through poems such as the eighth-century Gaelic poem Saltair na Rann and the fourteenth-century Pearl. At the present she is translating Book V of Villalpando's De Postrema Ezechielis Prophetae Visione and Newton's work on the Temple of Solomon from Latin into English.
The correct citation for this paper is:
Tessa Morrison, "Villalpando's Sacred Architecture in the Light of Isaac Newton's Commentary", pp. 79-91in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.
