Nexus Network Journal: Architecture and Mathematics Online
Nexus 2006, 7-9 June 2006, Genova, Italy
Maria Zack
Department of Mathematical, Information and Computer Sciences
Point Loma Nazarene University
3900 Lomaland Drive
San Diego, CA 92106 USA
Christopher Wren's St. Stephen Walbrook (photo by Maria Zack)
After the Great London Fire of 1666, Christopher Wren was appointed as a member of the group that was charged with rebuilding the City of London. Over the next several decades this massive reconstruction effort was lead by Wren and Robert Hooke. Both Wren and Hooke were founding members of the Royal Society and until the Great Fire both were best known for their work in mathematics, physics and astronomy. It is a curious fact of scientific history that Wren who was the Savilian Professor of Astronomy at Oxford, the man that Newton called one of the three "leading geometers of the day," is remembered for his work as an architect.
Wren has been credited with designing St. Paul's Cathedral as well as more than fifty parish churches that were built as part of the post fire restoration of the City of London. Current scholarship indicates that the parish churches were most likely designed by Wren, Hooke and others that worked in their office. However, based on Hooke's diaries and some parish vestry minutes, it is possible to link certain specific church designs with Wren or Hooke. Using the church architecture of Wren and Hooke as illustrations, this paper discusses the evidence for and against the notion that Wren's mathematical and scientific knowledge directly influenced his architectural designs.
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, "Are There Connections Between the Mathematical Thought and Architecture of Sir Christopher Wren?", pp. 171-180 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
Mark Wilson-Jones - Ancient Architecture and Mathematics: Methodology and the Doric Temple
Benamy Turkienicz, Rosirene Mayer - Oscar Niemeyer Curved Lines: Few Words, Many Sentences
Richard Talbot
63 Bedford Street
North Sheilds
Tyne and Wear NE29 0AR UK
The construction of a 16-sided mazzocchio. Drawing by Richard Talbot
This paper stems from a study of one of the most complex and well-known examples of early renaissance perspective drawings, the Chalice - a drawing that has become almost iconic within the history of perspective, although neither the author nor the exact date of its execution are certain. It is a study of both the design of the Chalice and of the geometry underlying its perspective construction and asks whether there is any relationship between its design - its elevation, and the geometric constructions and procedures that would have been required for its depiction.
Close examination of the drawing reveals the progression of its construction as well as the method of its perspective projection and suggests that the large octagonal mazzocchio was the first element to be drawn. Measurements taken from this and an associated mazzocchio drawing, 1756A in the Uffizi, show that their proportions - specifically the relationship between their octagonal section and their diameter are identical. Further measurement and reconstruction shows that these proportions have their origin in the geometric constructions necessary for the depiction of the mazzocchi.
The construction of the plan of the 32 sided mazzocchio, with its octagonal vertical cross section, would have required drawing a square, constructing an octagon within that square, further division to give 16 and then the 32 sections that describe the circumference. The paper argues that it is these constructions that relate directly to the elevation of the Chalice.
The central proposition of this paper is, therefore, that the design of the Chalice - its elevation, is not arbitrary. It is derived from the same geometric constructions and procedures required for the construction and depiction of the large octagonal mazzocchio.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. gained his BA in Fine Art at Goldsmiths' College, and his MA Sculpture at Chelsea School of Art. In 1980, he was awarded the Rome Scholarship in Sculpture and spent two years at the British School at Rome, travelling widely throughout Italy and Egypt. His drawings have been widely exhibited and he has recently completed a commission at Westminster Abbey. He currently holds an AHRC Fellowship in the Creative and Performing Arts at Newcastle University. His work can be seen at www.richardtalbot.org
The correct citation for this paper is:
Richard Talbot, "Design and perspective construction: Why is the Chalice the shape it is?", pp. 121-134 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
Helge Svenshon, Rudolf H.W. Stichel - Systems of Mondad' as Design Principle in the Hagia Sophia: Neo-Platonic Mathematics in the Architecture of Late Antiquity
Clara Silvia Roero
Dipartimento di Matematica, Università di Torino
Via Carlo Alberto, 10
10123 Turin (Torino) ITALY
The Narmer Palette, Egyptian, ca. 3000 B.C., an excellent approximation of the catenary curve.
During the course of centuries mathematics has interacted in many ways with culture and human activities, and among these a place of privilege has been reserved for art and architecture.
This paper discusses several examples of the existence of three levels of interaction between mathematics and art: the presence of a mathematical substrate in various archaeological and artistic relics from antiquity, the conscious or unconscious application by artists of mathematical principles whose theories that had not yet been fully developed and, finally, the relationship established by some mathematicians with artists and art theorists that permitted an awareness and acquisition of mathematical knowledge and rules that were then applied to artistic creations. The development of these three levels of interactions between mathematics and art can be a valid aid to the creation of a unified vision of the history of culture of peoples and civilizations, indicating various kinds of influence: technical-practical, theoretical-scientific, mystical-sacred, principles and customs, etc.
Indeed, in the wake of a long-term historiographic approach, new research perspectives have emerged recently that have been favourably received by art historians and critics.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. is Full Professor in the Department of Mathematics at the University of Turin, and President of the Italian Society of Historians of Mathematics.
The correct citation for this paper is:
Clara Silvia Roero, "Relationships between History of Mathematics and History of Art", pp. 105-110 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
K. Graham Pont
54 Birchgrove Road
Balmain NSW 2041 AUSTRALIA
Athens, Acropolis after 450 B.C. Perspective view, reconstruction by C.A. Doxiadis, 1972. Courtesy of the MIT Press.
In his doctoral thesis (1937), translated as 'Architectural Space in Ancient Greece' (1972), Constantinos Doxiadis argued that the apparently haphazard layout of Greek temple sites could be explained by a system of planning by 'polar coordinates'. From a fixed pole, usually at the ritual entrance, the planner could locate any building by measuring the distance to that building and the size of the angle between two vectors or radii, here sightlines from the viewer to the outer edges of that building.
Analysis of 29 ancient sites revealed two systems of ancient planning - the Doric, based on a 12-part division of the 360-degree visual field, and the Ionic, based on a 10-part division of that field. In both cases, buildings were carefully sited so that their outer edges (stylobates, cornices etc) appeared to the viewer at canonic angles of vision, such as 30, 60, 90 (Doric) and 18, 36, 72 (Ionic), thus creating a 'unified composition' of the visible landscape. Doxiadis' theory is testable and was confirmed by the discovery of a 30-degree angle between sightlines from the top western step of the Propylaea to the outer edges of the temple of Athene Nike.
A similar system of planning might have been used in Italy by augurs practising the 'Etruscan Rite' which was also based on a ritual division of the visual 'templum' (sacred space). Since Joseph Rykwert's reconstruction of the Etruscan Rite, in The Idea of a Town (1976), is confirmed by only one cardinally oriented and orthogonally planned city (Marzabotto), I suggest that the many irregular Italian sites (including Rome and Hadrian's Villa) might have been ritually planned by methods analogous to the Greek system and involving a 'Pythagorean' world-view based on an 'harmonic' division of space and time.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. is a philosopher specialising in aesthetics of music and architecture. He is currently working on the notation and interpretation of Handel's music, the irregularities of Greek and Gothic architecture, and the design philosophy of Walter Burley and Marion Mahony Griffin...
The correct citation for this paper is:
K. Graham Pont, "Inauguration: Ritual Planning in Ancient Greece and Italy", pp. 93-104 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
Michael Ostwald
School of Architecture and Built Environment
Faculty of Engineering and Built Environment
University of Newcastle
New South Wales, AUSTRALIA 2308
In recent years the development of computational algorithms for the transformation of shapes has made the process of producing curvilinear forms deceptively simple. Even the most banal CAD program can generate complex three dimensional shapes, and associated building designs, without the designer having to display any detailed knowledge of geometry or indeed the history of similar forms and their relative successes and failures. This paper asks whether such a situation in innately problematic or not?
The production of intricate architectural forms has historically occurred in an environment that is aware of the cultural, political or symbolic importance of the curved form. For example the archetypal Baroque compound curve found in the facade Borromini's S. Carlo alle Quattro Fontane has both regular sinusoidal flowing surfaces along with more dynamic syncopated curves constructed of broken oval segments. Such curves responded to the social, symbolic and phenomenological needs of the era and indeed, because of this, can be seen to have an ethical function (as argued by critics such as Ruskin or more recently Harries). The paper analyses a series of recent examples and experiments which have employed computer generated curvilinear geometric forms to interrogate the extent to which such architectural techniques, which rely on geometric transformation, can be seen as having and ethical foundation. Through this analysis the paper argues for the importance of geometry in architecture as being more than simply a formal tool, but rather a device which has wider significance and more important properties and potentialities.
About the author
Professor This email address is being protected from spambots. You need JavaScript enabled to view it. is Dean of Architecture at the University of Newcastle, a Visiting Professor at RMIT University and a Professional Research Fellow at Victoria University Wellington. His research into design history and philosophy, often with a secondary focus on geometry or computing, has been widely published and he has lectured in Australasia, Europe and North America. His recent books include The Architecture of the New Baroque (2006), Antipodean Structures (2007) and Residue: Architecture as a Condition of Loss (2007). In 2006 he was awarded the Mellon International Prize for humanities scholarship and in 2007 he was awarded a higher doctorate; the Doctor of Science. He is a member of the editorial board of the Nexus Network Journal.
The correct citation for this paper is:
Michael Ostwald, "Geometric Transformations and the Ethics of the Curved Surface in Architecture", pp. 77-92 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
Elena Marchetti
Department of Mathematics
Milan Polytechic
Piazza Leonardo da Vinci, 32
21033 Milan ITALY
Luisa Rossi Costa
Department of Mathematics
Milan Polytechic
Piazza Leonardo da Vinci, 32
21033 Milan ITALY
A rose window with cyclic symmetry from the Cathedral of Milan. Photograph by Luisa Rossi Costa and Elena
Marchetti Studying Arts and Architecture in connection with Mathematics to underline important geometrical properties, we considered the Duomo of Milan , one of the most important monuments in our city.
The construction of the Cathedral begun in the 1380s and continued, with delays, through the last century. Several different architects, artists and consultants (including Leonardo and Bramante) were asked to work on the design and to collaborate in the construction.
The Milan Cathedral, Gothic at the first glance, even if unusual, and completed by a Renaissance façade, looks to be complex but well-formed according to a proportioned superposition of different styles. Its majestic beauty conquers everyone, resident or non-resident alike, so as mathematicians, we tried to discover the secrets of that fascination in terms of aesthetic geometrical canons. Starting from the drawings by Cesare Cesariano (1475-1543), in which are evident ratios involving "metallic" numbers, we looked for different symmetries in the building, as well as in the decorative patterns of the floor and the stained-glass windows. In particular we analysed the arabesques in the coloured rose windows, each different from the others, but all described by dihedral or cyclic groups. The Duomo provided us with an opportunity to discover other mathematical peculiarities: meaningful curves in the inlaid marble decorations of the pavement and geometric figures in the windows recall forms even used in mechanical engineering.
Knowing of the wide cultural formation, at any given time, of the architects, engineers and collaborators of the so-called "Veneranda Fabbrica del Duomo" (venerable factory of the Cathedral), we believe that such harmonic construction does not arise by chance but is the natural synthesis of creativity and technical and geometrical knowledge.
About the authors
This email address is being protected from spambots. You need JavaScript enabled to view it., Associate Professor, has taught mathematics courses to architecture students at the Faculty of Architecture of the Politecnico di Milano since 1988. She has produced numerous publications in Italian and international scientific journals in the area of numerical integration. She has published many papers about the applications of mathematics to architecture and arts. The experience gained through intense years of teaching courses to architecture students led her to collaborate in some books dedicated to this topic, with multimedia support packages.
This email address is being protected from spambots. You need JavaScript enabled to view it., Associate Professor of Mathematical Analysis in the Engineering Faculties of the Politecnico di Milano, developed researches in Numerical and in Functional Analysis (the results are published in several papers). She contributed also to the creation of the first-level degree in Engineering via the Internet and of the e-learning platform M@thonline. Since 1998, involved in the relationship between High Schools and Politecnico, she wrote many papers in teaching methods, in connection between Art, Architecture and Mathematics to familiarize pupils with Mathematics, and collaborated in editing a book on symmetry, complete of a DVD support.
The correct citation for this paper is:
Elena Marchetti and Luisa Rossi Costa, "What Geometries in Milan Cathedral?", pp. 63-76 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
Ulrich Kortenkamp
Department of Computer Science
University of Education Schwäbisch Gmünd
Oberbettringer Straße 200
73525 Schwäbisch Gmünd
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Berlin's Alexanderplatz and the quasi-periodic tiling designed to pave it
In this paper we describe a mathematical approach to create an organic, yet efficient to create tiling for a large non-rectangular space, the Alexanderplatz in Berlin. We show how to use the refinement algorithm for Penrose tilings in order to create a polygonal tiling that consist of four different tiles and is quasi-periodic. We also derive, based on the refinement algorithm, bounds for the percentage of tiles of each type needed.
Another question that is addressed is whether it is possible to describe the calculated tiling in a linear form. Otherwise, it wouldn't be possible to use the tiling, as there must be a concise description suitable for the workers who lay out the concrete tiles.
About the author
This email address is being protected from spambots. You need JavaScript enabled to view it. is working in Mathematics, Computer Science, and Education of these disciplines. In his work in Education he is always looking for topics that exhibit the beauty of Mathematics and the usefulness of Computer Science, which is almost always true for mathematically supported architectural themes. He is also co-author of the interactive geometry software Cinderella, that constitutes a user-friendly approach to geometry with a strong mathematical foundation.
The correct citation for this paper is:
Ulrich Kortenkamp, "Paving the Alexanderplatz Efficiently with a Quasi-Periodic Tiling", pp. 57-62 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
